Related papers: Invariant Clusters for Hybrid Systems
We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…
The security of multivariate cryptosystems and digital signature schemes relies on the hardness of solving a system of polynomial equations over a finite field. Polynomial system solving is also currently a bottleneck of index-calculus…
Barrier certificates, a form of state invariants, provide an automated approach to the verification of the safety of dynamical systems. Similarly to barrier certificates, recent works explore the notion of closure certificates, a form of…
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify…
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
In this paper we present methods for the synthesis of polynomial invariants for probabilistic transition systems. Our approach is based on martingale theory. We construct invariants in the form of polynomials over program variables, which…
The combination of numerical integration and deep learning, i.e., ODE-net, has been successfully employed in a variety of applications. In this work, we introduce inverse modified differential equations (IMDE) to contribute to the behaviour…
We associate with each simple Lie algebra a system of second-order differential equations invariant under a non-compact real form of the corresponding Lie group. In the limit of a contraction to a Schr\"odinger algebra, these equations…
We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation…
We determine the rings of invariants in the symmetric algebra on the dual of a vector space V over the field of two elements, for the group G of orthogonal transformations preserving a non-singular quadratic form on V. The invariant ring is…
Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We…
It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the $G$--modules $W$ and $m$ copies of $V$ can be separated by polynomial…
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
Certain quantum topological invariants of three manifolds can be written in the form of the Gaussian sum. It is shown that such topological invariants can be approximated efficiently by a quantum computer. The invariants discussed here are…
Various partially ordered Grothendieck group invariants are introduced for general operator algebras and these are used in the classification of direct systems and direct limits of finite-dimensional complex incidence algebras with common…
Polynomial invariants of a group action often appear only in high degree, and in many representations the invariant ring imposes severe degree constraints before any nontrivial invariants can occur. In contrast, the larger class of unitary…
We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find…
Recently, Neural Ordinary Differential Equations has emerged as a powerful framework for modeling physical simulations without explicitly defining the ODEs governing the system, but instead learning them via machine learning. However, the…
We present an algorithm for finding all time-reversible systems within a given family of 2-dim systems of ODE's whose right-hand sides are polynomials. We also study an interconnection of time-reversibility and invariants of a subgroup of…