# Seminar on Nonlinear Analysis and its Applications: SONATA

**Topics discussed**

The aim of the seminar is to provide a forum for scientific debate on a broad range of issues in the field of nonlinear analysis and its applications. Particular attention will be paid to topological methods applying in nonlinear analisys as well as to applications of nonlinear analysis (with particular emphasis on nonlinear differential and integral equations) to physics, finance and economics, and other fields. The survey lectures are going to be addressed to a wide audience of researchers and Ph.D. students. We are planing to invite guests from various Polish academic institutions in order to get familiarized with recent developments in nonlinear analysis and its applications, which attract the attention of other research groups in Poland.

**Seminar dates**

The seminar is to be held once a month (on Friday) usually at 11:00am in room B1-37, of Collegium Mathematicum at Uniwersytetu Poznańskiego 4 in Poznań. Information on the exact location and date of the seminar will be announced by the Secretary in due course.

**22-04-2022 | Prof. J****ü**rgen Appell: *How to measure the noncompactness of operators*

**ü**rgen Appell:

*How to measure the noncompactness of operators*

The noncompactness of a linear operator may be measured by its essential norm, of a nonlinear operator by its measure of noncompactness. We illustrate this by means of two linear operators, viz. multiplication and substitution operators, in the space C of continuous functions, and in the space BV of functions of bounded variation. The main emphasis is put on examples and counterexamples. This is joint work with Simon Reinwand (Wurzburg), Laura Angeloni and Gianluca Vinti (Perugia, Italy), and Tomas Dominguez (Sevilla, Spain).

**22-04-2022 | Dr. Simon Reinwand: ***A brief introduction to the SIR model for pandemics, with data from Germany*

*A brief introduction to the SIR model for pandemics, with data from Germany*

The SIR model is one of the most basic models describing the spread and development of pandemics. We give a very brief introduction to this model, which assumptions are needed to get going and how one may derive the relevant system of ordinary differential equations. Moreover, we show how to estimate not only solutions but also the necessary parameters using real data from Germany measuring the beginning of the Corona pandemic in 2020. We also show how to derive some extremal values like the maximal number of simultaneously infected as well as the total number of infected during the entire pandemic. Finally, we point out the reason why politics have made the decisions they made and also the relevance of the terms ”basic reproduction number” and ”exponential growth” that have been very popular in the media.

**20-04-2020 | prof. PP dr hab. Jacek Żak: ***Multiple criteria decision making/aiding – methodological (mathematical) backgrond and practical application in transportation and logistics*

*Multiple criteria decision making/aiding – methodological (mathematical) backgrond and practical application in transportation and logistics*

The speech is an introduction to the Methodology of Multiple Criteria Decision Making/Aiding (MCDM/A). It presents the application of MCDM/A in transportation and logistics. Historical background and motivation for applying MCDM/A are presented. Classification and major features of multiple criteria decision problems are discussed. The multiple criteria decision making/aiding process, including its major phases and players, is demonstrated. Basic notions and principles of MCDM/A methodology (e.g. Pareto optimal solutions, Ideal and Nadir points, Pay-off Matrix, compromise solution, model of preferences, etc.) are defined. Classification and major features of MCDM/A methods are shown.

Several real life case studies are presented, to show the practical applicability of the presented methodology. The case studies cover 2-3 (time permitting) of the following problems:

- Multiple objective evaluation and ranking of the common carriers – transportation/logistics service providers.
- Multiple criteria analysis for the Football Fun Zone location problem.
- Redesign of the distribution system. The comparison of single and bi-criteria optimization
- MCDM/A – based redesign of the complex business process (e.g.: order fulfillment and product delivery).
- Multiple criteria selection of suppliers in different industries.
- Design and multiple criteria evaluation of global/ international logistics corridors.

The results of computational experiments are demonstrated.

*The seminar takes place at 11:00 in room B1-37.*

**02-03-2018 | prof. dr hab. inż. Jacek Jachymski: ***On three extensions of Cantor’s intersection theorem and their applications in metric fixed point theory*

*On three extensions of Cantor’s intersection theorem and their applications in metric fixed point theory*

In the talk there will be presented three extensions of Cantor’s intersection theorem in which, successively, there will be weakened the assumptions: the closedness of sets, the condition that a sequence of sets is descending and the convergence of diameters of sets to zero. The first result let us give alternative proofs of some fixed point theorems for discontinuous mappings. With the help of the second one it is possible to simplify significantly the proof of the Reich-Zaslavski fixed point theorem for nonself mappings. The third theorem concerns a sequence of subsets of the so-called superreflexive Banach space and implies the Browder-Göohde-Kirk theorem for nonexpansive mappings. The lecture is available for a wide group of listeners.

