Union-free Hypergraphs and Probability Theory

Union-free Hypergraphs and Probability Theory

Notice that the upper bound is an immediate consequence of (I<;J) :s;; 2n. To prove the lower bound we use an algebraic construction which is a modification of a construction of Babai and S6s [1]. How a family of sets can fail to have the union-free property? There are essentially two possibilities: (a) there are four distinct sets A, B, C, D with Au B =CuD. (b) there are three distinct sets A,...

Free Probability Theory and Free Diffusion

Free Probability Theory

Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the twenty years since its creation, fr...

Free Probability Theory

Now we come to one of the most important and inspiring realizations of freeness. Up to now we have realized free random variables by generators of the free group factors or by creation and annihilation operators on full Fock spaces. This was not very surprising because our definition of freeness was modelled according to such situations. But there are objects from a quite different mathematical...

Notes on Free Probability Theory

Lecture 1. Free Independence and Free Harmonic Analysis. 2 1.1. Probability spaces. 2 1.2. Non-commutative probability spaces. 3 1.3. Classical independence. 4 1.4. Free products of non-commutative probability spaces. 5 1.5. Free Fock space. 6 1.6. Free Central Limit Theorem. 8 1.7. Free Harmonic Analysis. 10 1.8. Further topics. 16 Lecture 2. Random Matrices and Free Probability. 17 2.1. Rando...