Here are the titles and abstracts of the seminars which were held in 2013:

Marcin Karczewski**Boltzmann kinetic equation**

(Continuation of the 2013-12-10 Nonlinear Seminar)

Marcin Karczewski**Boltzmann kinetic equation**

The aim of the seminar is to discuss the Boltzmann kinetic equation and its physical consequences. After gaining intuition we shall return to the earlier discussed VHP gas model.

Piotr Kasprzak**Application of formal power series to differential equations**

During the seminar we are going to discuss the Frobenius Method, that is, we are going to seek solutions to certain differential equations (not necessarily of integer-order) in the form of formal power (or Laurent) series. Several illustrative examples will be provided.

dr Aneta Sikorska-Nowak**Asymptotic stability of the integro-differential equation with delay**

(Continuation of the 2013-11-19 Nonlinear Seminar)

dr Aneta Sikorska-Nowak**Asymptotic stability of the integro-differential equation with delay**

Certainly, the Lyapunov direct method has been, for more than 100 years, the main tool for the study of stability properties of ordinary, functional, integro-differential and partial differential equations. Nevertheless, the applications of this method to problem of stability in differential equations with delay has encountered serious difficulties if the delay is unbounded or if the equation has unbounded terms. The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov’s direct method.

I will present the results on the asymptotic stability of solutions of nonlinear integro-differential equations with delay using Sadovskii Fixed Point Theorem and properties of the measure of noncompactness.

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

(Continuation of the 2013-10-29 Nonlinear Seminar)

Monika Naskręcka**Mathematical model of a competitive market with stocks.**

Equilibrium and stability are the most frequently investigated properties in the market equilibrium models. On global and local markets we usually observe the lack of equilibrium. Nevertheless, the original model does not consider the possibility of stocks. However, we observe this situation, when there exists an excess supply. We should consider then how the accumulated stocks affect the equilibrium (or disequilibrium) and the stability of the model. During the seminar I will present the competitive economy model and the model with stocks. I will explore how these stocks influence the path of the price trajectory, equilibrium conditions and price stability.

Piotr Kasprzak**Application of formal power series to differential equations**

(Continuation of the 2013-05-07 Nonlinear Seminar)

Piotr Kasprzak**Application of formal power series to differential equations**

During the seminar we are going to discuss the Frobenius Method, that is, we are going to seek solutions to certain differential equations (not necessarily of integer-order) in the form of formal power (or Laurent) series. Several illustrative examples will be provided.

Dr. hab. Aleksander Ćwiszewski**TITLE soon**

Prof. dr hab. Bronisław Jakubczyk (Institute of Mathematics, Polish Academy of Sciences)**Series of formally noncommutative variables versus functions and vector fields on manifolds**

Taylor coefficients of a function can be calculated as iterated derivatives along constant vector fields (versors), computed at a fixed point. If a constant vector fields is replaced by any fixed set of vector fields on a manifold, a similar operation leads to the definition of a formal series with noncommutative variables.

We will show some interesting facts connected with such series, in particular the so-called *realization theorem* and its consequences. In particular, we will prove that a controlled and observed dynamical system on a manifold can be represented by such a series. Another consequence is a new test of convergence of a series of noncommutative variables.

Jędrzej Sadowski**About integrate-and-fire models**

We will talk about two integrate-and-fire models.

The first one will be the neon bulb circuit, the second a japanese water toy.

Piotr Zdanowicz**Basic properties of formal series**

During the seminar we will discuss and prove some properties of formal series.

(Continuation of the 2012-11-20 and 2012-11-27 Nonlinear Seminar)

Marcin Borkowski**Some properties of the Henstock-Kurzweil Integral**

(Continuation of the 2013-01-22 and 2013-01-15 Nonlinear Seminar)

Marcin Borkowski**Some properties of the Henstock-Kurzweil Integral**

(Continuation of the 2013-01-15 Nonlinear Seminar)

Marcin Borkowski**Some properties of the Henstock-Kurzweil Integral**

During this seminar we will discuss some properties of the Henstock-Kurzweil Integral for an unbounded domain. In particular we shall prove that the functions of significantly bounded variation are multipliers of HK-integrable functions.

Monika Naskręcka**Homeomorphism of the space of polynomials and the space of their roots**

(Continuation of the 2012-12-11 and 2012-12-18 Nonlinear Seminar)