References:

- D. W. Boyd, J. S. W. Wong,
*Another proof of the contraction mapping principle*, Canad. Math. Bull.**11**(1968), 605-606. - F. E. Browder,
*Nonexpansive nonlinear operators in a Banach space*, Proc. Nat. Acad. Sci. U.S.A.**54**(1965), 1041-1044. - J. Jachymski,
*Around Browder’s fixed point theorem for contractions*, J. Fixed Point Theory Appl.**5**(2009), 47-61. - J. Jachymski,
*A Cantor type intersection theorem for superreflexive Banach spaces and fixed points of almost affine mappings*, J. Nonlinear Convex Anal.**16**(2015), 1055-1068. - S. Reich, A. J. Zaslavski,
*Convergence of iterates for a class of mappings of contractive type*, JP J. Fixed Point Theory Appl.**2**(2007), 69-78.

*The seminar takes place at 11:00 in room B1-37.*

**02-02-2018 | Prof. Dr. rer. nat. habil. Gerhard-Wilhelm Weber (współautorzy: Emel Savku, Ioannis Baltas, Emre Akdogan):** **Stochastic optimal control under a Markov regime-switching jump-diffusion model with delay in finance, economics and neuroscience**

**Stochastic optimal control under a Markov regime-switching jump-diffusion model with delay in finance, economics and neuroscience**

We contribute to modern Operational Research by hybrid, e.g., mixed continuous-discrete dynamics of stochastic differential equations with jumps and to its optimal control. These hybrid systems allow for the representation of random regime switches or paradigm shifts, and are of growing importance in science, especially, in biology, economics, finance and engineering. We introduce some new approaches to this area of stochastic optimal control and present results. One is analytical and bases on the finding of optimality conditions and, in certain cases, closed-form solutions.We further discuss aspects of differences in information, given by delay – a first form of memory – or partial information, and we give a short introduction into the involvement of the “human factor”, especially, through neuroscience and behavioral economics. The presentation ends with a conclusion and an outlook to future studies.

For the presentation time being limited, not all technical details can be given. Instead we refer the interested reader to the references stated subsequently.

References:

- N. Azevedo, D. Pinheiro and G.-W. Weber,
*Dynamic programming for a Markov–switching jump-diffusion*, Journal of Computational and Applied Mathematics**267**(2014) 1–19. - E. Savku and G.-W. Weber,
*A stochastic maximum principle for a Markov regime-switching jump-diffusion model with delay and an application to finance*, Journal of Optimization Theory and Applications, to appear. - B. Z. Temocin and G.-W. Weber,
*Optimal control of stochastic hybrid system with jumps: a numerical approximation*, Journal of Computational and Applied Mathematics**259**(2014) 443-451.

*The seminar takes place at 11:00 in room B1-37.*

**17-11-2017 | dr hab. Dobrosława Kasprowicz: ***Luminescence crystals*

*Luminescence crystals*

In the last few years a significant interest has been motivated in development of the novel rare earth doped crystals based upon infrared to visible frequency up-conversion with potential applications in various optoelectronic devices, e.g. visible up-conversion lasers, high density memories, 3D displays [1], optical amplifiers for fiber-optic communication [2], fluorescent lamps [3] or white light emission materials [4].

During the lecture the spectroscopic properties of \(KGd(WO_4)_2\) crystals doped with selected \(Pr^{3+}\), \(TM^{3+}\), \(Tb^{3+}\), \(Ho^{3+}\), \(ER^{3+}\) and/or \(YB^{3+}\) ions will be presented. It was shown that all investigated systems exhibited multicolor up-conversion fluorescence under 980 nm laser irradiation [5]. The investigated materials are very promising as a new generation energy converters with significant potential applications in novel optical devices.

