An Existence Theorem for Sum and Product Integrals

Proof of the Sum - Product Theorem

These are notes from a mini course on additive combinatorics given in Princeton University on August 23-24, 2007. The lectures were Boaz Barak (Princeton University), Luca Trevisan (University of California at Berkeley) and Avi Wigderson (Institute for Advanced Study, Princeton). The notes were taken by Aditya Bhaskara, Arnab Bhattacharyya, Moritz Hardt, Cathy Lennon, Kevin Matulef, Rajsekar Ma...

Notes on the sum product theorem

The Sum-Product Theorem: A Foundation for Learning Tractable Models

Inference in expressive probabilistic models is generally intractable, which makes them difficult to learn and limits their applicability. Sumproduct networks are a class of deep models where, surprisingly, inference remains tractable even when an arbitrary number of hidden layers are present. In this paper, we generalize this result to a much broader set of learning problems: all those where i...

An Equilibrium Existence Theorem

Bewley 14, Theorem l] proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of Arrow and Debreu [ 11. The assumptions on technology and preferences are natural and applicable in a wide variety of cases. The proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dim...

Sum Formula for Maximal Abstract Monotonicity and Abstract Rockafellar’s Surjectivity Theorem

In this paper, we present an example in which the sum of two maximal abstract monotone operators is maximal. Also, we shall show that the necessary condition for Rockafellar’s surjectivity which was obtained in ([19], Theorem 4.3) can be sufficient.