Related papers: Singular paths spaces and applications
This work derives closed-form expressions computing the expectation of co-presence and of number of co-occurrences of nodes on paths sampled from a network according to general path weights (a bag of paths). The underlying idea is that two…
Rough sheets are two-parameter analogs of rough paths. In this work the theory of integration over functions of two parameters is extended to cover the case of irregular functions by developing an appropriate notion of rough sheet. The main…
For generic maps from compact surfaces with boundary into the plane we develop an explicit algorithm for minimizing both the number of cusps and the number of components of the singular locus. More precisely, we minimize among maps with…
We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework…
The purpose of this paper is to show how Rees algebras can be applied in the study of singularities embedded in smooth schemes over perfect fields. In particular, we will study situations in which the multiplicity of a hypersurface is a…
Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as…
All random wave fields possess a network of phase singularities. We show that while the phase statistics within speckle patterns is generic, the statistics of the motion of phase singularities differs substantially for diffusive and…
We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices. Our main tool is the theory of regularity structures,…
In this chapter we provide a theoretically founded investigation of state-of-the-art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization…
A path-integral approach for the computation of quantum-mechanical propagators and energy Green's functions is presented. Its effectiveness is demonstrated through its application to singular interactions, with particular emphasis on the…
We consider a family of singular maps as an example of a simple model of dynamical systems exhibiting the property of robust chaos on a well defined range of parameters. Critical boundaries separating the region of robust chaos from the…
Using the methods originally developed for Random Matrix Theory we derive an exact mathematical formula for number variance (introduced in [4]) describing a rigidity of particle ensembles with power-law repulsion. The resulting relation is…
Soft lubrication has been shown to drastically affect the mobility of an object immersed in a viscous fluid in the vicinity of a purely elastic wall. In this theoretical study, we develop a minimal model incorporating viscoelasticity,…
We continue the approach in Part I \cite{duchong19} to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part II deals with driving…
Boundary integral equation methods are widely used in the solution of many partial differential equations. The kernels that appear in these surface integrals are nearly singular when evaluated near the boundary, and straightforward…
We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided $\alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairer's…
We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our…
We introduce a new approach to model and analyze \emph{Mobility}. It is fully based on discrete mathematics and yields a class of mobility models, called the \emph{Markov Trace} Model. This model can be seen as the discrete version of the…
We embed the rough integration in a larger geometrical/algebraic framework of integrating one-forms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated…
This study introduces a new unified structural framework for orbifold sigma models that incorporates twisted sectors, singularities, and smooth regions into a single algebraic object. Traditional approaches to orbifold theories often treat…