Here are the titles and abstracts of the seminars which were held in 2015:

Marcin Borkowski**On some fixed point theorem in ultrametric spaces**

We will discuss a theorem establishing the existence of a fixed point and a (generalized) sequence approximating this fixed point. The talk is based in a paper by S. Priess-Crampe and P. Ribenboim from 2013.

Mateusz Krukowski**Darbo type theorem for quasi-measures of non-compactness.**

This presentation introduces the notion of a quasi-measure of non-compactness. Such functions need not to be invariant in a convex hull in contrast to classical measures of non-compactness. Using the compactness criterion in the space ( are Banach spaces with ), we construct a quasi-measure of non-compactness and study its properties. Furthermore we prove the analog of the Darbo theorem using the measure of non-convexity. We also show the application of the shown tools to finding a fixed point of the Hammerstein operator.

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-11-17 Nonlinear Seminar)

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-10-20, 2015-11-03 and 2015-11-10 Nonlinear Seminar.)

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-10-20 and 2015-10-27 Nonlinear Seminar)

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

(Continuation of the 2015-10-20 Nonlinear Seminar)

Elżbieta Gajecka**Resampling methods in the time domain for time series with periodic and almost periodic structure.**

(The seminar will be held in room B3-38)

Statistical reasoning, in the case of time series, based on asymptotic distributions cannot always be a basis for effective statistical procedures. However unknown distributions of estimators or statistics can be approximated by the so-called resampling procedures. The idea of resampling methods is obtaining replications of the estimator and calculating the empirical distribution from those replications. The main question, to be answered, is whether this empirical distribution, called the resampling distribution, is close to the real distribution.

Intense research is being conducted towards resampling methods in nonstationary time series, in particular series with periodic and almost periodic structure. During this seminar various methods of resampling, especially subsampling, will be shown. The advantage of subsampling is its insensitivity to the shape of the asymptotic distribution. We will also show conditions of compatibility of resampling methods in the domain of time for -mixing or weakly dependent time series with periodic or almost periodic structure. Especially weak dependency gives new tools to analyse statistical procedures for very general data generating time series, together with periodic series with long memory and heavy tails. An example of a periodic model with heavy tails and long memory will be shown.

Paweł Kliber**Consumer choices using hyperbolic discounting - the model and Euler conditions.**

This presentation concerns an article in which the problem of choosing the consumption path for a consumer using hyperbolic discounting (he treats the present and the whole future differently) in a model with random rate of return is discussed. The model, along with assumptions and the most important properties of the solution are presented.

Piotr Kasprzak**Abel's integral equation**

(Continuation of the 2015-10-06 Nonlinear Seminar.)

Piotr Kasprzak**Abel's integral equation**

The aim of the talk is to discuss mathematical background for the theory of Abel's integral equation and to present some of its real world applications.

Adam Nawrocki**Doubly periodic functions**

(Continuation of the 2015-05-05 and 2015-04-28 Nonlinear Seminar)

Prof. Xiao-Xiong Gan

(Mathematics Department, Morgan State University, Baltimore, USA)**Some Remarks on Minkowski's Inequality**

The classical Minkowski's inequality has two different forms based on the values of the positive number :

- If , then ;
- If , then .

We provide a one-form Minkowski's inequality for all . A non-conjugate Hölder's inequality is also introduced. Some applications of the one-form Minkowski's inequality and non-conjugate Hölder's inequality are introduced too.

**The lecture will be delivered in English**

Patryk Kołacki**Sliding mode control in automatic control systems**

(Continuation of the 2015-05-12 Nonlinear Seminar)

Patryk Kołacki**Sliding mode control in automatic control systems**

In this seminar, I'm going to present you on the most popular control laws, i.e. sliding mode control. There will be discussed it's basic, canonical form and it's most popular applications. I also want to show a few examples which let you understand the idea of this control law.

