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Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…
In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new…
Thermodynamics (in concert with its sister discipline, statistical physics) can be regarded as a data reduction scheme based on partitioning a total system into a subsystem and a bath that weakly interact with each other. The ubiquity and…
Most nonlinear partial differential equation (PDE) solvers require the Jacobian matrix associated to the differential operator. In PETSc, this is typically achieved by either an analytic derivation or numerical approximation method such as…
We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the…
An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and…
For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation…
I describe a method for computer algebra that helps with laborious calculations typically encountered in theoretical microhydrodynamics. The program mimics how humans calculate by matching patterns and making replacements according to the…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations…
In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems,…
In this short letter, I present a powerful application in dimensionality reduction of the lesser-used Jacobi-Davidson algorithm for the generalized eigenvalue decomposition. When combined with matrix-free implementations of relevant…
The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale…
We present a program for the reduction of large systems of integrals to master integrals. The algorithm was first proposed by Laporta; in this paper, we implement it in MAPLE. We also develop two new features which keep the size of…
Ihe first author presented an efficient algorithm for computing involutive (and reduced Groebner) bases. In this paper, we consider a modification of this algorithm which simplifies matters to understand it and to implement. We prove…
These lectures give a brief introduction to the Computer Algebra systems Reduce and Maple. The aim is to provide a systematic survey of most important commands and concepts. In particular, this includes a discussion of simplification…
We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_tf)(x_0)(v_0)={1\over t} E f(x_t) \int_0^t \langle Y(x_s)(v_s),dB_t\rangle_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
Estimating vibrational entropy is a significant challenge in thermodynamics and statistical mechanics due to its reliance on quantum mechanical properties. This paper introduces a quantum algorithm designed to estimate vibrational entropy…
The family of left-to-right GCD algorithms reduces input numbers by repeatedly subtracting the smaller number, or multiple of the smaller number, from the larger number. This paper describes how to extend any such algorithm to compute the…