References:

- G. Lakshminarayana, H. Yang, J. Qiu,
*White light emission from \(Tm^{3+}\)*/*\(Dy^{3+}\) co-doped oxyfluoride germanate glasses under UV light excitation*, J. Solid State Chem.**182**(2009), 669-676. - K. Ouannes, M. T. Soltani, M. Poulain, G. Boulon, G. Alombert-Goget, Y. Guyot, A. Pillonnet, K. Lebbout,
*Spectroscopic properties of \(Er^{3+}\)-doped antimony oxide glass*, J. Alloys. Compd.**603**(2014), 132–135. - X. Min, M. Fang, Z. Huang, Y. Liu, C. Tang, X. Wu,
*Luminescent properties of white-light-emitting phosphor \(LaMgAl_{11}O_{19}:Dy^{3+}\)*, Mater. Lett.**125**(2014), 140-142. - J. H. Chung, S. Y. Lee, K. B. Shim, J. H. Ryu,
*White lighting upconversion in \(Tm^{3+}\)*/*\(Ho^{3+}\)*/*\(Yb^{3+}\) co-doped \(CaWO_4\)*, Appl. Phys. Express**5**(2012), 052602. - D. Kasprowicz, P. Głuchowski, B.M. Maciejewska, M. Chrunik, A. Majchrowski,
*Up-conversion luminescence of rare earth-doped \(KGd(WO_4)_2\) phosphors for tunable multicolour light generation*, New J. Chem.**41**(2017), 9847–9856.

*The seminar takes place at 11:00 in room B1-37*.

**17-11-2017 **| d**r hab. Tomasz Runka: ****From crystal structure to Raman spectrum**

**From crystal structure to Raman spectrum**

During the lecture the following issues will be discussed: crystal structure of bodies, crystal systems, Bravais lattices, point and space groups, crystals of perovskite structure, group theory in spectroscopic applications, Raman scattering.

On the example of two-component \(SAT_{1-x}:LA_x\) crystals of perovskite structure, the spectroscopic characterization of these materials in vibrational range will be presented. It will also be presented the correlation between group theory calculations and experimental technique such as Raman spectroscopy.

References:

- Ch. Kittel,
*Wstęp do fizyki ciała stałego*, Wydawnictwo Naukowe PWN, Warszawa 2011. - G. Turrell,
*Infrared and Raman spectra of crystals*, Academic Press, London, 1972. - D.I. Rousseau, R.P. Bauman, S.P.S. Porto,
*Normal mode determination in crystals*, J. Raman Spectrosc.**10**(1981), 253-290. - T. Runka, K. Łapsa, A. Łapiński, R. Aleksiyko, M. Berkowski, M. Drozdowski,
*Spectroscopic study of mixed oxide \(SAT_{1-x}:LA_x\) perovskite crystals*, J. Mol. Struct.**704**(2004), 281-285. - T. Runka, R. Aleksiyko, M. Berkowski, M. Drozdowski,
*Raman scattering study of \((SrAl_{0.5}Ta_{0.5}O_3)_{1-x-y}:(LaAlO_3)_x:(CaAl_{0.5}Ta_{0.5}O_3)_y\) solid solution crystals*, Cryst. Res. Technol.**40**(2005), 453-458.

*The seminar takes place at 11:00 in room B1-37*.

**10-03-2017 | dr Jarosław Mederski: ***Nonradial solutions of the Schrödinger equation with general*

*nonlinearity*

*Nonradial solutions of the Schrödinger equation with general*

*nonlinearity*

We study a nonlinear Schrödinger equation with a general nonlinearity satisfying Berestycki-Lions assumptions. A classical result states that there exist infinitely many radial solutions of the equation. We show that we can find infinitely many nonradial solutions even in case of the lack of compactness, e.g. when a Palais-Smale condition is not satisfied.

*The seminar takes place at 11:00 in room B1-37.*

**13-01-2017 | prof. dr hab. Krzysztof Chełmiński: ***Renormailized solutions in thermoplasticity*

*Renormailized solutions in thermoplasticity*

In this talk I will present recent results obtained in the theory of thermo-viscoelastic solids. Use is made throughout of the complete nonlinear heat conduction. For this problem, not much is known at present. In order to obtain mathematical results I introduce the audience to the concept of renormalized solutions.

*The seminar takes place at 11:00 in room B1-37.*

**09-12-2016 | dr Giselle Monteiro: ****Kurzweil-Stieltjes integral and some particular classes of functions**

**Kurzweil-Stieltjes integral and some particular classes of functions**

It is known that Kurzweil-Stieltjes integral generalizes the integrals of Riemann and Riemann-Stieltjes. With this in mind, when analyzing such a general integration theory, we can ask if it mimics some of the properties of the Riemann-Stieltjes integral with respect to functions. In this talk we provide a partial answer to this question.

