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In dealing with nonlinear systems, it is common to use numerical solutions. Unlike the careful behavior towards the numerical results in chaotic regions, the validity of numerical results in regions of transient chaos might not always be…
We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum-Shub-Smale model with square root. It…
A new symbolic algorithmic implementation of the general scheme of the exponentially convergent functional-discrete (FD-) method is developed and justified for the Sturm-Liouville problem on a finite interval for the Schr\"odinger equation…
A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial…
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…
For the class of polynomial skew products of C^2, we describe a rigorous computer algorithm which, for a given map F, will (1) build a model of the dynamics of F on its chain recurrent set, and (2) attempt to determine whether F is Axiom A.…
Numerous interesting properties in nonlinear systems analysis can be written as polynomial optimization problems with nonconvex sum-of-squares problems. To solve those problems efficiently, we propose a sequential approach of local…
We propose a symbolic execution method for programs that can draw random samples. In contrast to existing work, our method can verify randomized programs with unknown inputs and can prove probabilistic properties that universally quantify…
Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows…
Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. Machine learning -- specifically deep learning -- techniques have shown their…
In this chapter we present the sums of Hermitian squares approach to noncommutative polynomial optimization problems. This is an extension of the sums of squares approach for polynomial optimization arising from real algebraic geometry. We…
Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time…
The connection of Taylor maps and polynomial neural networks (PNN) to solve ordinary differential equations (ODEs) numerically is considered. Having the system of ODEs, it is possible to calculate weights of PNN that simulates the dynamics…
Automatic theorem proving with deep learning methods has attracted attentions recently. In this paper, we construct an automatic proof system for trigonometric identities. We define the normalized form of trigonometric identities, design a…
We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set $S$. The truncated cones of moments of measures supported on the set $S$ is dual to nonnegative polynomials on $S$,…
This paper presents a probabilistic model validation methodology for nonlinear systems in time-domain. The proposed formulation is simple, intuitive, and accounts both deterministic and stochastic nonlinear systems with parametric and…
We address the decision problem for sentences involving univariate functions constructed from a fixed Pfaffian function of order $1$. We present a new symbolic procedure solving this problem with a computable complexity based on the…
We have designed a new symbolic-numeric strategy to compute efficiently and accurately floating point Puiseux series defined by a bivariate polynomial over an algebraic number field. In essence, computations modulo a well chosen prime $p$…
We present an algorithm that enumerates all the minimal triangulations of a graph in incremental polynomial time. Consequently, we get an algorithm for enumerating all the proper tree decompositions, in incremental polynomial time, where…
Non-linear polynomial systems over finite fields are used to model functional behavior of cryptosystems, with applications in system security, computer cryptography, and post-quantum cryptography. Solving polynomial systems is also one of…