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5

I couldn't resist trying to code up the algorithm hinted at in my comment. Sage code at: https://pastebin.com/3ANkqfZp This is v2 of the code; version 1 used only prime-degree endomorphisms, and thus failed to distinguish between the orders $\mathbf{Z}[\sqrt{-15}]$ and $\mathbf{Z}[\tfrac{1 + \sqrt{-15}}{2}]$ (since the sets of primes arising as norms of ...

4

When they say "use the midpoint," they are treating each $y_n$ as an estimate of $\sqrt{5}$, and so in order to avoid the periodicity, they are attenuating the movement toward the next estimate $y_{n+1}$ by averaging the previous estimate with the next. So if the original recursion was $y_{n+1} = 5/y_n$, then we can reduce this oscillatory ...

2

Too long for comment; still working on it, but these observations ought to be useful already. An easy way to confirm that your $G$ has order $(p^2-1)(p^2-p)$ is to note that $X$ is the vector space of dimension $2$ over the field with $p$ elements, so $G$ is $\mathsf{GL}(2,p)$, the general linear group of degree $2$. The basis $(1,0)$, $(0,1)$ must map to a ...

2

I think the issue here is a confusion re: partial vs. total computable functions. (Below I fix some "appropriate ambient axiom system," say $\mathsf{ZFC}$ - so e.g. "theorem" means "$\mathsf{ZFC}$-theorem" and so on.) Something which is often insufficiently emphasized in my experience is the extent to which partial (= defined on ...

2

The short answer: gate number. What I mean by that is that is the number of physical gates used in the quantum algorithm is what is counted for complexity classes. Source: this is what I study and that is what we count for complexity purposes. If you want a little more information, you could try googling BQP. This is the set of problems which can be solved ...

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Some clarifications: I guess that you have the constraints $n\ge i$, $i \ge 0$ and $0\le y \le1$. $dy$ is just an indication that $Y$ is a continuous variable and that reported is a probability density. $n$ and $i$ indicate instead integer valued random variables $\lambda$ is just a positive real parameter As you noticed, to apply Gibbs sampling you need ...

1

Regarding complex version of IEEE-754 It is true that no complex version of IEE754 exists, but the same design rules as for IEEE-754 can be applied for complex arithmetic. However, this is more subtle than it seems in the first place. For example the direct cartesian implementation of the complex multiplication (i.e. $(a+ib)(c+id)$ does not work with $inf$. ...

1

A complex Inf has no need for a sign, but arithmetic should work similarly. Inf plus anything should be Inf, Inf multiplied by anything should be Inf unless multiplied by 0 in which case the result should be 0. I found a good resource for checking how various programming languages handle infinity here: https://rosettacode.org/wiki/...

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I have no references, but i'm actually coding in Julia these days, and the profiling tool allows me do dig the computational time up until functions Base.+() and Base.*(). For the computations i do (and this is not a generality), there are a lot more additions than multiplications in numbers, but a lot more multiplications than additions in computing time. ...

1

This is coming from \begin{aligned} \Vert f \Vert^2 &= f^Tf = [M^{-1}(S*x - T)]^T [M^{-1}(S*x - T)]\\ &=(S*x - T)^T (M^{-1})^TM^{-1}(S*x - T)\\ &=(S*x - T)^T MM^{-1}(S*x - T)\\ &=(S*x - T)^T (S*x - T) = \Vert S*x-T \Vert^2 \end{aligned}

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