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The method of equivariant moving frames on multi-space is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for…
One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators…
Computing spherical harmonic decompositions is a ubiquitous technique that arises in a wide variety of disciplines and a large number of scientific codes. Because spherical harmonics are defined by integrals over spheres, however, one must…
We discuss how to resolve generic skew-symmetric and generic symmetric determinantal singularities. The key ingredients are (skew-) symmetry preserving matrix operations in order to deduce an inductive argument.
By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved…
The point-splitting regularization technique for composite operators is discussed in connection with anomaly calculation. We present a pedagogical and self-contained review of the topic with an emphasis on the technical details. We also…
Chiral logarithms in $m_\pi^2$ are calculated at one loop, taking into account the leading contributions to flavor symmetry breaking due to staggered fermions. I treat both the full QCD case (2+1 light dynamical flavors) and the quenched…
Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial…
We extend the notion of regularized integrals introduced by Li-Zhou that aims to assign finite values to divergent integrals on configuration spaces of Riemann surfaces. We then give cohomological formulations for the extended notion using…
This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized…
Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in…
In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second…
We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a…
Through defining irreducible loop integrals (ILIs), a set of consistency conditions for the regularized (quadratically and logarithmically) divergent ILIs are obtained to maintain the generalized Ward identities of gauge invariance in…
Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave…
We construct a specific formalism for calculating the one-loop virtual corrections for standard model processes with an arbitrary number of external legs. The procedure explicitly separates the infrared and ultraviolet divergences…
We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
Symmetries play an critical role in finding analytic solutions to nonlinear differential equations. A symmetry is a mapping of the solutions of the differential equation into the solutions and have been studied extensively for over a…
The possibility of treating colour in one-loop amplitude calculations alike the other quantum numbers is briefly discussed for semi-numerical algorithms based on generalized unitarity and parametric integration techniques. Numerical results…