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Algebraic Aspects of Conditional Independence and Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Thomas Kahle, Johannes Rauh, Seth Sullivant
The goal of this chapter is to explore these questions and introduce tools from computational algebra for studying them. Our perspective is that, for a fixed type of random variable, the set of distributions that satisfy a collection of independence constraints is the zero set of a collection of polynomial equations. Solutions of systems of polynomial equations are the objects of study of algebraic geometry, and so tools from algebra can be brought to bear on the problem. The next section contains an overview of basic ideas in algebraic geometry which are useful for the study of conditional independence structures and graphical models. In particular, it introduces algebraic varieties, polynomial ideals, and primary decomposition. Section 3.3 introduces the ideals associated to families of conditional independence statements, and explains how to apply the basic techniques to deduce conditional independence implications. Section 3.4 illustrates the main ideas with some deeper examples coming from the literature. Section 3.5 concerns the vanishing ideal of a graphical model, which is a complete set of implicit restrictions for that model. This set of restrictions is usually much larger than the set of conditional independence constraints that come from the graph, but it can illuminate the structure of the model especially with more complex families of models involving mixed graphs or hidden random variables. Section 3.6 highlights some key references in this area.
The coupled-cluster formalism – a mathematical perspective
Published in Molecular Physics, 2019
A. Laestadius, F. M. Faulstich
There is a rich history of mathematical investigations addressing CC methods prior to the local analyses in [15–17]. To give a complete historical account is beyond the scope of this article. We therefore limit ourselves and mention only a few important results. As a system of polynomial equations, the CC equations can have real or if the cluster operator is truncated, complex solutions. Furthermore, using quasi-Newton–Raphson methods to compute solutions of non-linear equations can lead to divergence since the approximated Jacobian may become singular. This is, in particular, the case when strongly correlated systems are considered. These and other related aspects of the CC theory have been addressed by Živković and Monkhorst [18,19] and Piecuch et al. [20]. Significant advances in the understanding of the nature of multiple solutions of single-reference CC have been made by Živković and Monkhorst [19], Kowalski and Jankowski [21], and by Piecuch and Kowalski [22]. An interesting attempt to address the existence of a cluster operator and cluster expansion in the open-shell case was done by Jeziorski and Paldus [23]. We would also like to mention the coupled-electron pair approximation (CEPA) [24–27]. This approach was introduced as a size-consistent alternative to the CISD method that was achieved by modifying (through topological factors [28]) the CI equations to account for higher excitations. This makes CEPA non-variational (for an adapted variational formulation of CEPA see [29]). CEPA can be regarded as an approximation of the CC method and does not form a truncation hierarchy that converges to the full-CI limit [30].
General solution of spin-1 Ising model in the effective field theory approximation: critical temperatures and spontaneous magnetization
Published in Phase Transitions, 2022
In this respect, in the present paper we shall show that a closed algebraic solution valid for all values of the coordination number also exists in the framework of the single-site cluster EFT approximation of the spin-1 Ising model. Here, first of all, the system of two algebraic polynomial equations for the determination of the critical temperatures of the model for all values of the coordination number will be derived. Moreover, the existence of this closed algebraic solution of the problem will allow us to find a simple function that approximates critical temperatures of the model with high precision for all coordination numbers. In addition, a closed system of polynomial equations is also found for the spontaneous magnetization of the model.
Remainder and quotient without polynomial long division
Published in International Journal of Mathematical Education in Science and Technology, 2021
In conclusion, we can reduce the search for factors of degree 2 of a polynomial f (of degree n) to the search for the integral roots of a system of polynomial equations of degree less than or equal to n−1. These last searches can be empirically executed using the rational root theorem.