Related papers: A top-down approach to algebraic renormalization i…
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for…
An inverse elastic source problem with sparse measurements is of concern. A generic mathematical framework is proposed which incorporates a low- dimensional manifold regularization in the conventional source reconstruction algorithms…
We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to a robust…
We investigate a supersymmetric generalisation of topological recursion from two perspectives: algebraic and geometric. The algebraic side concerns a recursive structure encoded in modules of a super Virasoro algebra, and the geometric…
We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological…
We outline the proofs of several principal statements in conventional renormalization theory. This may be of some use in the light of new trends and new techniques (Hopf algebras, etc.) recently introduced in the field.
We consider multiple polylogarithms in a single variable at non-positive integers. Defining a connected graded Hopf algebra, we apply Connes' and Kreimer's algebraic Birkhoff decomposition to renormalize multiple polylogarithms at…
We present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology. We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape…
We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and Weber to tackle quasi-linear singular SPDEs. We show that this approach is consistent…
Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's…
We present a novel way of constructing reduced models for systems of ordinary differential equations. The reduced models we construct depend on coefficients which measure the importance of the different terms appearing in the model and need…
In order to derive a class of geometric-type deformations of post-Lie algebras, we first extend the geometrical notions of torsion and curvature for a general bilinear operation on a Lie algebra, then we derive compatibility conditions…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
We focus on functional renormalization for ensembles of several (say $n\geq 1$) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form $ \exp[-\mathrm{Tr}(V_1)\times\ldots\times…
Models of complex systems often consist of multiple interconnected subsystem/component models that are developed by multi-disciplinary teams of engineers or scientists. To ensure that such interconnected models can be applied for the…
We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state…
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable…
The algebraic method of renormalization is applied to the standard model of electroweak interactions. We present the most important modifications compared to theories with simple groups.
Multi-relational learning has received lots of attention from researchers in various research communities. Most existing methods either suffer from superlinear per-iteration cost, or are sensitive to the given ranks. To address both issues,…
Many surface reconstruction methods incorporate normal integration, which is a process to obtain a depth map from surface gradients. In this process, the input may represent a surface with discontinuities, e.g., due to self-occlusion. To…