G^N vs Probability Theory Foundations

G^N is a special calculator not just a game: This math app animates iterations of one or compositions of two multi-valued maps for free groups of in principle arbitrary finite rank in the abelian and non-commutative cases respectively (in this version the rank is restricted to values in between 2 and 11). Well-known examples of such multi-valued maps are permutations, for example the flip (x_1,x_2)->(x_2,x_1), in G^N we just write (1,2)->(2,1) and instead of the inverse we convenient flip the number. A more involved type of such a multi-valued map is the n+1 cycle defined by (1,...,n)->(-2-3-...-n-1,3,4,...,n,1), a non-trivial theorem. For example if n=2 this map is the hexagon (1,2)->(-2-1,1), this map applied twice reads (1,2)->(2,-2-1) and the third iteration is the identity map (1,2)->(1,2).

Probability Theory Foundations is an independent study app for learners who want structured practice in core probability ideas, problem-solving habits, and exam-style reasoning. Built for students, self-learners, and anyone reviewing fundamentals, the app includes 160 questions across Fundamentals, Practical Skills, Common Pitfalls, and Advanced Concepts. Every question includes an explanation so you can understand why an answer works, not just whether it was right. Features: - FSRS spaced-repetition intervals to help review at the right time - Weak-point heatmap to spot topics that need attention - Daily challenge for consistent practice - Mock exams for timed review - Offline practice with no login required - Explanations for every question Practice offline, no login required. Independent study app for Probability Theory Foundations practice. Not affiliated with, endorsed by, or sponsored by any company, certification body, exam operator, or vendor. All trademarks belong to their respective owners.