## Interný seminár oddelenia (2015/2016)

• Prednáška v rámci výskumného projektu SASPRO
13.6.2016 (9:30-11:00, SJ2P10)

On Kurzweil-Stieltjes integration and particular classes of functions
(Giselle Antunes Monteiro, University of São Paulo, Brazil)

In the late fifties, a theory of integration introduced by J. Kurzweil revealed to be more general than the Riemann and the Lebesgue integrals, including the Stieltjes type. Regarding Stieltjes integrals, it is well-known that the classes of continuous functions and functions of bounded variation are adjoint with respect to the Riemann-Stieltjes integral. In this presentation we analyse whether the Kurzweil-Stieltjes integral mimics such a property of the Riemann-Stieltjes integral with respect to functions. In particular, we study the classes of regulated functions and functions of bounded variation with respect to their integrability, providing then a partial answer to the question of adjoint classes of Kurzweil-Stieltjes integrable functions.

• Cyklus prednášok v rámci programu Erasmus+
17.5. a 19.5.2016 (10:45-12:15, SJ1S38)

Discontinuous Galerkin method and its applications
(Jiří Hozman, Technická univerzita v Liberci, ČR)

This lecture series is concerned with the theoretical analysis and practical applications of the discontinuous Galerkin method (DGM).

• At first, we introduce students with the concept of this method as a combination of finite volume and finite element methods. DGM is based on a piecewise polynomial but discontinuous approximation which provides robust and high-order accurate approximations of solutions resulting from the system of PDEs describing variety of real-world problems.
• Next, we apply the afore-mentioned approach to a scalar nonstationary convection-diffusion equation which represents a model problem for prospective applications. The resulting semi-discrete, and later fully discrete schemes are theoretically analysed and several numerical experiments are presented.
• Further, we extend the DGM technique to the inviscid/viscous compressible flow problems (CFP). The presented extension from a scalar case to the case of CFP is not straightforward and the special attention is paid to several features of the DGM, namely choice of boundary conditions, numerical flux, discretization of inviscid/viscous terms and other implementation aspects. This approach is also supplemented by numerical results.
• In the last part, we treat valuation of options, incorporating DGM technique into the numerical option pricing scheme for selected types of options and under specific market conditions. First, we mention the fundamentals of option pricing and introduce the appropriate one-factor Black-Scholes model describing the evolution of the simplest (vanilla) options. Next, we generalize this approach to basket options and to path-dependent ones such as Asian options. Finally, each afore-mentioned model is illustrated by practical experiments on real market data.

• Prednáška v rámci výskumného projektu SASPRO
16.5.2016 (15:00-16:00, SJ1S38)

Kurzweil’s and Dobrakov’s approaches to integration
(Giselle Antunes Monteiro, University of São Paulo, Brazil)

The main purpose of the lecture is to introduce into the research project where we aim to work with nonabsolute integrals via two different approaches: The first corresponds to a continuation of the research developed by M. Tvrdý and the author on the abstract Kurzweil-Stieltjes integral. The second approach concerns the Kurzweil-Stieltjes integral over sets aiming to investigate its connection with the Dobrakov integral.

• Cyklus prednášok v rámci programu Erasmus+
9.5.-12.5.2016 (9:00-10:30 a 11:00-12:30, SJ1S38)

Theory of distributions and its applications
(Svetlana Mincheva-Kamińska, Andrzej Kamiński, Universitet Rzeszowski, PL)

• Introduction to the functional and sequential theories of distributions. Fractional derivatives of functions.
• Convolution of functions and distributions. Conditions for the existence. Power functions as distributions on the positive cone.
• Product and other irregular operations on distributions. Fractional derivatives of distributions.
• Fourier transforms of tempered distributions and the exchange formulae. Applications of Fractional Calculus.

• predvianočný ústavný seminár 17.12.2015 (10:30, SJ2P09)
A survey of higher computability theory
(Dávid Tóth, Department of Pure Mathematics, University of Leeds, UK)

In Computability Theory if there exists a computer program that can decide if a number is in a chosen subset of the natural numbers, then such a set is called computable. In Higher Computability Theory computers (Turing machines) have available transfinite time and compute on the subsets of the ordinals. However, the main definitions in Higher Computability Theory start with Goedel's constructible universe $L$. Higher Computability Theory is at the intersection of Classical Computability Theory and Set Theory. The talk will focus on the core concepts in Higher Computability Theory and its interconnections with with other branches of Mathematics such as Effective Descriptive Set Theory, Hyperarithmetical Theory and $\alpha$-Computability Theory. One of the examples of the statements is Gandi-Spector theorem: a subset of the natural numbers is ${\Pi }_{1}^{1}$ iff it is ${\Sigma }_{1}^{0}$-definable over ${L}_{\omega _{1}^{\mathrm{CK}}}$, ${\omega }_{1}^{CK}$-level of Goedel's constructible universe where ${\omega }_{1}^{CK}$ is the first uncomputable ordinal.

