Related papers: Multiple polylogarithms, a regularisation process …
In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which…
The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to…
Via a unified geometric approach, a class of generalized trigonometric functions with two parameters are analytically extended to maximal domains on which they are univalent. Some consequences are deduced concerning radius of convergence…
The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the…
Regularization approaches have demonstrated their effectiveness for solving ill-posed problems. However, in the context of variational restoration methods, a challenging question remains, which is how to find a good regularizer. While total…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study…
Balayage of measures with respect to classes of all subharmonic or harmonic functions on an open set of a plane or finite-dimensional Euclidean space is one of the main objects of potential theory and its applications to the complex…
The present paper establishes boundedness of $[m-\frac n2+\frac 12]$ derivatives for the solutions to the polyharmonic equation of order $2m$ in arbitrary bounded open sets of $\RR^n$, $2\leq n\leq 2m+1$, without any restrictions on the…
When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular…
We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function…
This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges…
We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link…
Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional…
We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…
Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional…
We derive a divergence formula for a group of regularization methods with an L2 constraint. The formula is useful for regularization parameter selection, because it provides an unbiased estimate for the number of degrees of freedom. We…
Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic…