Any two 1 by 1 real matrices commute.  This is in general not the case for 2 by 2 real matrices.  However, if A, B, C, and D are any 2 by 2 real matrices, then ABCD - ABDC - ACBD + ACDB + ADBC - ADCB - BACD + BADC + BCAD - BCDA - BDAC + BDCA + CABD - CADB - CBAD + CBDA + CDAB - CDBA - DABC + DACB + DBAC - DBCA - DCAB + DCBA = 0.  This identity is the first instance of a general result of Amitsur and Levitski; I will explain a simple graph-theoretic proof due to Swan.

TBA

This is a joint OxPDE and Numerical Analysis seminar. 

Professor Jinchao Xu will talk about; 'Neural Networks and Classical Numerical Methods: A Theoretical Perspective'

This talk compares neural network-based methods with classical numerical methods from a theoretical perspective. Through several representative examples, we examine both the potential and the limitations of deep neural networks in scientific computing and, more broadly, in machine learning. We begin by comparing ReLU deep neural networks with polynomials and piecewise polynomial spaces, focusing on their structures and expressive power. We then revisit the curse of dimensionality and discuss whether deep neural networks truly offer advantages over traditional numerical methods for high-dimensional problems. Next, we consider the use of deep neural networks for solving partial differential equations, with particular emphasis on the challenge of achieving high accuracy. Finally, we examine multigrid methods and explore whether their underlying principles can help us better understand, design, and train deep neural network models with possible implications for broader AI applications.

This is a Joint OxPDE & Numerical Analysis Seminar