A multivariate polynomial f is decomposable if f is a composition f=u∘ h , where u is a univariate polynomial of degree greater than one and h is a multivariate polynomial. We consider the problem of computing the nearest decomposable polynomial to a given multivariate polynomial with respect to the 2-norm. By using the relationship between the decomposability of a polynomial and the rank of a structured matrix corresponding to the polynomial, we reduce this problem to the structured low-rank approximation problem. Then we propose algorithms for computing an approximate decomposition of a multivariate polynomial based on the structured low-rank approximation problem.
Cancer is a major cause of mortality, yet the mechanisms governing its growth, progression, and elimination remain incompletely understood. Mathematical modeling not only clarifies these complex processes but also provides clinicians with predictive insights into cancer cell behavior that support informed decision-making. This paper presents a computational algorithm for simulating cancer cell growth. The model first employs ordinary differential equations (ODEs) to describe temporal dynamics and is then extended to partial differential equations (PDEs) to capture spatial heterogeneity in tumor development and immune interactions. The proposed method applies stabilized moving least squares (MLS) functions within a Galerkin framework in both time and space to approximate the solution. Unlike conventional MLS approaches, which lose stability as node spacing decreases, the stabilized MLS ensures accuracy and robustness regardless of point distribution. Through polynomial fitting via weighted local least squares, the stabilized MLS technique achieves precise functional approximation with low computational cost. The algorithm is computationally efficient, straightforward to implement, and compatible with standard computer systems. To validate the proposed technique, two comprehensive numerical cases based on cancer cell growth models are provided, demonstrating the methods efficiency, accuracy, and its consistency with observed cancer phenomena.
This work is devoted to the symbolic computation of centralizers of ordinary differential operators (ODOs), in the ring of differential operators. Starting with an operator L of order n and the order 𝔪 of a non-trivial operator in its centralizer, which is not a multiple of n , a finite set of generators of a subalgebra of the centralizer is obtained, maximal of a certain rank R , the greatest common divisor of all orders of its elements. The true rank r of the centralizer is unknown to start unless (n,𝔪)=1 since 1≤ r≤ R≤ (n,𝔪) , and remains unknown unless our algorithm returns R=1 . Ours is a direct approach based on solving the systems of equations of the stationary Gelfand-Dickey (GD) hierarchies, 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 computable differential field. In addition, by considering parametric coefficients, we develop an algorithm to generate families of ODOs with non-trivial centralizer, whose coefficients belong to a previously chosen differential field. Our algorithms are implemented in SageMath.
In this work, we address the problem of expressing sequences of d -orthogonal polynomials in terms of either orthogonal polynomials or the canonical basis. We derive a general recurrence relation that enables the symbolic recursive computation of these coefficients. Several results are established to investigate the impact of the Appell character, as well as the symmetry and d -symmetry of the polynomial sequences, on these coefficients. Additionally, we deduce some results concerning the zeros of d -orthogonal polynomials.
