On the graph condition regarding the FF-inverse cover problem

In their paper titled "On $F$-inverse covers of inverse monoids", Auinger and Szendrei have shown that every finite inverse monoid has an $F$-inverse cover if and only if each finite graph admits a locally finite group variety with a certain property. We study this property and prove that the class of graphs for which a given group variety has the required property is closed downwards in the minor ordering, and can therefore be described by forbidden minors. We find these forbidden minors for all varieties of Abelian groups, thus describing the graphs for which such a group variety satisfies the above mentioned condition

Daganatos sejtek rezisztenciáját gátló vegyületek fejlesztése = Development of compounds targeting multidrug resistant cancer

A korszerű daganatellenes terápia jelentős sikerei ellenére a kemoterápiával szemben fellépő rezisztencia (multidrog rezisztencia, MDR) továbbra is megoldásra váró klinikai kihívás. Számos rosszindulatú megbetegedés, valamint az áttétet adó daganatok hatékony kezelése a terápia során rendszerint kialakuló MDR hatás miatt a mai napig nem megoldott. A rezisztens fenotípus gyakran társul az ABC-transzporterek családjába tartozó fehérjék emelkedett expressziójával. E család legismertebb képviselője a Pgp (ABCB1) membránfehérje, mely az ATP energiáját felhasználva megakadályozza a citosztatikus vegyületek sejten belüli felhalmozódását. A farmakogenomikai megközelítés révén lehetővé válik a személyre szabott gyógyítás, a daganatos megbetegedések molekuláris profiljához igazított kemoterápiás kezelés. A kutatás fő célja az volt, hogy a korábban kidolgozott farmakogenomikai módszer segítségével olyan ?MDR-inverz? vegyületeket fedezzünk fel, melyek szelektíven elpusztítják az egyébként multidrog rezisztens sejteket. Fontosabb eredményeink a következő pontokban összegezhetők: (i) módszerünk számos további MDR-inverz vegyületet azonosított; (ii) a szerkezetek analízise lehetővé tette QSAR modellek felállítását; (iii) javaslatot tettünk a vegyületek hatásmechanizmusára. Távlati tervünk, hogy a megismert MDR-inverz vegyületekből kiindulva originális gyógyszerkutatást folytassunk a rákos sejteket szelektíven pusztító molekulák preklinikai fejlesztése céljából. | Despite considerable advances in drug discovery, resistance to chemotherapy confounds the effective treatment of cancer patients. Cancer cells can become resistant to a single drug or they may acquire broad cross-resistance to mechanistically and structurally unrelated drugs (multidrug resistance (MDR)). ATP-Binding Cassette (ABC) proteins comprise the largest protein family, many members of which are of immediate medical importance and relevant to human health. The application of pharmacogenetics has the potential to improve the management of patients, particularly by providing the molecular basis for choosing among the increasing number of chemotherapeutic agents available for the treatment. The major aim of this project was to apply a pharmacogenomic approach to discover ?MDR-inverse? compounds that selectively kill multidrug resistant cancer cells. The results can be summarized as follows: (i) we identified a series of MDR-inverse compounds; (ii) we delineated structural features associated with their cytotoxic activity; (iii) we proposed a mechanism of action for the toxicity of newly identified MDR1-inverse compounds. Our future aim is to establish the framework for the preclinical development of the most promising MDR-inverse molecules, setting the stage for a fresh therapeutic approach that may eventually translate into improved patient care

Inverse monoids and immersions of 2-complexes

It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathcal C$, we need to replace the fundamental group of $\mathcal C$ by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex

Inverse monoids of partial graph automorphisms

A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well

Convolution of second order linear recursive sequences II.

summary:We continue the investigation of convolutions of second order linear recursive sequences (see the first part in [1]). In this paper, we focus on the case when the characteristic polynomials of the sequences have common root