Related papers: Dimensional analysis using toric ideals
This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular,…
The basic concepts of classical mechanics are given in the operator form. The dynamical equation for a hybrid system, consisting of quantum and classical subsystems, is introduced and analyzed in the case of an ideal nonselective…
We review, from a didactic point of view, the definition of a toric section and the different shapes it can take. We'll then discuss some properties of this curve, investigate its analogies and differences with the most renowned conic…
For studying the objectivity and the quality of a given form of the Periodic system as a single whole we compare plots of functions presenting properties of elements in pairs of periods. Using mathematical statistics we introduce a…
Canonical variables for the Poisson algebra of quantum moments are introduced here, expressing semiclassical quantum mechanics as a canonical dynamical system that extends the classical phase space. New realizations for up to fourth order…
Classical vector analysis is the predominant formalism used by engineers of computational electromagnetism, despite the fact that manifold as a theoretical concept has existed for a century. This paper discusses the benefits of manifolds…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial…
It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a topological semiring replacement for the space of continuous valuations on a topological ring. This requires building the theory of…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…
Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection,…
A new, coordinate-free (geometric) approach to multivariate statistical analysis. General multivariate linear models and linear hypotheses are defined in geometric form. A method of constructing statistical criteria is defined for linear…
This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic…
Principal component analysis is a versatile tool to reduce dimensionality which has wide applications in statistics and machine learning. It is particularly useful for modeling data in high-dimensional scenarios where the number of…
Correspondence identifies relationships among objects via similarities among their components; it is ubiquitous in the analysis of spatial datasets, including images, weather maps, and computational simulations. This paper develops a novel…
We explore the positive geometry of statistical models in the setting of toric varieties. Our focus lies on models for discrete data that are parameterized in terms of Cox coordinates. We develop a geometric theory for computations in…