Related papers: Integration-by-parts identities in FDR
We present a general approach for evaluating a large variety of three-dimensional Fourier transforms. The transforms considered include the useful cases of the Coulomb and dipole potentials, and include situations where the transforms are…
Feynman integral reduction based on intersection theory provides an alternative to the traditional integration-by-parts method, yet its practical application has been constrained by the large number of variables required in the computation.…
We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gr\"obner basis method to determine the basis for…
The inverse design of photonic integrated circuits (PICs) presents distinctive computational challenges, including their large memory requirements. Advancements in the two-photon polymerization (2PP) fabrication process introduce additional…
We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the…
Normals with unknown parameters (NUP) can be used to convert nontrivial model-based estimation problems into iterations of linear least-squares or Gaussian estimation problems. In this paper, we extend this approach by augmenting factor…
We formulate a method to find the meromorphic solutions of higher-order recurrence relations in the form of the sum over poles with coefficients defined recursively. Several explicit examples of the application of this technique are given.…
In a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting…
Property Directed Reachability (PDR) is a widely used technique for formal verification of hardware and software systems. This paper presents an incremental version of PDR (IPDR), which enables the automatic verification of system instances…
We explain the construction of Forcer, a FORM program for the reduction of four-loop massless propagator-type integrals to master integrals. The resulting program performs parametric IBP reductions similar to the three-loop Mincer program.…
An automated treatment of iterated integrals based on letters induced by real-valued quadratic forms and Kummer--Poincar\'e letters is presented. These quantities emerge in analytic single and multi--scale Feynman diagram calculations. To…
We present a simple method which simplifies the evaluation of the on-shell multiple box diagrams reducing them to triangle type ones. For the $L$-loop diagram one gets the expression in terms of Feynman parameters with $2L$-fold…
The two point integrals contributing to the self energy of a particle in a three dimensional quantum field theory are calculated to two loop order in perturbation theory as well as the vacuum ones contributing to the effective potential to…
A recursive method is given for finding generating functions which enumerate rooted hypermaps by number of vertices, edges and faces for any given number of darts. It makes use of matrix-integral expressions arising from the study of…
We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an…
We present a new method for the numerical evaluation of arbitrary loop integrals in dimensional regularization. We first derive Mellin-Barnes integral representations and apply an algorithmic technique, based on the Cauchy theorem, to…
This article is the third and last of a series presenting an alternative method to compute the one-loop scalar integrals. It extends the results of first two articles to the infrared divergent case. This novel method enjoys a couple of…
We provide a sufficient condition for avoiding squared propagators in the intermediate stages of setting up differential equations for loop integrals. This condition is satisfied in a large class of two- and three-loop diagrams. For these…
Interval arithmetic is hardly feasible without directed rounding as provided, for example, by the IEEE floating-point standard. Equally essential for interval methods is directed rounding for conversion between the external decimal and…
Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we…