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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
Commutative algebra is the study of systems of polynomial equations, and algebraic geometry is the study of geometric properties of their solutions. Both are rich fields with many deep results. This section only gives a very coarse introduction to the basic facts that hopefully makes it possible for the reader to understand the phenomena and algorithms discussed in later parts of this chapter. For a more detailed introduction, the reader is referred to the standard textbook [7].
A dynamical proof of the van der Corput inequality
Published in Dynamical Systems, 2022
Nikolai Edeko, Henrik Kreidler, Rainer Nagel
Let be a right cancellative semigroup. Moreover, let and be right Følner nets for ,E be a pre-Hilbert module over a unital commutative -algebra A, a representation as Markov operators, a representation such that is -dominated for every , and a state for every .Then for every .
On the definition of matrix multiplication
Published in International Journal of Mathematical Education in Science and Technology, 2020
According to Edwards and Ward (2008) ‘mathematical definitions frequently do have a history – that is, they can and do evolve’ (p. 224). Undoubtedly, one of the most reliable sources to search for some answers, especially about the philosophy of generating mathematical concepts and definitions, is the history of mathematics. So, we want to pay attention to the philosophy of the usual definition of matrix multiplication and its applications in mathematical textbooks and the reason for the complicated form of that definition. A glance at the history of mathematics shows that algebra of matrices, as a non-commutative algebra, was introduced by Arthur Cayley in 1857 (see Eves, 1990). He invented matrices during the study of a linear transformation () which transforms the ordered pair (x, y) into (x′, y′) as follows:
Exergetic port-Hamiltonian systems: modelling basics
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Markus Lohmayer, Paul Kotyczka, Sigrid Leyendecker
Vector fields are (isomorphic to) derivations on the commutative -algebra of smooth functions with pointwise multiplication. The Leibniz rule says that for some fixed we have and thus it asserts that is a vector field.