Related papers: Theoretical and Numerical Study of Self-Organizing…
We perform a perturbative calculation of the physical observables, in particular pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator, and the classical Hamiltonian for the PT-symmetric cubic…
The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to…
We study the predictability of large events in self-organizing systems. We focus on a set of models which have been studied as analogs of earthquake faults and fault systems, and apply methods based on techniques which are of current…
Generic open quantum systems are notoriously difficult to simulate unless one looks at specific regimes. In contrast, classical dissipative systems can often be effectively described by stochastic processes, which are generally less…
The conventional Langevin equation offers a mathematically convenient framework for investigating open stochastic systems interacting with their environment or a bath. However, it is not suitable for a wide variety of systems whose dynamics…
We study a quantum oscillator interacting and back-reacting on a classical oscillator. This can be done consistently provided the quantum system decoheres, while the backreaction has a stochastic component which causes the classical system…
We consider the Langevin dynamics of a many-body system of interacting particles in $d$ dimensions, in a very general setting suitable to model several out-of-equilibrium situations, such as liquid and glass rheology, active self-propelled…
A fast algorithm to study one-dimensional self-gravitating systems, and, more generally, systems that are Lagrangian integrable between collisions, is presented. The algorithm is event-driven, and uses a heap-ordered set of predicted future…
In this paper, we study two kinds of singular optimal controls (SOCs for short) problems where the systems governed by forward-backward stochastic differential equations (FBSDEs for short), in which the control has two components: the…
In Bender and Dokuchaev (2013), we studied a control problem related to swing option pricing in a general non-Markovian setting. The main result there shows that the value process of this control problem can be uniquely characterized in…
We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an…
An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian…
We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a…
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators…
Dense Associative Memory (DAM) models generalize the classical Hopfield model by incorporating n-body or exponential interactions that greatly enhance storage capacity. While the criticality of DAM models has been largely investigated,…
Environmentally-induced superselection or "einselection" has been proposed as an observer-independent mechanism by which apparently classical systems "emerge" from physical interactions between degrees of freedom described completely…
The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et…
The question how complex systems become more organized and efficient with time is open. Examples are, the formation of elementary particles from pure energy, the formation of atoms from particles, the formation of stars and galaxies, the…
From tumour invasion to cell sorting and animal territoriality, many biological systems rely on nonlocal interactions that drive complex spatial organisation. Partial differential equations (PDEs) with nonlocal advection are increasingly…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…