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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].
Variants of the Simplex Method
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
In one sense, the self-dual parametric algorithm is a variation of the primal and dual simplex algorithms. In another sense, it introduces a completely new idea for solving problems. We want to solve problem P. Create an infinite set of problems P(θ), parameterized by θ, such that P(0)=P and P(t) is easy to solve for some easily determined value t. (P(θ) should be a continuous function of θ in the sense that both the objective and constraint functions are continuous functions of θ.) Starting from θ=t, reduce |θ| to 0 continuously, tracking the solution to P(θ) as θ changes. When θ reaches 0, problem P is solved. Is P too hard to solve directly? Change it to an easy problem, solve that, then change the problem back to P. That may seem like a crazy idea, but it can work. This kind of algorithm is called a homotopy method. Homotopy methods have been used successfully to find equilibria [83],[84],[85],[259] and to solve systems of polynomial equations [190],[254] and various other nonlinear problems [148].
Optimization of Thermal-Flow Processes in a System of Conjugate Cooling Towers
Published in Heat Transfer Engineering, 2020
Paweł Regucki, Marek Lewkowicz, Renata Krzyżyńska
Equation (5) provides explicit formula for the unknowns qi: in terms of the unknown parameter λ. Here, both branches of the solution corresponding to the signs “+” or “-” have to be taken into account. The condition given by Eq. (6) can now be treated as a non-algebraic equation with one variable λ only. Various numerical methods easily provide good numerical approximations of some solutions of such equations. The question of whether all possible solutions have been found is far more difficult. For the objective functions considered in this paper, we have used two approaches, each leading to a set of solutions, the completeness of which can be proven in a mathematically rigorous way. One approach is based on advanced techniques from numerical algebra and uses Groebner bases, a technique applicable to systems of polynomial equations with rational coefficients, as in Eq. (5). The other seeks all the solutions of Eq. (7) by extending the methods usually used to find all the roots of univariate polynomials to the algebraic functions. The details will not be provided in this paper.
Polynomial semantics for modal logics
Published in Journal of Applied Non-Classical Logics, 2019
Juan C. Agudelo-Agudelo, Santiago Echeverri-Valencia
In Agudelo-Agudelo, Agudelo-González, and García-Quintero (2016), an abstract and general definition of whether a propositional logic is characterised by polynomials is given, what allows to see the interpretation of logics by means of polynomials not only as a method for solving logical problems, but as an algebraic semantics alternative to the usual algebraic characterisation of propositional logics. Under such definition, it is shown that every finite-valued propositional logic (including those with several designated truth values, which are not included in the previous works) and even some paraconsistent propositional logics, which are not characterisable by finite-valued matrices, can be characterised by polynomials with coefficients on finite fields. As a direct consequence, several logic questions can be solved by using the theory of Gröbner bases. Moreover, an algorithm to solve systems of polynomial equations over finite fields is given, and such a method is used to find models and counter-models for sets of formulas.
Multidimensional realisation theory and polynomial system solving
Published in International Journal of Control, 2018
Philippe Dreesen, Kim Batselier, Bart De Moor
We have aimed to keep the exposition as simple and accessible as possible, requiring only the most elementary notions of linear algebra and state-space system theory. Throughout the paper, systems of polynomial equations are used to represent multidimensional difference equations (using a multi-indexed Z-transform). We assume that the difference equations define scalar signals that vary in n independent indices. Furthermore, we assume that the corresponding systems of polynomials describe zero-dimensional solution sets in the projective space, a fact that our approach will also reveal as the dimensions of certain null spaces will stabilise in the case the solution set is zero-dimensional. With some abuse of terminology, at times we may refer to an overdetermined system of multidimensional difference equations as its representation as a polynomial system, or vice versa.