arXiv subjects

Antonio Jiménez-Pastor

Publications and source records attributed to Antonio Jiménez-Pastor.

9 recordsLinked to original sources

arXiv2026

Gröbner bases of Burchnall-Chaundy ideals for ordinary differential operators

The correspondence between commutative rings of ordinary differential operators (ODOs) and algebraic curves was established by Burchnall and Chaundy, Krichever and Mumford, among many others. To make this correspondence computationally effective, in this paper we aim to compute the defining ideals of spectral curves, Burchnall-Chaundy (BC) ideals. We provide an algorithm to compute a Gröbner basis of a BC ideal. The point of departure is the computation of the finite set of generators of a maximal commutative ring of ODOs, which was implemented by the authors in the package dalgebra of SageMath. The algorithm to compute BC ideals has been also implemented in dalgebra. The differential Galois theory of the corresponding spectral problems, linear differential equations with parameters, would benefit from the computation on this prime ideal, generated by constant coefficient polynomials. In particular, we prove the primality of the differential ideal generated by a BC ideal, after extending the coefficient field. This is a fundamental result to develop Picard-Vessiot theory for spectral problems.

math.AC↗

arXiv2025

Computing almost commuting bases of ODOs and Gelfand-Dickey hierarchies

Almost commuting operators were introduced in 1985 by George Wilson to present generalizations of the Korteweg-de Vries hierarchy, nowadays known as Gelfand-Dickey (GD) hierarchies. In this paper, we review the formal construction of the vector space of almost commuting operators with a given ordinary differential operator (ODO), with the ultimate goal of obtaining a basis by computational routines, using the language of differential polynomials. We use Wilson's results on weighted ODOs to guarantee the solvability of the triangular system that allows to compute the homogeneous almost commuting operator of a given order in the ring of ODOs. As a consequence, the computation of the equations of the GD hierarchies is achieved without using pseudo-differential operators. A new package in SageMath called \texttt{dalgebra} has been designed to perform symbolic calculations in differential domains. The algorithms to calculate the almost commuting basis and the GD hierarchies in the ring of ODOs are implemented in SageMath, and explicit examples are provided.

math.AC↗

arXiv2025

Effective computation of centralizers of ODOs

This work is devoted to computing the centralizer $Z (L)$ of an ordinary differential operator (ODO) in the ring of differential operators. Non-trivial centralizers are known to be coordinate rings of spectral curves and contain the ring of polynomials $C [L]$, with coefficients in the field of constants $C$ of $L$. We give an algorithm to compute a basis of $Z (L)$ as a $C [L]$-module. Our approach combines results by K. Goodearl in 1985 with solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchy, which after substituting the coefficients of $L$ become linear, and whose solution sets form a flag of constants. We are assuming that the coefficients of $L$ belong to a differential algebraic extension $K$ of $C$. In addition, by considering parametric coefficients we develop an algorithm to generate families of ODOs with non trivial centralizer, in particular algebro-geometric, whose coefficients are solutions in $K$ of systems of the stationary GD hierarchy.

math.RA↗

arXiv2024

Approximate Constrained Lumping of Chemical Reaction Networks

Gaining insights from realistic dynamical models of biochemical systems can be challenging given their large number of state variables. Model reduction techniques can mitigate this by decreasing complexity by mapping the model onto a lower-dimensional state space. Exact constrained lumping identifies reductions as linear combinations of the original state variables in systems of nonlinear ordinary differential equations, preserving specific user-defined output variables without error. However, exact reductions can be too stringent in practice, as model parameters are often uncertain or imprecise -- a particularly relevant problem for biochemical systems. We propose approximate constrained lumping. It allows for a relaxation of exactness within a given tolerance parameter $\varepsilon$, while still working in polynomial time. We prove that the accuracy, i.e., the difference between the output variables in the original and reduced model, is in the order of $\varepsilon$. Furthermore, we provide a heuristic algorithm to find the smallest $\varepsilon$ for a given maximum allowable size of the lumped system. Our method is applied to several models from the literature, resulting in coarser aggregations than exact lumping while still capturing the dynamics of the original system accurately.

cs.CE↗

arXiv2024

Forward and Backward Constrained Bisimulations for Quantum Circuits using Decision Diagrams

