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Our aim in this paper is twofold. Firstly, we develop a new asymptotic theory for Floquet exponents. We consider a linear system of differential equations with a time-periodic coefficient matrix. Assuming that the coefficient matrix depends…
Consider the differential equation ${ m\ddot{x} +\gamma \dot{x} -x\epsilon \cos(\omega t) =0}$, $0 \leq t \leq T$. The form of the fundamental set of solutions are determined by Floquet theory. In the limit as $m \to 0$ we can apply WKB…
Floquet's Theorem is a celebrated result in the theory of ordinary differential equations. Essentially, the theorem states that, when studying a linear differential system with $T$-periodic coefficients, we can apply a, possibly complex,…
Continual learning in artificial neural networks is fundamentally limited by the stability--plasticity dilemma: systems that retain prior knowledge tend to resist acquiring new knowledge, and vice versa. Existing approaches, most notably…
Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves this is analogous to a…
Multi-Agent Reinforcement Learning involves agents that learn together in a shared environment, leading to emergent dynamics sensitive to initial conditions and parameter variations. A Dynamical Systems approach, which studies the evolution…
The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised,…
In the Determinant Maximization problem, given an $n\times n$ positive semi-definite matrix $\bf{A}$ in $\mathbb{Q}^{n\times n}$ and an integer $k$, we are required to find a $k\times k$ principal submatrix of $\bf{A}$ having the maximum…
Nonequilibrium topological matter has been a fruitful topic of both theoretical and experimental interest. A great variety of exotic topological phases unavailable in static systems may emerge under nonequilibrium situations, often…
Discrete dynamics arise naturally in systems with broken temporal translation symmetry and are typically described by first-order recurrence relations representing classical or quantum Markov chains. When memory effects induced by hidden…
We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite…
In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are…
We consider a misspecified optimization problem that requires minimizing a function f(x;q*) over a closed and convex set X where q* is an unknown vector of parameters that may be learnt by a parallel learning process. In this context, We…
We address the problem of policy evaluation in discounted Markov decision processes, and provide instance-dependent guarantees on the $\ell_\infty$-error under a generative model. We establish both asymptotic and non-asymptotic versions of…
This is the continuation of previous article. For subspaces $M^n(t)$ and $M^{n-m}(t)$ which are invariant manifolds of the differential equation under consideration we build a change of variables which splits this equation into a system of…
Recent work [4] analyses the local convergence of Adam in a neighbourhood of an optimal solution for a twice-differentiable function. It is found that the learning rate has to be sufficiently small to ensure local stability of the optimal…
Characterizing time-periodic Hamiltonians is pivotal for validating and controlling driven quantum platforms, yet prevailing and unadjusted reconstruction methods demand dense time-domain sampling and heavy post-processing. We introduce a…
Adaptive optimization algorithms, particularly Adam and its variant AdamW, are fundamental components of modern deep learning. However, their training dynamics lack comprehensive theoretical understanding, with limited insight into why…
In this paper we study a generalized class of Maxwell-Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the…
Q-learning has long been one of the most popular reinforcement learning algorithms, and theoretical analysis of Q-learning has been an active research topic for decades. Although researches on asymptotic convergence analysis of Q-learning…