相关论文: Relative invariants, difference equations, and the…

This short book is the result of various master and summer school courses I have taught. The objective is to introduce the readers to mathematical control theory, both in finite and infinite dimension. In the finite-dimensional context, we…

Three versions of the Weak Law of Large Numbers are proposed for weakly dependent and generally speaking non-equally distributed random variables, with finite or possibly infinite expectations.

For a family of stochastic differential equations, we investigate the asymptotic behaviors of its corresponding Picard's iteration, establishing convergence results in terms of relative entropy. Our convergence results complement the…

The first part deals with piecewise fractional linear maps with three branches. Given a map $T$ a map $S$ is called a related map if some branches of $T$ are replaced by a 'flipped' branch, namely a branch of $1-T$. The main question is if…

In this note, we give a remark on the proof of Lemma 3 in Lipshitz's paper "The diagonal of a D-Finite power series is D-Finite". This remark is motivated by the observation that the statement from line -8 to -3 on page 375 of that paper…

The paper that is commented by Touchette contains a computational study which opens the door to a desirable generalization of the standard large deviation theory (applicable to a set of $N$ nearly independent random variables) to systems…

These lecture notes consist of three chapters. In the first chapter we present oracle inequalities for the prediction error of the Lasso and square-root Lasso and briefly describe the scaled Lasso. In the second chapter we establish…

The main purpose of the paper is to demonstrate that condition of invariance with respect to the Legendre transformations allows effectively isolate the class of integrable difference equations on the triangular lattice, which can be…

This contribution presents a comprehensive analysis of Colombeau (-type) algebras in the range between the diffeomorphism invariant algebra introduced in Part I and Colombeau's original algebra. Along the way, it provides several…

A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie…

The gauge-invariance properties and singularity elimination of the modified perturbation theory for QCD introduced in previous works, are investigated. The construction of the modified free propagators is generalized to include the…

Chapter 4 of my book [1] is dedicated to Lie-B\"acklund group analysis of various types of differential equations. The Russian edition of this book had an Addendum (pages 262-266) containing a summary of new results (obtained by the end of…

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat…

We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criteria for the…

With respect to earlier investigations, the theory of multi-component, concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and extended, including subsystems with nonzero density on the boundary and subsystems with…

Assuming that the differential field $(K,\delta)$ is differentially large, in the sense of Le\'on S\'anchez and Tressl, and "bounded" as a field, we prove that for any linear differential algebraic group $G$ over $K$, the differential…

We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a non-linear generalization…

In this paper, we proved P(n,3), which is an important part of the DDVV conjecture. The general case will be treated in the next version of the paper.

The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when…

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to…