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We study a class of systems of quasilinear differential inequalities associated to weakly coercive differential operators and power reaction terms. The main model cases are given by the $p$-Laplacian operator as well as the mean curvature…
A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator) $H_{q}$ relevant to the fluctuation spectrum and poles of the order-$q$ power spectrum is proposed. The ``partition…
In the paper we consider the linear inverse problem that consists in recovering the initial state in a first order evolution equation generated by a skew-adjoint operator. We studied the well-posedness of the inversion in terms of the…
This paper deals with some nonlinear problems which exponential and biexponential decays are involved in. A proof of the quasiconvexity of the error function in some of these problems of optimization is presented. This proof is restricted…
The evolution of any factorized time-reversible symplectic integrators, when applied to the harmonic oscillator, can be exactly solved in a closed form. The resulting modified Hamiltonians demonstrate the convergence of the Lie series…
We consider an infinite planar straight strip perforated by small holes along a curve. In such domain, we consider a general second order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation…
Properties of compositions and convex combinations of averaged nonexpansive operators are investigated and applied to the design of new fixed point algorithms in Hilbert spaces. An extended version of the forward-backward splitting…
In this paper, we introduce the notion of a characteristic operator for closable linear operators and explore their connected spectral properties via equivalence. Additionally, we develop an explicit scheme for constructing characteristic…
We propose a simple proof of the exponential convergence to equilibrium for ultrafast diffusion equations in $\mathbb{R}^n$. Our approach, based on the direct use of Poincar\'e inequality, gets rid of the optimal transport arguments used in…
We propose an approach to obtaining explicit estimates on the resolvent of hypocoercive operators by using Schur complements, rather than from an exponential decay of the evolution semigroup combined with a time integral. We present…
This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class…
We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a…
We study higher-order small-noise fluctuation expansions for the overdamped Langevin dynamics in a quartic double-well potential. Assuming that the initial data admits a suitable expansion structure, we obtain a strong dynamical expansion…
The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators…
Before we proposed an algebraic technics for the Hamiltonian approach to the evolution systems of partial differential equations, including systems with constraints. Here we further develop this approach and present the defining system of…
In a recent paper, Ling et al. investigated the over-parametrized Deep Equilibrium Model (DEQ) with ReLU activation. They proved that the gradient descent converges to a globally optimal solution at a linear convergence rate for the…
This paper introduces a generalization of the well-known Riccati recursion for solving the discrete-time equality-constrained linear quadratic optimal control problem. The recursion can be used to compute the solutions as well as optimal…
In this paper we aim to generalize results obtained in the framework of fractional calculus by the way of reformulating them in terms of operator theory. In its own turn, the achieved generalization allows us to spread the obtained…
Explicit diagrammatic calculation of evolution equations for high-twist correlation functions is a challenge already at one-loop order in QCD coupling. The main complication being quite involved mixing pattern of the so-called…
For an open quantum system described by the Lindblad equation, full characterization of its dynamics typically needs the knowledge of the Liouvillian spectrum and correlation functions. Solving the Liouvillian spectrum and correlation…