We use functional calculus methods to investigate qualitative properties of C0semigroups that are preserved by time-discretization methods. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. Preservation of contractivity (or norm-bound) of the semigroup is investigated in the Banach space ...

The paper presents a new result for the existence and uniqueness of solutions of the system of nonlinear algebraic equations obtained from the application of the BDF methods to stii initial value problems in ordinary diierential equations. Expressed in terms of the local truncation error, starting errors in the BDF and contractivity properties of the diierential equation, the result is used to ...

We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalized convergence proofs along explicit rates convergence. More specifically, we define new notion being ψ-weakly modulus, and present series abstract theorems which both generalize unify known results from the literature. Rates are fo...

Discretizations of the Langevin diffusion have been proven very useful for developing and analyzing algorithms sampling stochastic optimization. We present an improved non-asymptotic analysis Euler-Maruyama discretization diffusion. Our does not require global contractivity, yields polynomial dependence on time horizon. Compared to existing approaches, we make additional smoothness assumption, ...

In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shad light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative ...

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Abstract. We introduce and study a notion of Orlicz hypercontrac-tive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, ...

We consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different, possibly non-adjacent edges. This generalizes the setting that is common in the literature, where the only considered interactions take place at the boundary, i. e., in the nodes of the network. We discuss well-posedness of the associated initial value probl...

Quantum correlations in a composite system can be measured by resorting to a geometric approach, according to which the distance from the state of the system to a suitable set of classically correlated states is considered. Here we show that all distance functions, which respect natural assumptions of invariance under transposition, convexity, and contractivity under quantum channels, give rise...