• 15.12.2015
A general model for various cohesive frictional contacts
(Roman Vodička, Stavebná fakulta, TUKE)

A general model for a large variety of cohesive contacts with friction between visco-elastic bodies is presented. A semi-implicit time discretisation which is numerically stable and convergent and which advantageously decouples the solved system is proposed. After a spatial discretisation by the symmetric Galerkin boundary element method, it enables an efficient numerical implementation by sequential quadratic programming algorithms. Several solved problems are presented to assess the applicability of the proposed model for a wide range of engineering structures.

• 8.12.2015
One extension of Itô integral
(Jozef Kiseľák)

The celebrated Wiener-Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus.The first version of this theorem was proved by Wiener in 1938. Later Itô (1951) showed that in the Wiener space setting the expansion could be expressed in terms of iterated Itô integrals. It is a convenient starting point for the introduction of several important stochastic concepts, including the Skorohod integral (1975, named after the Ukrainian mathematician Anatoliy Skorokhod). This integral may be regarded as an extension of the Itô integral to integrands which are not necessarily Ft-adapted (unpredictable processes). It has other important properties, e.g. it is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations and an infinite-dimensional generalization of the divergence operator from classical vector calculus. We discuss the advantages of this concept introduced by techniques mentioned above.

• 1.12.2015
O novej algebrickej charakterizácii diskrétneho Sugenovho integrálu
(Jozef Pócs, Matematický ústav SAV)

Predstavíme novú vlastnosť diskrétneho Sugenovho integrálu, ktorá sa dá tiež chápať ako jeho charakterizácia. Táto vlastnosť, kompatibilita vzhľadom na kongruencie na [0, 1], podčiarkuje dôležitosť Sugenovho integrálu v multirozhodovacích procesoch.

• 10.11.2015
Zvonku merateľné Lp-priestory
(Ondrej Hutník)

V prednáške sa budeme venovať konštrukciám a výsledkom aktuálneho článku Do, Y., Thiele, C.: Lp-theory for outer measures and two themes of Lennart Carleson united. Bull. Amer. Math. Sci. (2015), ktoré sú zaujímavé a podnetné v súvislosti s rozvíjanou teóriou neaditívnych integrálov.

• 27.10.2015
(Tomáš Gregor, Matematický ústav SAV)

Odvodíme vlnovú rovnicu pre pozdĺžne kmitanie. Riešením jednorozmernej vlnovej rovnice vznikne Fourierov rad, ktorý budeme skúmať z pohľadu teórie multi-polarity.

• 20.10.2015
Spojitosť bez topológie?
(Stanislav Krajči, Ústav informatiky PF UPJŠ)

Ukážeme dve ekvivalentné podmienky k definícii spojitosti reálnej funkcie v bode zľava založené na usporiadaní. Jednu z nich zovšeobecníme pre ľubovoľnú usporiadanú množinu a ukážeme jej ekvivalenciu s definíciou spojitosti "zo severozápadného kvadrantu" v prípade komplexných funkcií.

• 13.10.2015
Slabý P-ideál a QN-priestor
(Jaroslav Šupina)

V prednáške predstavíme ideálovú verziu QN-priestoru, tzv. J QN-priestor. Ukážeme, že ideál J na $\omega$ obsahuje izomorfnú kópiu ideálu Fin×Fin na $\omega ×\omega$ práve vtedy, keď každý topologický priestor je J QN-priestor. Ak J neobsahuje izomorfnú kópiu ideálu Fin×Fin, tak Baireov priestor $\omega \omega$ nie je J QN-priestor. Dokážeme tiež niekoľko výsledkov súvisiacich s ideálovou verziou $S$1(Γ,Γ)-priestoru. Ako dôsledok dostaneme, že ideálová verzia Scheepersovej domnienky neplatí pre ideály obsahujúce izomorfnú kópiu Fin×Fin.

• 6.10.2015
3-polarita, farby a výpočty
(Zuzana Hurňáková, študentka matematiky PF UPJŠ)

Príspevok prezentuje výsledky letnej pracovnej stáže autorky na Matematickom ústave SAV v Košiciach u doc. Halušku na tému multipolarity a matematického modelu popisu farieb. Ukážeme niekoľko praktických výpočtov o použiteľnosti študovaného modelu pre praktické účely.

• 29.9.2015
Hudobné akcie dihedrálnych grúp
(Ondrej Hutník)

V prednáške objasníme, ako sa dá hudba interpretovať pomocou dihedrálnej grupy rádu 24 (t.j. grupy symetrií pravidelného 12-uholníka). Zameriame sa na klasické kompozičné techniky transpozície a inverzie bežné napríklad v Bachovej zbierke Umenie fúgy a ukážeme, že tieto hudobné transpozície a inverzie predstavujú z matematického hľadiska symetrie pravidelného 12-uholníka. Spomenieme tiež hudobno-teoretický neo-Riemannovský pohľad prostredníctvom akcie grupy pôsobiacej na durové a molové trojzvuky. Vyslovíme prekvapujúce zistenie, že tieto dve akcie dihedrálnej grupy rádu 24 na triády sú duálne. Spomínané akcie grupy a ich dualitu ilustrujeme na príkladoch známych hudobných diel Beethovena, Pachelbela, Wagnera a pod.