wds-m 2015 book of abstracts
TRANSCRIPT
Charles University in PragueFaculty of Mathematics and Physics
24th Annual Student Conference
Week of Doctoral Students 2015
Book of Abstracts
of the
Week of Doctoral Studentsof the School of Mathematics 2015
June 8, 2015
Sokolovská 83186 75 Praha 8
Editace v LATEXu Mirko Rokyta, s použitím grafické úpravy Andreje Živcáka a Petra Pošty.
http://www.karlin.mff.cuni.cz/~rokyta/WDS-M/2015/http://www.mff.cuni.cz/veda/konference/wds/
Vytisklo Reprostredisko MFF UK Prahac© 2015 MFF UK Praha
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Preface
In the beginning of 2014, the Management of the Faculty of Mathematics and Physicsdecided that the traditional conference of PhD students called the WDS (Week ofDoctoral Students) would not be organized as an activity of the entire faculty. In-stead, the decision as to whether to organize the conference or not was left to therespective Schools (of Computer Science, of Mathematics, and of Physics).
Already for the second year since then the School of Mathematics organizes thisWDS-M (Week of Doctoral Students of the School of Mathematics, http://www.karlin.mff.cuni.cz/~rokyta/WDS-M/2015/), this time as a one-day conference,in the framework, and as a continuation of, the (24th) WDS of the Faculty of Math-ematics and Physics (http://www.mff.cuni.cz/veda/konference/wds/).
This year 16 students have registered as active participants to the conference. Webelieve that this event, which takes place in the “mathematical” Karlín building of thefaculty, will attract the attention of the students but also of the broad mathematicalaudience. We thus encourage all of those interested in the scientific activities of ourdoctoral students to attend this meeting.
Prague, June 8, 2015
doc. RNDr. Mirko Rokyta, CSc.Vice-Dean for MathematicsFaculty of Mathematics and PhysicsCharles University Prague
Contents
4M1 – Algebra, teorie císel a matematická logikaJan Horácek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4M2 – Geometrie a topologie, globální analýza a obecné strukturyAdam Bartoš . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Petr Zima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4M3 – Matematická analýzaMartin Francu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Martin Krepela . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Vít Musil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Bretislav Skovajsa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4M4 – Pravdepodobnost a matematická statistikaMgr. Ing. David Coufal, PhD. . . . . . . . . . . . . . . . . . . . . . . . . . 10Petr Coupek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Karel Kadlec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4M6 – Vedecko-technické výpoctyFilip Roskovec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4M8 – Obecné otázky matematiky a informatikyTereza Bártlová . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Tereza Kovárová . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Michal Zamboj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4M9 – Pravdepodobnost a statistika, ekonometrie a financní matematikaJakub Vecera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4F11 – Matematické a pocítacové modelováníPetr Vágner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Network codingMgr. Jan Horácek
E-mail: [email protected] studia: 4M1 – Algebra, teorie císel a matematická logikaRocník: 1.Školitel: prof. RNDr. Aleš Drápal, CSc., DSc.
AbstractNetwork coding is a new powerful technique used in networking and it
has been studied extensively as a part of coding theory. Network coding,as opposed to traditional error-correcting codes for communication throughone channel, assumes a richer infrastructure – a network. We describe mainprinciples behind network coding and explore its algebraic background.A random linear network coding scheme, which is one of the most practicalimplementation of network coding, is presented.
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Maximal connected topologiesMgr. Adam Bartoš
E-mail: [email protected] studia: 4M2 – Geometrie a topologie, globální analýza a obecné strukturyRocník: 1.Školitel: prof. RNDr. Petr Simon, DrSc.
AbstractIf P is a property of topological spaces, we say that a topological space
〈X ,τ〉 (or the corresponding topology τ) is maximal P if it has the prop-erty P , but no strictly finer topology τ∗ > τ has the property P . We areinterested in the case where P means connectedness, i.e. in maximal con-nected topologies. There are examples of Hausdorff maximal connectedspaces as well as examples of Hausdorff connected spaces having no finertopology that is maximal connected. The problem of existence of a regularmaximal connected topology remains open.
We will summarize the history of the problem of existence of maximalconnected spaces with prescribed separation axioms, present known facts,and illustrate them on several examples.
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Prolongation of Killing equationsMgr. Petr Zima
E-mail: [email protected] studia: 4M2 – Geometrie a topologie, globální analýza a obecné strukturyRocník: 1.Školitel: doc. RNDr. Petr Somberg, Ph.D.
