Doctoral School Mathematics and Scientific Computing

Abstract: In this study, we consider a degenerate reaction-diffusion system that models the interactions among the prey, the predator, and the predator's natural enemy. The global existence and the uniform boundedness in time of the solution are proved. Next, we present the stability analysis of the constant equilibria, in which we identify a sufficient condition under which Turing instability does not occur and prove the nonlinear stability of these equilibria.

Abstract: A classical result due to Banach says that for a basis in a Banach space, the partial sum operators are uniformly bounded. A basis with respect to a filter is a generalization of a basis, where the ordinary convergence of series is replaced by convergence of partial sums with respect to a filter. The talk is devoted to the following natural question: given a free filter  F, what restrictions on the norms of partial sums of an F-basis does one obtain, including their possible rates of growth?

Abstract: We study the maximal gap among the dilates {α a_n}_{n∈ N} mod 1 for lacunary sequences satisfying the Hadamard gap condition.   For any fixed dimension d>1 and Lebesgue-almost every α∈[0,1]^d, every convex body of volume at least (log N)²/N contains a point of this dilated set for all large N.  We extend the result to general measures satisfying a suitable Fourier decay condition.   The bounds are optimal up to logarithmic factors. Our theorem recovers as a special case the recent result in dimension d=1, [S., Adv. Math., 2024]