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Source-code generation methods may be used to produce the large number of separate base cases desirable to implement this strategy efficiently. [ 12 ] The generalized version of this idea is known as recursion "unrolling" or "coarsening", and various techniques have been proposed for automating the procedure of enlarging the base case.
Alternatively, replace the 3n + 1 with n ′ / H(n ′) where n ′ = 3n + 1 and H(n ′) is the highest power of 2 that divides n ′ (with no remainder). The resulting function f maps from odd numbers to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, f k (n) = 1).
[22] [23] [24] The running time of the Quine–McCluskey algorithm grows exponentially with the number of variables. For a function of n variables the number of prime implicants can be as large as 3 n / n {\displaystyle 3^{n}/{\sqrt {n}}} , [ 25 ] e.g. for 32 variables there may be over 534 × 10 12 prime implicants.
How to Solve It suggests the following steps when solving a mathematical problem: . First, you have to understand the problem. [2]After understanding, make a plan. [3]Carry out the plan.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
A variant of the 3-satisfiability problem is the one-in-three 3-SAT (also known variously as 1-in-3-SAT and exactly-1 3-SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has exactly one TRUE literal (and thus exactly two ...
Python supports normal floating point numbers, which are created when a dot is used in a literal (e.g. 1.1), when an integer and a floating point number are used in an expression, or as a result of some mathematical operations ("true division" via the / operator, or exponentiation with a negative exponent).
The Zen of Python is a collection of 19 "guiding principles" for writing computer programs that influence the design of the Python programming language. [1] Python code that aligns with these principles is often referred to as "Pythonic". [2] Software engineer Tim Peters wrote this set of principles and posted it on the Python mailing list in ...