Mathematics Days in Sofia – 2026

Section “Analysis”

Participants

Invited Speaker

  • Milena Stanislavova, University of Alabama at Birmingham, USA

Contributors

  • Dragan Djordjevic, University of Niš, Serbia
  • Emilia Bazhlekova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
  • Hristo Sendov, University of Western Ontario, USA
  • Mikhail Shkolnikov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
  • Miroslav Marinov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
  • Pando Georgiev, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
  • Virginia Kiryakova, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria
  • Yulian Tsankov, Sofia University “St. Kliment Ohridski”, Bulgaria

Program and Abstracts

In this work we examine the dynamics of a model for oscillons in 1-dimensional space-time field theories with a cubic nonlinearity. We utilize a reduction of the model to first and third harmonics, which leads to a reduced partial differential equation (PDE) system whose steady states are candidates for the original PDE oscillons. We analyze the steady states of this model and their stability using tools from index theory. We develop suitable functionals needed for the study of such stationary states, as well as an analogue of the famous Vakhitov-Kolokolov criterion for a quantity whose change of monotonicity reflects a change of stability. Then, we test the relevant predictions, over the full range of oscillon frequencies, through systematic numerical computations of both the reduced model, its steady states and stability, and also of the original PDE model, identifying its time-periodic oscillon solution. Our results yield some significant connections with previous studies, but also some fundamental new insights both on the reduced system and the dynamics of the original system.

Weconsider inversions of unbounded linear operators on Banach and Hilbert spaces. Results related to the ordinary inverse as well as to the various types of generalized inverses will be presented. Some applications to operator equations and spectral theory will be explined.

The multinomial Mittag-Leffler function plays a crucial role in the study of evolution equations with multiple fractional derivatives. In this talk the Prabhakar-type generalization of the multinomial Mittag-Leffler function is discussed. Its complete monotonicity is established by the use of Bernstein functions’ technique. Based on this property, generalized fractional calculus operators with multinomial Prabhakar-type singular Sonin kernels are introduced and studied.

We extend bounds, proved by R. C. Thompson in 1966, on the sum of the j-th largest eigenvalues of the (n−1)×(n−1) principal matrices of an n×n Hermitian matrix. Our bounds are stronger than just summing up Thompson’s bounds. We achieve the extensions as a corollary of a more general result giving bounds on the zeros of the generalized derivatives of polynomials with real roots. We use the extended bounds to obtain majorization relations hips between the eigenvalues of all m×m principal matrices of an n×n Hermitian matrix. These majorization relationships imply both a well-known majorization result by Schur and the well-known Szasz’s inequalities.

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In this work we find an explicit solution of a nonlocal boundary value problem for the one dimensional fractional diffusion equation with the Caputo time-derivative. The specified boundary conditions lead to two sequences of eigenvalues of the corresponding spectral problem. We prove uniqueness and existence of a classical solution. Then, using the operational calculus approach of Dimovski, we obtain an explicit Duhamel-type representation, which contains non-classical convolution products.