Streszczenie. W latach pięćdziesiątych Gagliardo wykazał, że dla obszaru $\Omega$ z
regularnym brzegiem operator śladu z przestrzeni Sobolewa
$W^1_1(\Omega)$ do przestrzeni $L^1(\partial \Omega)$ jest surjekcją.
Zatem naturalne jest pytanie o istnienie prawego odwrotnego operatora
do operatora śladu. Petree udowodnił, że w przypadku półpłaszczyzny
$\mathbb{R}x\mathbb{R}_{+}$ nie istnieje prawy odwrotny operator do
operatora śladu. Podczas referatu przedstawię prosty dowód twierdzenia
Petree, który wykorzystuje tylko pokrycie Whitney'a danego obszaru
oraz klasyczne własności przestrzeni Banacha. Następnie zdefiniujemy
operator śladu z przestrzeni Sobolewa $W^1_1(K)$, gdzie $K$ jest
płatkiem Kocha. Przez pozostałą część mojego referatu skonstruujemy
prawy odwrotny do operatora śladu na płatku Kocha. W tym celu
scharakteryzujemy przestrzeń śladów jako przestrzeń Arensa-Eelsa z
odpowiednią metryką oraz skorzystamy z twierdzenia Ciesielskiego o
przestrzeniach funkcji hölderowskich.
Entropy Weighted Regularisation: A General Way to Debias Regularisation Penalties
Olof Zetterqvist (University of Gothenburg/Chalmers)
Lasso and ridge regression are well established and successful models for variance reduction and, for the lasso, variable selection. However, they come with a disadvantage of an increased bias in the estimator. In this seminar, I will talk about our general method that learns individual weights for each term in the regularisation penalty (e.g. lasso or ridge) with the goal to reduce the bias. To bound the amount of freedom for the model to choose the weights, a new regularisation term, that imposes a cost for choosing small weights, is introduced. If the form of this term is chosen wisely, the apparent doubling of the number of parameters vanishes, by means of solving for the weights in terms of the parameter estimates. We show that these estimators potentially keep the original estimators’ fundamental properties and experimentally verify that this can indeed reduce bias.
Przedstawione zostaną dwie interpretacje uogólnionych ścieżek Dycka. Jedna z nich bazuje na artykule M.T.L. Bizleya z 1954 roku, w którym ścieżki Dycka zdefiniowane zostały jako ścieżki na punktach kratowych płaszczyzny, spełniające określone warunki. Drugi punkt widzenia, oparty na pracy P. Duchona z 2000 roku, opisuje ścieżki Dycka jako słowa szczególnej postaci, napisane w dwuliterowym alfabecie. Celem rozważań jest wyznaczenie równań, które spełnia funkcja tworząca D(t) dla zbioru słów Dycka wolnych od faktorów. Przedstawiony zostanie wzór ogólny, który opisuje funkcje tworzące dla przypadku, gdy jedna z liter ma waluację nieparzystą, a druga równa -2.
Virtual combination of relatively quasiconvex subgroups and separability properties
Ashot Minasyan
Quasiconvex subgroups are basic building blocks of hyperbolic groups, and relatively quasiconvex subgroups play a similar role in relatively hyperbolic groups. If $Q$ and $R$ are relatively quasiconvex subgroups of a relatively hyperbolic group $G$ then the intersection $Q \cap R$ will also be relatively quasiconvex, but the join $\langle Q,R \rangle$ may not be. I will discuss criteria for the existence of finite index subgroups $Q’ \leqslant_f Q$ and $R’ \leqslant_f R$ such that the ``virtual join’’ $\langle Q’, R’ \rangle$ is relatively quasiconvex. This is closely related to separability properties of $G$ and I will present applications to limit groups, Kleinian groups and fundamental groups of graphs of free groups with cyclic edge groups. The talk will be based on joint work with Lawk Mineh.
Coupled Volterra integral equations with blowing up solutions
Wojciech Mydlarczyk (PWr)
Abstract: In the talk, coupled nonlinear Volterra type integral equations
will be considered. We will focus on criteria for the existence of
positive solutions,
expressed in terms of the generalized Osgood condition. The global
behavior of the
solutions, especially the conditions when they experience blow up will
be also discussed.
How do we randomly sample an infinite sequence from a first order structure? What model theoretic properties might hold on almost all random sequences? Can we do probability theory in this setting in a meaningful way? This talk takes these questions seriously. Building on the body of work around Keisler measures, we construct a plausible framework to engage with probabilistic phenomena. This is joint work with James Hanson.
Extremes of multivariate locally-additive Gaussian random fields
Pavel Ievlev (Université de Lausanne)
In this talk, I am going to present some of my recent results in joint work with Nikolai Kriukiv on the extremes of multivariate Gaussian random fields. I will begin with the 2019 paper by K. Dębicki, E. Hashorva, and L. Wang, which laid the groundwork for further investigations in the area of multivariate Gaussian extremes. I will explain that some of the assumptions of this paper may not hold in cases that are practically important, and I will discuss how these issues can be amended by considering second-order contributions — I will clarify this terminology during the talk. Next, we will explore what is, in a sense, the simplest extension of these results from processes (indexed by R) to fields (indexed by R^n), which we refer to as 'locally-additive'. As an application of this extension, I will present an exact asymptotic result for the probability that a real-valued process first hits a high positive barrier and then a low negative barrier within a finite time horizon.
Perfectly meager sets in the transitive sense and the Hurewicz property
Piotr Szewczak (UKSW)
We work in the Cantor space with the usual group operation +. A set X
is perfectly meager in the transitive sense if for any perfect set P
there is an F-sigma set F containing X such that for every point t the
intersection of t+F and P is meager in the relative topology of P. A
set X is Hurewicz if for any sequence of increasing open covers of X
one can select one set from each cover such that the chosen sets
formulate a gamma-cover of X, i.e., an infinite cover such that each
point from X belongs to all but finitely many sets from the cover.
Nowik proved that each Hurewicz set which cannot be mapped continuously
onto the Cantor set is perfectly meager in the transitive sense. We
answer a question of Nowik and Tsaban, whether of the same assertion
holds for each Hurewicz set with no copy of the Cantor set inside. We
solve this problem, under CH, in the negative.
This is a joint work with Tomasz Weiss and Lyubomyr Zdomskyy.
The research was funded by the National Science Centre, Poland and the
Austrian Science Found under the Weave-UNISONO call in the Weave
programme, project: Set-theoretic aspects of topological selections
2021/03/Y/ST1/00122