Efficient methods for the simulation of quantum circuits on classic computers are crucial for their analysis due to the exponential growth of the problem size with the number of qubits. Here we study lumping methods based on bisimulation, an established class of techniques that has been proven successful for (classic) stochastic and deterministic systems such as Markov chains and ordinary differential equations. Forward constrained bisimulation yields a lower-dimensional model which exactly preserves quantum measurements projected on a linear subspace of interest. Backward constrained bisimulation gives a reduction that is valid on a subspace containing the circuit input, from which the circuit result can be fully recovered. We provide an algorithm to compute the constraint bisimulations yielding coarsest reductions in both cases, using a duality result relating the two notions. As applications, we provide theoretical bounds on the size of the reduced state space for well-known quantum algorithms for search, optimization, and factorization. Using a prototype implementation, we report significant reductions on a set of benchmarks. In particular, we show that constrained bisimulation can boost decision-diagram-based quantum circuit simulation by several orders of magnitude, allowing thus for substantial synergy effects.

quant-ph↗

arXiv2024

Factorial Basis Method for q-Series Applications

The Factorial Basis method, initially designed for quasi-triangular, shift-compatible factorial bases, provides solutions to linear recurrence equations in the form of definite-sums. This paper extends the Factorial Basis method to its q-analog, enabling its application in q-calculus. We demonstrate the adaptation of the method to q-sequences and its utility in the realm of q-combinatorics. The extended technique is employed to automatically prove established identities and unveil novel ones, particularly some associated with the Rogers-Ramanujan identities.

cs.SC↗

arXiv2023

Proper Lumping for Positive Bilinear Control Systems

Positive systems naturally arise in situations where the model tracks physical quantities. Although the linear case is well understood, analysis and controller design for nonlinear positive systems remain challenging. Model reduction methods can help tame this problem. Here we propose a notion of model reduction for a class of positive bilinear systems with (bounded) matrix and exogenous controls. Our reduction, called proper positive lumping, aggregates the original system such that states of the corresponding reduced model represent non-negative linear combinations of original state variables. We prove a characterization result showing that the reductions by proper positive lumping are exactly those preserving the optimality of a suitable class of value functions. Moreover, we provide an efficient polynomial-time algorithm for the computation of the minimal lumping. We numerically evaluate our approach by applying it to a number of benchmark case studies.

math.OC↗

arXiv2022

The Factorial-Basis Method for Finding Definite-Sum Solutions of Linear Recurrences With Polynomial Coefficients

The problem of finding a nonzero solution of a linear recurrence $Ly = 0$ with polynomial coefficients where $y$ has the form of a definite hypergeometric sum, related to the Inverse Creative Telescoping Problem of [14][Sec. 8], has now been open for three decades. Here we present an algorithm (implemented in a SageMath package) which, given such a recurrence and a quasi-triangular, shift-compatible factorial basis $\mathcal{B} = \langle P_k(n)\rangle_{k=0}^\infty$ of the polynomial space $\mathbb{K}[n]$ over a field $\mathbb{K}$ of characteristic zero, computes a recurrence satisfied by the coefficient sequence $c = \langle c_k\rangle_{k=0}^\infty$ of the solution $y_n = \sum_{k=0}^\infty c_kP_k(n)$ (where, thanks to the quasi-triangularity of $\mathcal{B}$, the sum on the right terminates for each $n \in \mathbb{N}$). More generally, if $\mathcal{B}$ is $m$-sieved for some $m \in \mathbb{N}$, our algorithm computes a system of $m$ recurrences satisfied by the $m$-sections of the coefficient sequence $c$. If an explicit nonzero solution of this system can be found, we obtain an explicit nonzero solution of $Ly = 0$.

cs.SC↗

arXiv2022

Exact linear reduction for rational dynamical systems

Detailed dynamical systems models used in life sciences may include dozens or even hundreds of state variables. Models of large dimension are not only harder from the numerical perspective (e.g., for parameter estimation or simulation), but it is also becoming challenging to derive mechanistic insights from such models. Exact model reduction is a way to address this issue by finding a self-consistent lower-dimensional projection of the corresponding dynamical system. A recent algorithm CLUE allows one to construct an exact linear reduction of the smallest possible dimension such that the fixed variables of interest are preserved. However, CLUE is restricted to systems with polynomial dynamics. Since rational dynamics occurs frequently in the life sciences (e.g., Michaelis-Menten or Hill kinetics), it is desirable to extend CLUE to the models with rational dynamics. In this paper, we present an extension of CLUE to the case of rational dynamics and demonstrate its applicability on examples from literature. Our implementation is available in version 1.5 of CLUE at https://github.com/pogudingleb/CLUE.

q-bio.QM↗