Mathematical modeling of cancer cells populations behavior

Abstract:  The talk will describe how mathematical modeling of biological processes can address the important questions which can not be solved by experimental approaches. Two examples from cancer biology will be discussed: (1) the phenomenon of Cancer Stem Cells population stabilization and (2) the phenomenon of post-irradiation induction of Cancer Stem Cells. Cancer Stem Cells (CSC) is a small population in heterogeneous cancer cells population which is resistant to conventional cancer therapies (chemo- or radio therapies), and thus responsible for tumor relapses. The model considers different possible modes of stem and non-stem cancer cells population behavior in different conditions with different sets of experimentally measured parameters. The results obtained by the model allow determination of time-varying corridors of probabilities for different cell fates of CSC in a given experimental system, and determination of cell-cell communication factors influencing these time-varying probabilities of cell behavior scenarios. The analysis of these results provides as theoretical insights into the phenomena of CSC behavior, so a set of biomedical suggestions, essential for cancer therapy.

Simple Elliptic Singularities and Generalized Slodowy Slices

Abstract: We will present a relation between the family of simple elliptic singularities which are complete intersections and finite dimensional Lie algebras. We will extend the concept of Slodowy slices to the case of these singularities.

On proper holomorphic maps

Abstract: The image of a proper holomorphic map from an open Riemann surface is an analytic subvariety of the ambient space. We will explain a method for constructing such maps, and we will present some recent related results.

From Krasner’s corpoid and Bourbaki-Krasner’s graded rings to Krasner-Vuković’s paragraded rings

Abstract:

Marc Krasner investigating valued fields and observing a connection between them and their valuation rings using equivalence of valuations came to the fundamental notion of a corpoid in a series C.R. of Academy of Sciences of Paris notes (1944-45), which represents the origin of the development forerunner of the general graded theory – a theory of homogroupoids, anneids and moduloids – more general than one given by Bourbaki (Algèbre, Chap II Paris, 1962), since neither the associativity, nor the commutativity nor the existence of a neutral element is assumed in the set of grades, where corpoid, as a special case of an anneid, is viewed
as a homogeneous part of a graded field. So, the abstract notion of corpoid led Krasner to a development of general graded structures (Anneaux gradués géneraux, 1980)

Since the category of graded structures (groups, rings, modules) has no the property of closure with respect to the direct product and the direct sum, it was for M. Krasner and myself a motivation to go further with generalizations and to introduce the notion of paragraded structures: groups, rings, modules, which appeared for the first time, late in 1980s, in a series Krasner- Vuković’s Proceedings Japan Academy notes and in our monograph Structures Paragraduée (groupes, anneaux, modules) (Queen’s Papers in pure and Applied Mathematics, Kingston, ONT. Canada). As we have already noted, these structures resolves the formentioned problem of closure. In this presentation we are particularly interested in the structure of a paragraded ring and their radicals, and we are going to present some results on the different types of paragraded radicals, introduced in my joint papers with my pupils, in the class of the paragraded rings. More precizely we will present prime and Jacobson radicals, discuss the general Kurosh-Amitsur theory of radicals of paragraded rings, characterise paragraded normal radicals and prove that all special paragraded radicals of paragraded rings can be described by the appropriate class of their paragraded modules and finally provide information about ongoing work regarding paragraded Brown-McCoy radicals, i.e. inspired by Halberstadt results about Jacobson’s radicals of graded rings, to introduce two versions of paragraded Brown-McCoy radicals: the Brown- McCoy radical and the large Brown-McCoy radical of a paragraded rings and to prove that the large Brown-McCoy radical of a paragraded ring coincides with the largest homogeneous ideal contained in the classical Brown-McCoy radical of that ring.

Application of fractional calculus in solving some deterministic and stochastic PDEs

Abstract: Fractional calculus has proven to be very useful in studying many real-life problems. It is especially very effective when problems of consideration include memory effect issues. Therefore, the application of fractional calculus is widespread in many different scientific areas. Here we present a specific approach in solving some PDEs by using fractional calculus. We consider reaction-advection-diffusion equations with space fractional derivatives, inhomogeneous fractional evolution equations, time and time-space fractional wave equations with variable coefficients, and finally, stochastic fractional heat equation with variable thermal conductivity
and multiplicative noise.

In all cases mentioned, we prove that there exists a unique solution to the problem within a certain Colombeau generalized function space. These results are obtained by using the theory of generalized uniformly continuous semigroups of operators.