Polynomial equations for additive functions I

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered $$\sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x^{q_{i}})= 0 \qquad \left(x\in \mathbb{F}\right),$$ where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_{i}, g_{i}\colon \mathbb{F}\to \mathbb{C}$ are additive functions and $p_i, q_i$ are positive integers for all $i=1, \ldots, n$.Comment: 28 page

Differential equation approximations of stochastic network processes: an operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE

A kötelező gépjármű-felelősségbiztosítás rövid története és alapvető jellemzői

Insurance against civil liability in respect of the use of motor vehicles (motor insurance) is ofspecial importance for European citizens, whether they are policyholders or victims of anaccident. It is also a major concern for insurance undertakings as it constitutes an importantpart of non-life insurance business in the Community. Motor insurance also has an impact onthe free movement of persons and vehicles. The first part of the study deals with the originand the brief history of insurance, beginning with the ancient Rome. The second part tries todescribe the main characteristic features of the motor insurance including its legalbackground – with respect to the directives of the European Parliament and the Council, andthe Hungarian legislation –, its subjects, the conclusion and termination of the contract.Insurance against civil liability in respect of the use of motor vehicles (motor insurance) is ofspecial importance for European citizens, whether they are policyholders or victims of anaccident. It is also a major concern for insurance undertakings as it constitutes an importantpart of non-life insurance business in the Community. Motor insurance also has an impact onthe free movement of persons and vehicles. The first part of the study deals with the originand the brief history of insurance, beginning with the ancient Rome. The second part tries todescribe the main characteristic features of the motor insurance including its legalbackground – with respect to the directives of the European Parliament and the Council, andthe Hungarian legislation –, its subjects, the conclusion and termination of the contract