**Acknowledgement**: Research financed by the SASPRO Programme, co-financed by the European Union and the Slovak Academy of Sciences.

*The seminar takes place at 11:00 in room B1-37.*

**28-10-2016 | prof. AGH dr hab. Piotr Oprocha: ***Hereditarily indecomposable continua and entropy*

*Hereditarily indecomposable continua and entropy*

One-dimensional hereditarily indecomposable continua are mathematical objects of complicated structure, and their discovery is a part of history of Polish mathematical school. It is known since 1980s that such continua may arise as attractors in discrete dynamical systems on surfaces. Since then some insight into the topic has been made, however many questions still remain unanswered.

The aim of this lecture is introduction into the above topic and survey of some more recent results.

*The seminar takes place at 11:00 in room B1-37.*

**06-05-2016 | prof. PG dr hab. Grzegorz Graff: ***Shub conjecture – how smoothness generates periodic points*

*Shub conjecture – how smoothness generates periodic points*

Shub conjecture was formulated in the 70s of the twentieth century. It concerns the question of how quickly the number of periodic points for smooth self-maps of compact manifolds can grow. In particular, if a manifold under consideration is the sphere \(S^2\), periodic points are isolated, and the map has degree \(d\), where \(|d|>1\), then the conjecture states that an increase of the number of periodic points must be at least exponential.

Until now, no-one has been able to prove the Shub conjecture. Nevertheless, it was shown that it is true in some special cases. The aim of the talk is to formulate conjecture, sketch the known partial results and discuss possible prospects for its solution.

The lecture will be presented in a form accessible to the widest possible range of listeners, with the emphasis on geometric intuition.

*The seminar takes place at 11:00 in room B1-37.*

**08-04-2016 | prof. dr hab. Jerzy Motyl: ***Order-convex selections of multifunctions and their applications in control theory*

*Order-convex selections of multifunctions and their applications in control theory*

Let \(X\) be a Banach space and let \((Y,\preceq)\) be a Banach lattice. During the talk we will introduce a new class of multifunctions \(F\colon X \to 2^Y\), that is, the so-called “upper separated multifunctions”, and we will investigate the problem of the existence of their order-convex selections. We will also present some applications of the established results to differential and differential-stochastic inclusions. In particular, certain new results concerning the existence of solutions, stability and upper-lower bounds for the solutions sets will be discussed. In the second part of the talk, we will show how to apply the selection theorems to deterministic and stochastic optimal control problems.

References:

- J. Motyl,
*Caratheodory convex selections of set-valued functions in Banach*, Topol. Methods Nonlinear Anal.

lattices**43**(2014), 1-10. - J. Motyl,
*Stochastic retarded inclusion with Caratheodory-upper separated*, Set-Valued Var. Anal. (2016), to appear, DOI: 10.1007/s11228-015-0324-9.

multifunctions - J. Motyl,
*Caratheodory-convex selections of multifunctions and their applications*, J. Nonlinear Convex Anal., accepted for publication. - J. M. Bismut,
*Conjugate convex functions in optimal stochastic control*, J. Math. Anal. Appl.**44**(1973), 384-404. - R. T. Rockafellar,
*Conjugate convex functions in optimal control and the*, J. Math. Anal. Appl.

calculus of variations**32**(1970), 174-222. - R. T. Rockafellar,
*Duality in optimal controls*, Math. Control Theory, Lecture Notes in Math.**680**(1978), 219-257

*The seminar takes place at 11:00 in room B1-37.*

**04-03-2016 | prof. UZ dr hab. Mariusz Michta: ***Set-valued Ito integral – properties and applications*

*Set-valued Ito integral – properties and applications*

The talk deals with notions of set-valued stochastic integrals which are main tools both in the theory of set-valued stochastic equations and inclusions. In the presentation we discuss among others the problem of integrable boundedness of such integrals which is one of the most important feature in potential applications. Such property is equivalent to boundedness of square integrable selections for such set-valued integral. Unfortunately up to now there were no correct proofs of such property. Surprisingly, also negative answers to this problem were not explained correctly. Hence the problem seemed to be undetermined. We shall show that in general the answer is negative. We shall provide several relatively simple examples both in convex and nonconvex valued case. Finally, we describe some ideas how to overcome the problem of unboundedness of set-valued Ito’s integrals in applications to the theory of set-valued stochastic equations and inclusions.

