Related papers: On a List of Ordinary Differential Equations Probl…
In this work we discuss a Hamiltonian system of ordinary differential equations under Dirichlet boundary conditions. The system of equations in consideration features a mixed (concave-convex) power nonlinearity depending on a positive…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
These are lecture notes for the AGRA II school, which took place in August 2015 at Universidad de San Antonio Abad del Cusco (Per\'u). They are geared towards graduate students and young researchers. I. Modular forms and Shimura curves (R.…
This is a colloquium style pedagogical introduction to the paradigm of large extra dimensions. To be published in the Proceedings of the Workshop "Crossing the boundaries: Gauge dynamics at strong coupling," (May 14 - 17, 2009,…
The Monte Carlo method is a thriving and mathematically beautiful numerical technique used extensively, nowadays, to deal with many demanding problems in diverse fields. Here, we present an iterative Monte Carlo algorithm to work out very…
This celebratory article contains a personal and idiosyncratic selection of a few open problems in discrete probability theory. These include certain well known questions concerning Lorentz scatterers and self-avoiding walks, and also some…
Relations between differential calculi, quantum groups, integrable systems, and q-analysis are studied. Some new Hirota type formulas are established for qKP along with variations on classical Hirota formulas.
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_\lambda$} -\mathrm{div}(A(x)Du)=c_\lambda(x)u+( M(x)Du,Du)+h(x),\qquad u\in…
In the present paper, we give some examples of stochastic differential equations which have delicateness in the Markov and strong Markov properties, the uniqueness locally in time and globally in time, and initial conditions. Moreover, we…
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as…
These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and…
We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second order equation in divergence form with discontinuous coefficient. Our concern is to estimate the solutions with explicit constants,…
Some interesting (periodic!) solutions of certain systems of $4$ nonlinear Ordinary Differential Equations $dx_{n}\left( t\right) /dt=P_{2}^{\left( n\right) }\left[ x_{m}\left( t\right) \right] /\left[ x_{1}\left( t\right) +x_{2}\left(…
The present overview and gentle introduction to Grassmannian calculus and some of its applications to probability collects the notes of a mini-course given by the authors at the Brazilian School of Probability, August 5-9, 2024, in…
We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained…
In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [13].
We consider the problem of bounding the dimension of the linear system of curves in ${\bf P}^2$ of degree $d$ with prescribed multiplicities $m_1,...,m_n$ at $n$ general points (\cite{Hir1},\cite{Hir2}). We propose a new method, based on…
Personal recollections on theoretical particle physics in the years when the Standard Theory was formed. In the background, the remarkable development of Italian theoretical physics in the second part of the last century, with great…
The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years during my research. I believe that it is worth to think about them and, hopefully, it will be possible either to solve…