## Intern seminr oddelenia (2017/2018)

• 26.4.2018
Some new facts about the generalized Sugeno integral
(Michal Boczek, Posko)

We give two-sided attainable bounds of Jensen type for the generalized Sugeno integral of any measurable function. The results extend the previous results of Romn-Flores et al. for increasing functions and Abbaszadeh et al. for convex and concave functions. We also introduce a new functional called integral via iterating process having its origin in scientometric indices.

• 5.4.2018
Approximate invertibility in non-unital commutative Banach algebras
(Egor Maximenko, IPN, Mxico)

The concept of approximate identities is a standard tool in non-unital normed algebras. Let A be a normed algebra. A net $(e_j)_{j\in J}$ in $\mathcal{A}$ is an \emph{approximate identity} if for every element a of $\mathcal{A}$ the nets $(ae_j)_{j\in J}$ and $(e_j a)_{j\in J}$ converge to a. We say that an element x of $\mathcal{A}$ is \emph{approximately right invertible} if there exists a net $(r_j)_{j\in J}$ such that the net $(xr_j)_{j\in J}$ is an approximate identity. In this talk we show criteria of approximate invertibility in several non-unital algebras, including the algebra $C_0(\mathbb{R})$ of continuous functions vanishing at infinity, the convolution algebra $L^1(\mathbb{R})$, the small disk algebra and the algebra of compact operators. We also study relations between the approximately invertible elements, the density of the principal ideals and the topological zero divisors. The results presented in this talk are obtained jointly with Kevin Esmeral and Ondrej Hutnk. The talk is supported by the Erasmus+ mobility grant and partially by IPN-SIP projects.

• 15. a 22.3.2018
Kognitvne integrly
(Anton Hovana)

Motivated by some cognition styles, a new type of Pre-Aggregation Functions called Cognitive Integrals with its generalized forms are discussed. Rather than affected only by selected capacity, Generalized Cognitive Integrals consider other two parameters, the Cognitive Strength and the involved Semicopula. Our main focus is on Cognitive Strength, by which the integrals values will be monotonic decreasing with respect to our Cognitive Strength. After some appropriated adaptations, we later propose the concept of Adapted Cognitive Integrals with two equivalent forms of itself. Not only being a type of the Pre-Aggregation Functions, we prove that Adapted Cognitive Integrals are indeed one new type of Aggregation Functions, still equipped with the two new parameters like in Cognitive Integrals.

• 8.3.2018
Modelovanie zvislost pomocou kopl
(Katarna Luivjansk)

Kopuly s vhodnm nstrojom na modelovanie dt z rznych zdrojov. V prspevku prezentujeme ukky modelovania zvislosti medzi trhom akci a trhom dlhopisov pomocou kopula modelov.

• 22.2. a 1.3.2018
Diskrtny Choquetov integrl a size
(Lenka Halinov, Jana Borzov)

Diskutujeme interpretciu sizeovej verzie Choquetovho integrlu na konenom zkladnom priestore pomocou aktulneho algoritmu vpotu super level miery. Uvedieme niekoko prkladov, v ktorch porovnme tento nov prstup s klasickm Choquetovm integrlom a OWA.

• 11.12.2017
Kluvnkova vzorkovacia veta
(Ondrej Hutnk)

V krtkosti si pripomenieme histriu Shannonovej vzorkovacej vety a Kluvnkovho fundamentlneho prspevku v oblasti abstraktnej harmonickej analzy, ktor dnes nesie jeho meno. Ide o zoveobecnenie Shannonovej vety z relnej osi do loklne kompaktnch abelovskch grp.

• 4.12.2017
Tempered distributions on positive octant
(Smiljana Jaksic, Belehrad)

We start this lecture by the characterization of the spaces through Laguarre expansions which motivates us to construct the kernel theorem.