*The seminar takes place at 11:00 in room B1-37.*

**27-11-2015 | prof. UW dr hab. Marek Bodnar: ***Nonnegativity of solutions, stability of steady states and Hopf bifurcation in models of biochemical reactions with time delay*

*Nonnegativity of solutions, stability of steady states and Hopf bifurcation in models of biochemical reactions with time delay*

There are different mathematical models of biochemical reactions: a simple models consisting of a few ordinary differential equations, a complex systems of tens or hundreds of ODEs, differential equations with time delay, partial differential equations, stochastic differential equations. With the use of such equations, simple models of protein production and degradation are modelled as well as more complex pathway signalling processes. Models that consist of few equations can be analysed mathematically. However, more complex models can be usually studied only numerically. During my talk I present the influence of time delay on dynamics of biochemical models and problems that can arise as a consequence of the use of delay differential equations. On the basis of a very simple gene expression model, I show how a naive introduction of time delay into a model may lead to false results. Next, I talk about Hes1 gene expression model proposed by Monk in 2003 and I show the method that allows to prove global stability of the steady state by comparing the dynamics of delay differential equations with the dynamics of an appropriate discrete system.

*The seminar takes place at 11:00 in room B1-37.*

**05-06-2015 | prof. Xiao-Xiong Gan: ***JIT-transportation model and certain emergency management*

*JIT-transportation model and certain emergency management*

The JIT-transportation model requires that all demanded goods should be transported to their destinations on schedule, at zero or minimal destination-storage and minimal transportation cost. Certain emergency management problems are relevant to the JIT-transportation problem and therefore could be treated by applying a special JIT-transportation model to them. We provide a Emergency transportation model with its algorithm which treats the transportation problem during the rescue mission after a catastrophe occurred. This Emergency transportation model represents a plan which ships a number of commodities from numerous emergency control centers or distribution centers to numerous locations and requires that all rescuing commodities including goods, equipment and personnel to be shipped to their destinations in time or having minimum deviation of the scheduled times with the minimum cost of the transportation.

*The seminar takes place at 11:00 in room B1-37.*

**22-05-2015 | dr hab. inż. Izabela Lubowiecka: ***Biomechanical aspects of ventral hernia treatment: modelling, simulation, experiment*

*Biomechanical aspects of ventral hernia treatment: modelling, simulation, experiment*

Annually 30000 of ventral hernia operations are performed in Poland. Together with groin hernia cases the number reaches 70000. These are the most common scheduled medical interventions. Hernia orifice can be sewed up by a surgeon, if the orifice is sufficiently small; otherwise an implant is used to cover the orifice. The implant, in the form of a membrane, is fixed in the fascia of abdominal wall. Selection of the operation method, as well as an implant and its fixation selection is not a trivial task. Evidence of this is e.g. number of the sickness recurrences, reaching 20%, according to some sources.

Ventral hernia, operated with a usage of implant is a complex mechanical system containing abdominal wall fascia, joints and implanted surgical mesh. The biomechanics of this system is determined by the parameters of its components like e.g., properties of human tissue, implanted mesh and method of its fixation as well as the mechanical performance of the repaired abdominal wall in the context of extreme mechanical deformation resulting from the human physiological movements.

The problem of recurrences and quality of the patient’s life after surgery is an issue which requires interdisciplinary approach combining mechanics and medicine with the use of available mathematical tools. This is to recognise the mechanics of a healthy and operated human abdomen in order to identify the causes of joints failure and to prevent its appearance. The research includes mathematical modelling of the implant together with the constitutive model of its material and simulation of the behaviour of the whole complex system. Verification of the modelling and simulation is performed on the experimental basis. A significant difficulty of the analysis is its non-linear nature.

*The seminar takes place at 11:00 in room B1-37.*

**10-04-2015 | prof. PG dr hab. Grzegorz Graff: ***Topological invariants in action: fixed point indices of iterations in magnetohydrodynamics*

*Topological invariants in action: fixed point indices of iterations in magnetohydrodynamics*

Fixed point index is an important topological invariant applied to detect fixed points. In case of periodic points, a more powerful device is the whole sequence of indices of iterations.