AbstractKilling equations are certain systems of partial differential equations
arising naturally in (pseudo-)Riemannian and Spin-geometry. The systemsare always overdetermined and hence the existence of nontrivial solutionsis a rare phenomenon. But once the solutions exist, they provide valuableinformation about geometry of the underlying manifold. In particular, theequations have a strong intrinsic relationship with curvature.
Effective tool for approaching the Killing equations is the so called pro-longation procedure. It transforms the system of differential equations intoa larger equivalent system of simpler nature. This transformation providesboth theoretical and practical advantage. Firstly, it almost immediatelyyields an upper bound for the dimension of the space of solutions. Sec-ondly, in some specific examples the transformed system can be directlyintegrated yielding an explicit formula for the solutions.
The Killing equations come in different types depending on the quantitywhich constitutes the domain of the equation. The most prominent exam-ple are the Killing vector fields. They directly correspond to infinitesimalsymmetries of the underlying manifold. As such, they are widely knownand used in the fields of (pseudo-)Riemannian geometry and also theoret-ical physics. In this talk I will focus on a more complicated type whosedomain is a quantity called spinor-valued differential forms. To my bestknowledge, systematic study of this case is original work.
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Higher-order Sobolev-type imbeddingson Carnot-Carathéodory spaces
Mgr. Martin Francu
E-mail: [email protected] studia: 4M3 – Matematická analýzaRocník: 3.Školitel: prof. RNDr. Luboš Pick, CSc., DSc.
AbstractSufficient condition for higher-order Sobolev-type imbeddings on bou-
nded domains of Carnot-Carathéodory spaces is established for class ofrearrangement-invariant function spaces. The condition takes form of aone-dimensional inequality for suitable integral operators depending on theisoperimetric function relative to the Carnot-Carathéodory structure of therelevant sets. General results are then applied to particular Sobolev spacesbuild upon Lebesgue, Lorentz and Orlicz spaces on John domains in Heis-enberg group. In the case of Heisenberg group and first-order Sobolev-typeimbedding, the condition is shown to be necessary as well.
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Iterating bilinear inequalitiesMgr. Martin Krepela
E-mail: [email protected] studia: 4M3 – Matematická analýzaRocník: 4.Školitel: prof. RNDr. Luboš Pick, CSc., DSc.
AbstractConsider a bilinear operator constructed as a product of two ordinary
Hardy operators. Then it is possible to obtain weighted inequalities in-volving this operator, by iteration and using the classical weighted Hardyinequalities. In this way, one gets various results in form of inequalities fornonnegative, decreasing functions or other classes. The idea applies easilyto various Hardy-type bilinear or multilinear operators, but may as well beused to obtain estimates for convolution-type operators.
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Optimal Orlicz domains in Sobolev embeddingsMgr. Vít Musil
E-mail: [email protected] studia: 4M3 – Matematická analýzaRocník: 1.Školitel: prof. RNDr. Luboš Pick, CSc., DSc.
AbstractFor a given Banach function space Y (Ω), we study the question whether
there exists an optimal (i.e. largest) Orlicz space LA(Ω) satisfying the em-bedding
W mLA(Ω) → Y (Ω),
where Ω stands for a bounded domain in Rn, n ≥ 2, and W mLA(Ω) is anOrlicz-Sobolev space.
In the general setting, where in the place of LA(Ω) there is a rearrange-ment-invariant (r.i.) Banach function space (i.e. space where functions hav-ing the same distribution function have the same norm), such questionswere investigated using the method of reducing the Sobolev embeddingsto the boundedness of an appropriate modification of the weighted Hardyoperator. In the setting of r.i. spaces, the optimal domain and the optimaltarget spaces always exist and they are then explicitly described.
However, for certain specific applications such as to the solution ofpartial differential equations, it is often useful to investigate the optimalityof spaces in Sobolev-type embeddings restricted to the context of Orliczspaces. It turns out that the situation in this setting is significantly differentthan in the broader sense of r.i. spaces. For instance even the existence ofan optimal Orlicz domain may fail.
In the talk we present an answer to the question in the special case whenthe target space is chosen from the class of the so-called Marcinkiewiczendpoint spaces. This is not as restrictive as it may seem since the mostcustomary cases are covered. As a consequence of the method used in theapproach we obtain similar results also for Sobolev trace embeddings ofdifferent orders and on various domains in Rn at once.