• 30.11.2017
A simplified theory of distributions for engineering applications and time-frequency analysis
(Hans G. Feichtinger, Viede)

While classical Fourier Analysis has its basis in integration theory, using the Lebesgue integral in order to define the Fourier transform of an integrable function as a pointwise defined, continuous function, already the inverse Fourier transforms requires some approximations. The same is true for the proof of the Plancherel Theorem, showing that L2(Rd) is mapped isometrically onto itself by the Fourier (extended) transform. One then usually goes on to define topological vector spaces of test functions, such as the Schwartz space, in order to then extend the Fourier transform to e.g. tempered distributions in the sense of Laurent Schwartz. But this is a highly non-trivial path, and at the end it is not even convincing when it comes to the discussion of the approximation of the true Fourier transform of an integrable function by means of FFT-based algorithms.
Time-frequency analysis and in particular Gabor Analysis require other function spaces in order to e.g. describe continuous dependency of the dual Gabor atom on the lattice constants in the time-frequency plane. There is a Banach space of test functions, the so-called Segal algebra S0(Rd), which is much easier to handle than the Schwartz space. Together with the Hilbert space L2(Rd) and the dual space S'0(Rd) they form a Banach Gelfand triple which conveniently describes the Fourier transform in all its variants, and w*-convergence helps to give various transitions (e.g. from the periodic case to the non-periodic case) a correct mathematical way.

• 30.11.2017
On the second order impulsive periodic problem at resonance
(Martina Langerov, Plze)

We consider the periodic problem for the second order differential equation with impulses in the derivative at fixed times. We study the resonance problems and formulate general sufficient condition for the existence of a solution in terms of the asymptotic properties of both nonlinear restoring force and nonlinear impulses which generalizes the classical Landesman-Lazer condition. Our condition also implies the existence results for some open problems with vanishing and oscillating nonlinearities.

• 28.11.2017
The role of the metaplectic group for time-frequency analysis: applications to the fractional Fourier transform
(Maurice de Gosson, Viede)

We review the basics of the theory of the metaplectic group, which plays an important role in harmonic analysis and its applications (quantum mechanics and time-frequency analysis). We thereafter show that the notion of fractional Fourier transform can be very precisely and easily expressed in terms of metaplectic operators obtained from the rotation group. We give a few consequences previously obtained in collaboration with F. Luef.

• 31.10., 7.11. a 14.11.2017
Size-based super level measures on discrete space
(Jana Borzov, Lenka Halinov, Jaroslav upina)

We continue in the investigation of a concept of size introduced by Y. Do and C. Thiele. Our focus is a computation of corresponding super level measure, a key component of size application, on discrete space, i.e., a finite set with discrete topology. We found critical numbers which determine the change of a value of super level measure and we present an algorithm for super level measure computation based on these numbers.

• 17.10. a 24.10.2017
Comonotonicity of functions and its generalizations
(Anton Hovana, Ondrej Hutnk)

We consider relationships between three classes of functions: (i) comonotone functions, (ii) *-associated functions and (iii) positively dependent functions. We present examples and first observations on the topic.

• 3.10. a 10.10.2017
On possibility and necessity measures
(Jana Borzov)

Pozrieme sa detailnejie na aktulny lnok Chen et al. Possibility and necessity measures and integral equivalence z hadiska monho zoveobecnenia pre miery a integrly zaloen na sizeoch. Prezentujeme prv pozorovania a prklady pre kontrukciu veobecnho algoritmu.

• 19.9.2017 a 26.9.2017
Inequalities for seminormed integral
(Anton Hovana, Ondrej Hutnk)

Universal integrals, introduced in 2010, have attracted an increasing attention due to their promising efficiency in solving various mathematical, engineering and behavioral-science problems as well as real-world applications. In the last few years, many papers have been published regarding basic properties, convergence theorems and various inequalities for these integrals as non-additive counterparts to the well-known inequalities for the classical (additive) integrals. Since the relevant literature has expanded significantly, we summarize the majority of the results about integral inequalities for the class of universal integrals related to semicopula in peer-reviewed journals and conferences found in the available literature so far.