Magnetic field lines are often analysed in a flux tube, basic geometric structure used to modelling astrophysical magnetic field, such as that which occurs in the atmosphere of the sun. For decades, there are still attempts to describe the evolution of the field in such tubes, the first concepts, according to which the field should be reduced over time to a very simple form were not confirmed in experiments and simulations. The lecture will present the topological restrictions on the evolution of the field, expressed in terms of the fixed point index. The possibility of using topological invariants in this aspect has been extensively studied in recent years by G. Hornig and his collaborators.

The lecture will be presented in a form accessible to the widest possible range of listeners, with an emphasis on geometric intuition.

*The seminar takes place at 11:00 in room B1-37.*

**09-01-2015 | prof. UJK dr hab. Grzegorz Łysik: ***Summability of formal solutions to partial differential equations*

*Summability of formal solutions to partial differential equations*

One of the main problems arising in the analytic theory of partial differential equations is a characterization of data given on a manifold [laxtex]S[/latex] for which a solution of a boundary value problem is an analytic function in a variable normal to [laxtex]S[/latex]. In general, one can easily obtain formal power series solutions in a variable normal to [laxtex]S[/latex], and by the well-known Cauchy-Kowalevska theorem this formal solution is convergent if [laxtex]S[/latex] is not the characteristic variety of the equation. In other cases formal solutions need not to be convergent. At this point there arise natural questions:

- what is the meaning of a formal solution,
- is it an asymptotic expansion of an actual solution,
- can and how the actual solution be constructed from the formal one.

In case of ordinary differential equations the answers to those questions were given in 80-ties and 90-ties of the XX century by the (multi)summability theory. On the other hand in the case of partial differential equations the study of those problems started at the end of the XX century and, besides linear equations with constant coefficients in two variables, practically there are no general results. During the lecture we shall survey solutions to those problems for some classes of partial differential equations. A special attention will be paid on heat type equations.

*The seminar takes place at 11:00 in room B1-37.*

**12-12-2014 | dr hab. Piotr Maćkowiak i dr Piotr Kasprzak: ***Applications of functions of bounded variation to image processing*

*Applications of functions of bounded variation to image processing*

Image processing methods (in particular, the recovery of the real image from its degraded by a noise or blurred version) are widely used to analyse satellite images, X-ray computed tomography images, archival photographs, etc. It turns out that many of these methods are based on very sophisticated mathematical tools. During the seminar, we will briefly discuss the mathematical ideas behind the image processing. Particular attention will be paid to the application of functions of bounded variation to the above mentioned problems. In addition, several illustrative examples will also be provided.

*The seminar takes place at 11:00 in room B1-37.*

**07-11-2014 | dr Łukasz Woźny: ***Fixed points of monotone operators and time-consistent policies of quasi-hyperbolic consumers*

*Fixed points of monotone operators and time-consistent policies of quasi-hyperbolic consumers*

We discuss a generalization of Tarski/Markowski fixed point theorem for monotone operators. Our generalization concerns order-theoretic completeness assumption as well as existence of an increasing selection for the fixed point correspondence of a parametrized version of our operator. We present also some applications to time-consistent Markov policies of quasi-hyperbolic consumers under uncertainty.

*The seminar takes place at 11:00 in room B1-37.*

**03-10-2014 | prof. Gennaro Infante: ***New criteria for the existence of multiple solutions in cones*

*New criteria for the existence of multiple solutions in cones*

We present new sufficient conditions for the existence of multiple fixed points for a map between ordered Banach spaces. An interesting feature of this approach is that we do not require conditions on two boundaries, but rather on one boundary and a point with some extra information on the monotonicity of the nonlinearity on a certain set. We apply our results to prove the existence of at least two positive solutions for a nonlinear boundary value problem that models a the displacement of a beam subject to some feedback controllers.

*The seminar takes place at 11:00 in room B1-37.*

**21-03-2014 | Prof. UMK dr hab. Grzegorz Gabor: ***Viable and periodic solutions in problems with barrier*

*Viable and periodic solutions in problems with barrier*

The talk concerns the problem of the existence of solutions of differential equations (or inclusions) remaining in a prescribed set of constraints K when this set is not invariant. In this case there exists an exit set, which trajectories leave the set through. Good topological properties of the exit set with respect to those of K imply the existence of viable, even stationary, solutions in K (Ważewski retract method, Conley index). After these preliminary results we will be interested in the case where all trajectories of the flow escape from K, but there is a barrier M in the exit set which bounces some of trajectories. We then obtain an impulsive problem with the impulse function I from M to the state space. We will give sufficient topological conditions for the existence of viable and periodic trajectories in K. A possibility of applying the fixed point index on ANRs and the Conley index for multivalued discrete dynamical systems will be presented.