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Generalized ordinary differential equationsin metric spaces
Mgr. Bretislav Skovajsa
E-mail: [email protected] studia: 4M3 – Matematická analýzaRocník: 1.Školitel: prof. RNDr. Jan Malý, DrSc.
AbstractRecent years have revealed that many types of differential problems
can be studied in the context of metric spaces i.e. without any prior linearstructure. The theory of generalized ordinary differential equations has notbeen considered in this setting due to heavy reliance on integration overthe target space. The versatility of the Henstock-Kurzweil integral allowsthis theory to cover a very wide range of problems; extending it to metricspaces puts all of them into a new perspective. This talk will be concernedwith the basic principles behind the aforementioned generalization as wellas the latest results and their expected impact.
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Kernel methods in particle filterMgr. Ing. David Coufal, PhD.
E-mail: [email protected] studia: 4M4 – Pravdepodobnost a matematická statistikaRocník: 2.Školitel: prof. RNDr. Viktor Beneš, DrSc.
AbstractThe particle filter is a tool that extends the use of the filtering techniques
to non-linear and non-Gaussian processes. The work of the particle filterdraws on sampling techniques in order to construct random measures thatapproximate conditional distributions of interest. These distributions arecalled the filtering distributions. The carriers of the random measures (infact, the sampled points) are called particles. It is known that the construct-ed approximations converge to the filtering distributions with the increasingnumber of the particles generated.
In the talk, we aim on the construction of kernel estimates of the densi-ties of the filtering distributions (the filtering densities). We will show thatin spite of the non-i.i.d. character of the samples generated by the filter thekernel estimates still converge to the filtering densities. We will discuss theconditions and the character of the convergence along with the extensionof the results to the convergence of derivatives of the estimates.
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Cylindrical Volterra processes andstochastic evolution equations
Mgr. Petr Coupek
E-mail: [email protected] studia: 4M4 – Pravdepodobnost a matematická statistikaRocník: 2.Školitel: prof. RNDr. Bohdan Maslowski, DrSc.
AbstractThe aim of this talk is to present some results regarding a mild solution
of a linear stochastic evolution equation of the form
dXt = AXt dt +φ dBt , t ∈ [0,T ], (1)X0 = x,
which is driven by a cylindrical Volterra process – an infinite dimension-al analogue to processes which admit the canonical representation Xt =∫ t
0 K(t,r)dWr, where W denotes the standard Brownian motion and K is asuitable kernel. Our principal examples include the heat and wave equa-tions and thus we introduce two sets of sufficient conditions under whichthe mild solution of (1) is a well-defined mean-square continuous Gaus-sian process. These findings extend our previous results and generalizethe model with the driving process being a cylindrical fractional Brownianmotion of Hurst parameter H ≥ 1/2 (i.e. including the standard Brownianmotion).
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Ergodic control for infinite dimensional Lévy processesMgr. Karel Kadlec
E-mail: [email protected] studia: 4M4 – Pravdepodobnost a matematická statistikaRocník: 3.Školitel: prof. RNDr. Bohdan Maslowski, DrSc.
AbstractIn this contribution, weak solutions of stochastic evolution equations
driven by square integrable infinite dimensional Lévy processes are stud-ied. The commonly known Ito formula is directly applicable only to thestrong solutions of the stochastic evolution equations. The assumption ofthe existence of the strong solution is however too restrictive, therefore theIto formula in a suitable form for the weak solution of the stochastic evo-lution equation with the Lévy noise is given. The LQ stochastic optimalcontrol problem as well as the corresponding ergodic control problem areformulated. Some ergodic control results in the case of square integrableLévy noise are solved.
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Gauss-Radau polynomial reconstruction of discontinuousGalerkin solutionsMgr. Filip Roskovec
E-mail: [email protected] studia: 4M6 – Vedecko-technické výpoctyRocník: 1.Školitel: prof. RNDr. Vít Dolejší, Ph.D., DSc.
AbstractThe space-time discontinuous Galerkin (STDG) method offers a robust
and highly efficient approach to numerical solution of nonstationary PDEequations. We exploit the Gauss-Radau polynomial to design a reconstruc-tion of the STDG solution, which is conforming with respect to time. Thisreconstruction can be conveniently used to show a relationship between thetime discontinuous Galerkin method and a special class of Runge-Kuttamethods. Due to this relation we can better understand the convergencebehaviour of time discontinuous Galerkin method, especially the increasedorders of convergence in the endpoints of the time intervals and also inthe roots of Gauss-Radau polynomial. Furthermore, we propose a methodhow to employ this reconstruction to a posteriori estimation of the error ofSTDG method.
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Interactive science centersMgr. Tereza Bártlová
E-mail: [email protected] studia: 4M8 – Obecné otázky matematiky a informatikyRocník: 3.Školitel: prof. RNDr. Luboš Pick, CSc., DSc.
AbstractThe education of mathematics is usually associated with school teach-
ing. However, the education is not just about the volume of knowledge,but mainly about a permanent mental mastery of skills and abilities to usethem. An additional education of mathematics outside the classroom is thusdeveloped in completely unconventional environments such as interactivescience centers. Such centers have been appearing during the last few yearsaround the world. In these centers, favorable conditions are created for theeducation of children. The children therefore can learn new skills whileenjoying themselves in their spare time.
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Tiling proofs of combinatorial identitiesinvolving Fibonacci, Lucas and Pell numbers
Mgr. Tereza Kovárová
E-mail: [email protected] studia: 4M8 – Obecné otázky matematiky a informatikyRocník: 1.Školitel: RNDr. Antonín Slavík, Ph.D.
AbstractWe present three combinatorial identities involving the well-known Fi-
bonacci numbers, Lucas numbers and Pell numbers. These identities willbe proved by means of a combinatorial argument. For that purpose we usethe relationship between the Fibonacci numbers and the number of all pos-sible tilings of a rectangle 1× n with 1× 1 and 1× 2 polyominoes, thenthe relationship between the Lucas numbers and the number of all possibletilings of a circular board composed of n cells with curved 1×1 and 1×2polyominoes, and finally the relationship between the Pell numbers and thenumber of all possible tilings of a rectangle 1× n with blue 1× 1, green1×1 and 1×2 polyominoes.
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Synthetic approach to Chasles theoremfor timelike ruled surface
Mgr. Michal Zamboj
E-mail: [email protected] studia: 4M8 – Obecné otázky matematiky a informatikyRocník: 1.Školitel: Mgr. Lukáš Krump, PhD.
AbstractChasles theorem provides a simple method for constructing tangent
planes to a timelike ruled surface. Let Σ be a timelike ruled surface, and lbe a line of Σ. Then points on l are in projectivity with tangent planes toΣ in these points. The theorem is highly applicable in descriptive geome-try. We will present a synthetic approach to the proof of the theorem in thespecial case for the 2nd degree surfaces, as well as some further remarkson the synthetic approach to geometry in the projective extension of theEuclidean 3–dimensional space. For such purpose, we will introduce somegraphic tools in software Wolfram Mathematica 10 and Geogebra 5.
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Central limit theorem for Gibbsian U-statisticsof facet processesMgr. Jakub Vecera
E-mail: [email protected] studia: 4M9 – Pravdepodobnost a statistika, ekonometrie a financní matemati-kaRocník: 1.Školitel: prof. RNDr. Viktor Beneš, DrSc.
AbstractSpecial case of a Gibbsian facet process on a fixed window with a dis-
crete orientation distribution and with increasing intensity of the underly-ing Poisson process is studied. All asymptotic moments for interactionU-statistics are calculated and using the method of moments the centrallimit theorem is derived.
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Optimization of fuel cellsMgr. Petr Vágner
E-mail: [email protected] studia: 4F11 – Matematické a pocítacové modelováníRocník: 1.Školitel: prof. Ing. František Maršík, DrSc.
AbstractAn optimization of the fuel cells is a topic which importance grows
with the fuel cells spread. Therefore, proper theoretical tools are necessaryfor correct interpretation of the experimental data and meaningful proposalof future development. The Exergy analysis is the most used theoreticaltool for evaluation of the fuel cells efficiency, even though the fuel cellsdo not meet the Exergy analysis basic assumption, namely an isothermalboundary assumption. This inadequacy has been investigated in the arti-cle [1], where a new efficiency evaluation has been introduced. The newapproach is illustrated on a simple solid oxide fuel cell model. The demon-stration shows a substantial difference between the new approach and theExergy analysis in the case of non-isothermal boundary.
[1] M. Pavelka, V. Klika, P. Vágner, and F. Maršík, Applied Energy 137, 158 (2015), ISSN 0306-2619, DOI:
10.1016/j.apenergy.2014.09.071
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