In this paper, we try to clarify some of the questions related to a key concept in multivariate polynomial solving algorithm over a finite field: the degree of regularity. By the degree of regularity, here we refer to a concept first presented by Dubois and Gama, namely the lowest degree at which certain nontrivial degree drop of a polynomial system occurs. Currently, it is somehow commonly accepted that we can use this degree to estimate the complexity of solving a polynomial system, even though we do not have systematic empirical data or a theory to support such a claim. In this paper, we would like to clarify the situation with the help of experiments. We first define a concept of solving degree for a polynomial system. The key question we then need to clarify is the connection of solving degree and the degree of regularity with focus on quadratic systems. To exclude the cases that do not represent the general situation, we need to define when a system is degenerate and when it is irreducible. With extensive computer experiments, we show that the two concepts, the degree of regularity and the solving degree, are related for irreducible systems in the sense that the difference between the two degrees is indeed small, less than 3. But due to the limitation of our experiments, we speculate that this may not be the case for high degree cases.

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This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and ``quantum geometries.'' The dynamics is described by a novel variational principle, called the causal action principle. In addition to the basics, the book provides all the necessary mathematical background and explains how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields. The focus is on getting a mathematically sound connection between causal fermion systems and physical systems in Minkowski space. The book is intended for graduate students entering the field, and is furthermore a valuable reference work for researchers in quantum field theory and quantum gravity.

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We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrdinger equation. Traditional Gaussian beam type methods for the Schrdinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al.[S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrdinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi