In this thesis we present the formalization of three principal results that are the Jordan normal form of a matrices, the Bolzano-Weierstraß theorem, and the Perron-Frobenius theorem. To formalize the Jordan normal form, we introduce many concepts of linear algebra like block diagonal matrices, companion matrices, invariant factors, ... The formalization of Bolzano-Weierstraß theorem needs to d...

We establish a stochastic nonlinear analogue of the PerronFrobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space (Ω,F , P ) and a random mapping D(ω, ·) : R+ → R+. Under assumptions of monotonicity and homogeneity of D(ω, ·), we prove the existence of scalar and vector measurable functions α(ω) > 0 an...

Let K be a closed convex cone with dual K∗ in a finite-dimensional real Hilbert space V . A positive operator on K is a linear operator L on V such that L (K) ⊆ K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. We say that L is a Z-operator on K if 〈L (x), s〉 ≤ 0 for all (x, s) ∈ K ×K such that 〈x, s〉 = 0. The Z-operators are generalizat...

The paper presents basic concepts of a new type of algorithm for the numerical computation of what the authors call the essential dynamics of molecular systems. Mathematically speaking, such systems are described by Hamiltonian diierential equations. In the bulk of applications, individual trajectories are of no speciic interest. Rather, time averages of physical observables or relaxation times...

We study ergodicity of bounded, sub-additive and non-negatively homogeneous maps on finite dimensional spaces which we call upper transition operators. We show that ergodicity coincides with the necessary and sufficient condition for a generalised Perron-Frobenius theorem for upper transition operators. We show that ergodicity is equivalent with regular absorningness of the upper transition ope...

In this survey we deal with transcendental meromorphic and entire functions. We thoroughly discuss in context close relations between topological pressure conformal measures for geometric potentials. For two more special classes of functions, namely dynamically semi-regular those the class \(\mathcal {D}\) that have negative spectrum, then also discuss, by means appropriate transfer (or Perron-...

The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the essential dynamics of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times ...