*The seminar takes place at 11:00 in room B1-37*.

**07-03-2014 | dr Tomasz Cieślak: ***Vortex sheet spirals as vorticity generating 2D Euler flows*

*Vortex sheet spirals as vorticity generating 2D Euler flows*

In the beginning I will discuss a need to consider a singular solutions of 2d Euler equation generating the vorticity and the role of the space \(H^{-1}(\mathbb R^2)\) in that issue. Next, I will show that any object satisfying the so-called Prandtl similitude laws must be an element of \(H^{-1}(\mathbb R^2)\), and state the examples of the objects, known to physicists, satisfying the above mentioned Prandtl similitude laws. At the end, I will provide a short proof of continuous embedding of the space of Morrey type measures, with a support in a bounded set, in \(H^{-1}(\mathbb R^2)\), where \(n \geq 2\). It is based on the methods introduced to examine the vortex spirals.

*The seminar takes place at 11:00 in room B1-37.*

#### 31-01-2014 | prof. dr hab. Wojciech Okrasiński: **History of an inequality: from applications of mathematics to the theory**

**History of an inequality: from applications of mathematics to the theory**

Some nonlinear mathematical models lead us to the studies concerning solutions of a nonlinear Volterra integral equation. An integral inequality has appeared during considerations on the existence of solutions to that equation. That inequality is still studied and generalized.

*The seminar takes place at 11:00 in room B1-37.*

#### 08-11-2013 | dr Justyna Signerska: **Models of neuron activity with an almost periodic drive**

**Models of neuron activity with an almost periodic drive**

We will be concerned with mathematical analysis of integrate-and-fire systems, which are used in modelling of neural activity, electrical circuits or cardiac rhythms and arrhythmias. In these usually one-dimensional models, continuous dynamics govern by the differential equation is interrupted by the so-called threshold-reset behaviour, which is supposed to mimic spiking (generation of action potential) in real neurons.

The problem is to describe the sequence of consecutive spikes as iterations of some map, called *firing map*, and the sequence of interspike-intervals as displacement sequence along a trajectory of this map and to investigate dynamical properties of the firing map. If the function on the right-hand side of the differential equation is periodic in time variable (and regular enough), the question reduces to analysis of orientation preserving circle homeomorphisms/diffeomorphisms. Almost periodic forcing is a natural generalization of the periodic case and, simultaneously, a greater mathematical challenge.

We will investigate linear models with the input function almost periodic in the sense of Bohr or Stepanov and try to extend the results to even more general case of almost periodic functions in the Lebesgue measure.

*The seminar takes place at 11:00 in room B1-37.*

**25-10-2013 | dr inż. Łukasz Płociniczak: ***Anomalous diffusion in building materials. Model its and analytical solutions.*

*Anomalous diffusion in building materials. Model its and analytical solutions.*

Diffusion is one of the most common and important natural phenomena. It is a “random” spatial dispersion of some particles in its medium. The remarkable fact about the diffusion is its universality – almost on every scale some kind of diffusion is observed. We see it in fragrant particles dispersion, bacteria movement, contaminant transport and mergers of galaxies at cosmological scales. The diffusion equation is a classical description of these phenomena. Albeit of a spectacular precision and accuracy of the classical diffusion equation, a number of researches has lately reported about some discrepancies between theory and experiment. Especially this anomalous diffusion has been observed in porous media like those in building materials. In that case a self-similar profile has been observed only for a different space-time scaling. To model this feature a number of methods have been proposed, notably by the use of fractional derivative. We have obtained a simple, analytical approximation to the equation of nonlinear fractional diffusion. The high accuracy of the formula has been verified numerically. The method that we used is based on a series approximation of the Erdelyi-Kober fractional operator. After series truncation an ordinary differential equation can be obtained an solved. Our results are almost identical to those obtained numerically by other Authors, but our analytical form is an additional advantage that is very valuable in experimental work.

*The seminar takes place at 11:00 in room B1-37.*