Related papers: Guessing With Quadratic Differential Equations
Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important…
The software package developed in the MS thesis research implements functions for the intelligent guessing of polynomial sequence formulas based on user-defined expected sequence factors of the input coefficients. We present a specialized…
The technique of guessing can be very fruitful when dealing with sequences which arise in practice. This holds true especially when guessing is performed algorithmically and efficiently. One highly useful tool for this purpose is the…
A holonomic system of linear partial differential equations is, roughly speaking, a system whose solution space is finite dimensional. A distribution that is a solution of a holonomic system is called a holonomic distribution. We give…
Given finitely many consecutive terms of an infinite sequence, we discuss the construction of a polynomial difference equation that the sequence may satisfy. We also present a method to seek a candidate polynomial differential equation for…
Holographic Predictive Search (HPS) is a novel approach to search-based hologram generation that uses a mathematical understanding of the optical transforms to make informed optimisation decisions. Existing search techniques such as Direct…
Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. R{\'e}my showed that the holonomic equation associated with binary trees yields an efficient linear random generator of binary…
We show how the continuous Almkvist-Zeilberger algorithm can be used to efficiently discover and prove differential equations satisfied by generating functions of sequences defined as integrals of powers of C-finite polynomial sequences…
It was recently conjectured that every component of a discrete-time rational dynamical system is a solution to an algebraic difference equation that is linear in its highest-shift term (a quasi-linear equation). We prove that the conjecture…
Scaling inference compute in large language models (LLMs) through repeated sampling consistently increases the coverage (fraction of problems solved) as the number of samples increases. We conjecture that this observed improvement is…
A variation of Zeilberger's holonomic ansatz for symbolic determinant evaluations is proposed which is tailored to deal with Pfaffians. The method is also applicable to determinants of skew-symmetric matrices, for which the original…
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as…
Linear homogeneous recurrence equations with polynomial coefficients are said to be holonomic. Such equations have been introduced in the last century for proving and discovering combinatorial and hypergeometric identities. Given a field K…
The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. We apply the holonomic gradient method in the case of the conditional Poisson or multinomial distribution…
Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among…
With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger's "holonomic systems approach" in a broader mathematical sense is less common in the field of q-series and basic hypergeometric…
In this paper we discuss three symbolic approaches for the generation of a finite difference scheme of a partial differential equation (PDE). We prove, that for a linear PDE with constant coefficients these three approaches are equivalent…
Analytical solutions to nonlinear differential equations -- where they exist at all -- can often be very difficult to find. For example, Duffing's equation for a system with cubic stiffness requires the use of elliptic functions in the…