Let’s look at Python’s map function. What does this function do? Here is the syntax:
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map(function, iterable, ...) |
It takes a function called function and applies it to each element of iterable, returning the result as an iterable. For example:
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myList = [1,2,3,4] myList = map(lambda x:x**2 ,myList) for i in myList: print(i) |
The output of this program is:
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1 4 9 16 |
Let’s assume you have a polynomial over a finite field $\F_q$, defined in Sage. How can you tell whether it’s a permutation polynomial? That is, when is the corresponding function $\F_q\to\F_q$ bijective? This is how you might have a polynomial over $\F_q$ defined in Sage:
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R.<x> = GF(q) S.<t> = PolynomialRing(R) f = 1 + x*t^2 |
Here, the variable $x$ refers the element $x$ in […]
Let $\F_q$ be a finite field. For any function $f:\F_q\to \F_q$, there exists a polynomial $p\in \F_q[x]$ such that $f(a) = p(a)$ for all $a\in \F_q$. In other words, every function from a finite field to itself can be represented by a polynomial. In particular, every permutation of $\F_q$ can be represented by a polynomial. […]
Here is an old classic from linear algebra: given an $ n\times n$ matrix $ A = (a_{ij})$, the determinant of $ A$ can be calculated using the permuation formula for the determinant: $ \det(A) = \sum_{\sigma\in S_n} (-1)^\sigma a_{1\sigma(1)}\cdots a_{n\sigma(n)}$. Here $ S_n$ denotes the permutation group on $ n$ symbols and $ (-1)^\sigma$ […]