Related papers: Towards a Basis for Planar Two-Loop Integrals
We present a simplified variant of the integrand reduction algorithm for multiloop scattering amplitudes in $d = 4 - 2\epsilon$ dimensions, which exploits the decomposition of the integration momenta in parallel and orthogonal subspaces,…
We address a number of conjectures about the ground state O(1) loop model, computing in particular two infinite series of partial sums of its entries and relating them to the enumeration of plane partitions. Our main tool is the use of…
In this paper we study theoretical properties of inverter-based microgrids controlled via primary and secondary loops. Stability of these microgrids has been the subject of a number of recent studies. Conventional approaches based on…
Using evaluation at appropriately chosen points, we propose a Gr\"obner basis free approach for calculating the secondary invariants of a finite permutation group. This approach allows for exploiting the symmetries to confine the…
Algorithmic computation in polynomial rings is a classical topic in mathematics. However, little attention has been given to the case of rings with an infinite number of variables until recently when theoretical efforts have made possible…
In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high…
Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order models of parametrized partial differential equation problems. With speedups that can reach several orders of…
A perturbative approach to quantum field theory involves the computation of loop integrals, as soon as one goes beyond the leading term in the perturbative expansion. First I review standard techniques for the computation of loop integrals.…
In string theories, one-loop scattering amplitudes are characterized by integrals over genus-one surfaces using the Kronecker-Eisenstein series. A recent methodology proposed a genus-one basis formed from products of these series of chain…
A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail. The method is based on a repeated application of the functional relations proposed by the author.…
Recently, a new approach for high loop integrals has been proposed in \cite{Huang:2024nij}, where the whole parameter integration has been divided into two parts: a one-loop-like integration and the remaining parameter integration. In this…
This paper formalize the existence's proof of first-integrals for any second order ODE, allowing to discriminate periodic orbits. Up to the author's knowledge, such a powerful result is not available in the literature providing a tool to…
Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential…
We introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one-dimensional anharmonic oscillators. We present several methods of analytically fixing the nonlinear variational…
An algorithm for calculating two-loop propagator type Feynman diagrams with arbitrary masses and external momentum is proposed. Recurrence relations allowing to express any scalar integral in terms of basic integrals are given. A minimal…
I will present a new method for thinking about and for computing loop integrals based on differential equations. All required information is obtained by algebraic means and is encoded in a small set of simple quantities that I will…
A number theoretic algorithm is given for writing gauge theory amplitudes in a compact manner. It is possible to write down all details of the complete $L$ loop amplitude with two integers, or a complex integer. However, a more symmetric…
We revisit the familiar construction of one-loop scattering amplitudes via generalized unitarity in light of the recently understood properties of loop integrands prior to their integration. We show how in any four-dimensional quantum field…
We propose a general algorithm of constructing an extended formulation for any given set of linear constraints with integer coefficients. Our algorithm consists of two phases: first construct a decision diagram $(V,E)$ that somehow…
We use unitarity techniques to compute the two-loop non-planar corrections to the Sudakov form factor and the four-point amplitude in ABJM theory. We start by reconstructing non-planar integrals from two-particle cuts in three dimensions.…