Adam Nawrocki**Doubly periodic functions**

(Continuation of the 2015-04-28 Nonlinear Seminar)

Adam Nawrocki**Doubly periodic functions**

During this seminar we shall discuss the notion of doubly periodic functions and almost periodic functions in the context of elliptic functions.

dr Przemysław Chełminiak**What connects a biological cell with a partial diffusion equation.**

Tracking individual molecules in living biological cells has become possible today by a new imaging technique called single-molecule spectroscopy. Interestingly, a diffusion of such tiny nano-molecules as proteins, nucleic acids or semiflexible polymers within a dense cellular environment is slower than ordinary Brownian diffusion of the micro-particles suspended in water. Their spatial development inside the cell, characterized by the time or ensemble averaged mean squared displacements, does not grow linearly in time. Moreover, these two averaging procedures are not equivalent when the measurement time is long compared to the characteristic time scale of the diffusion process. Therefore, the erratic motion of the subcellular molecules can not be perceived as a typical Brownian motion, but it must be considered in terms of the anomalous diffusion (subdiffusion). From a formal point of view, the anomalous diffusion can be described by a widely applicable stochastic process known as the continuous-time random walk model. We apply this method to construct the so called fractional Fokker-Planck equation which is an extended version of the subdiffusion equation in an external force field and close to thermal equilibrium. Using the methods of fractional calculus we find a few special solutions of this fractional partial differential equation and show that the mean squared displacement of a subdiffusing particle scales with time according to the power-law.

dr Giselle Monteiro**New convergence theorem for the abstract Kurzweil-Stieltjes integral**

In theory of Riemann integral, the impact of the Bounded Convergence Theorem, also called Arzela or Arzela-Osgood or Osgood Theorem, is comparable to the importance of the Lebesgue Dominated convergence Theorem in the theory of the Lebesgue integration. In this talk we are concerned with the abstract Kurzweil-Stieltjes integral, that is, the Stieltjes type integral for functions with values in a Banach space introduced by S. Schwabik. Our aim is to present the Bounded Convergence Theorem in this general setting.

Piotr Maćkowiak**A combinatorial lemma ant it's consequences**

(Continuation of the 2015-03-17 Nonlinear Seminar)

Marcin Borkowski**On a generalisation of a metric space**

(Continuation of the 2015-03-10 Nonlinear Seminar)

Piotr Maćkowiak**A combinatorial lemma ant it's consequences**

During this seminar we shall introduce a combinatorial lemma stating the existence for particular triangulations of the -dimensional simplex with a particular labeling of vertices of such triangulations with numbers of a chain of simplices connecting a "vartex" with a "base" such that for any simplex from that chain each element of the set is a label of some vertex of that simplex. We shall show, that (simple) consequences of this lemma are:

- the existence of zeros of the multidimensional function of excess demand
- the existence of continuum zeros of the parametrised multidimensional function of excess demand
- the well-known Sperner lemma

Marcin Borkowski**On a generalisation of a metric space**

During this seminar we shall give the axioms of a , which generalises the notion of metric spaces. We shall give examples of and prove some of their properties.

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

(Continuation of the 2015-01-13, 2015-01-20 and 2015-01-27 Nonlinear Seminar)

dr Jacek Gulgowski**Compactness in spaces of functions of bounded variation.**

The studies on the existence of fixed points of mappings in Banach spaces often involve methods connected with the Schauder fixed point theorem or the Leray-Schauder degree. Both approaches require the considered mappings to be completely continuous, and therefore, if one wants to work in spaces of functions of bounded variation, it is crucial to understand the mutual relations between and spaces. During the seminar we are going to discuss certain results on compact embeddings of such spaces and characterise the complete continuity of some nonlinear integral operators acting in spaces.

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

(Continuation of the 2015-01-13 and 2015-01-20 Nonlinear Seminar.)

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

(Continuation of the 2015-01-13 Nonlinear Seminar.)

Jędrzej Sadowski**Quasicrystals and almost periodic patterns**

During this seminar we shall introduce the definition of almost periodic patterns, their basic properties and connection to quasicrystals.