Why does the Euclidean algorithm terminate? Explain the effect of multiplication by - 1 by observing that |-1| = 1 and arg (-1) = pi. Why are complex numbers, not an ordered field? Given the linear operator L = D^2(D + 1)(D^2 + 9). Write down the general solution of Ly =...
Why does the Euclidean algorithm terminate? Is the function one to one and onto? Explain why or why not. f: \mathbb{R} \rightarrow \mathbb{R} defined by f(x) = { -1 if x less than 0 and x if x \geq 0 } Why do residuals add up to zero?
Before we go on I should explain that for purposes of exposition I’m mostly not going to use the full system that’s in ChatGPT; instead I’ll usually work with a simpler GPT-2 system, which has the nice feature that it’s small enough to be able to run on a standard desktop com...
And every time one does this, different random choices will be made, and the text will be different—as in these 5 examples:It’s worth pointing out that even at the first step there are a lot of possible “next words” to choose from (at temperature 0.8), though their probabilities f...
I wonder if the ancient greeks stopped when they got a remainder in the Euclidean Algorithm? They didn't even have numbers, they used letters to represent quantities, Hindu-Arabic number system and invention (discovery?) of zero came much later. I guess if recent means 13th or 14th century...
And why does it work? My purpose here is to give a rough outline of what’s going on inside ChatGPT—and then to explore why it is that it can do so well in producing what we might consider to be meaningful text. I should say at the outset that I’m going to focus on the big...
As far as my understanding goes, the current Turf.js implementation does not calculate a "geodesic buffer" (which would calculate the correct shape on the ellipsoid/geoid), nor an "euclidean buffer", which would have to be calculated in the projection plane (and therefore take the map ...
Oceanic archipelagos typically harbour extensive radiations of flowering plants and a high proportion of endemics, many of which are restricted to a single island (Single Island Endemics; SIEs). The Azores represents an anomaly as overall levels of endem
Why does the long division of polynomials work? Why is that when it comes to diagonal it must be divided into 2? Why does the Euclidean algorithm terminate? Why are unit rates useful? Why is empty set a subset of power set? Why do we use synthetic division?
Why does any number raised to the 0 power equal 1? Why is there no commutative property for subtraction or division? Why does the Euclidean algorithm terminate? Explain how and why does P^(-1/4) Q/(P^(-1)Q^6) become P^(-1/4) Q(P^(-1)Q^6)^(-1)? Why does it go from ...