For the English version of the introduction click here
A BME Matematika Intézet Analízis és Differenciálegyenletek Tanszékének közös Alkalmazott Analízis Szemináriuma 2016. őszén indult Faragó István (Differenciálegyenletek Tanszék) kezdeményezésére az MTA-ELTE Numerikus Analízis és Nagy Hálózatok Kutatócsoporttal együttműködésben. A szeminárium célja, hogy elősegítse egy alkalmazott analízissel (funkcionálanalízis, differenciálegyenletek, numerikus módszerek) foglalkozó kutatói kör kialakítását az intézeten belül. A szemináriummal fórumot szeretnénk biztosítani az alkalmazott analízissel foglalkozó matematikusok és az analízist alkalmazó kutatók számára az együtt gondolkodásra. További cél az érdeklődő hallgatók (MSc, PhD) bevonása a kutatói munkába.
Szemináriumunk 2017-től felvette a Farkas Miklós Alkalmazott Analízis Szeminárium nevet. Ezzel szeretnénk emléket állítani egyetemünk egykori tanszékvezető matematikaprofesszorának, aki elindította egyetemünkön a matematikus-mérnök képzést, és a stabilitáselmélet valamint a biomatematika terén elért jelentős tudományos eredményeivel ill. könyveivel nagyban hozzájárult az alkalmazott matematika erősödéséhez.
A covid járvány miatt három évig szünetelt a szeminárium. 2023. szeptemberétől indult újra a korábban említett szervezetek utódjainak (BME Analízis és Operációkutatás Tanszék, HUN-REN-ELTE Numerikus Analízis és Nagy Hálózatok Kutatócsoport) szervezésében.
The talks are in English on a regular basis. / Az előadásokat angol nyelven tartjuk.
Organizers / Szervezők: 
Faragó István1,2,3, 
Karátson János1,2,3 ,
Horváth Róbert1,3 ,
Mincsovics Miklós1,3 , 
Svantnerné Sebestyén Gabriella1,3 (1BME, 2ELTE, 3HUN-REN-ELTE NUMNET)
Tentative program of the semester: 5 December - Research reports of PhD students
Next seminar: 28 November, Thursday, 10:15, BME H306
Ágnes Kovács1,2, 
Tamás Pfeil2 (1Department of Biostatistics (University of Veterinary Medicine), 2Department of Applied Analysis and Computational Mathematics (ELTE))
Mathematical modeling of immunochemical reactions
The enzyme-linked immunosorbent assay (ELISA) is a commonly employed technique in laboratory blood testing that uses antibody-antigen reactions to detect specific biomolecules. Recently immunologists have developed a dual-titration variant of ELISA designed to quantify antibody reactivity more accurately. Our goal is to develope and analyze mathematical models based on ordinary differential equations to describe the dynamics of these immunochemical reactions. Additionally, we use these models to evaluate lab measurement data obtained from the newly introduced variant of the ELISA method.
Previous seminars
14 November 2024
Giuseppe Habib (Department of Applied Mechanics BME, MTA-BME Lendület Global Dynamics Research Group)
Challenges in Global Dynamics: Dynamical Integrity Assessment and (small)-Data-Driven Prediction
One of the key challenges of nonlinear dynamical systems is that it can be difficult to exclude the possibility of overlooked behaviors. This is because properties such as the stability of a steady-state solution are only local, meaning they only hold true in a small region around the solution in the phase space. This presents a significant challenge from an engineering standpoint, as stability analysis alone is insufficient for determining a system's robustness against external perturbations, a property known as dynamical integrity. Accurate dynamical integrity assessment is crucial for ensuring safe operation. However, existing methods for evaluating dynamical integrity often face challenges related to computational efficiency and complexity.
This seminar will provide examples of locally stable dynamical systems and an overview of current approaches to dynamical integrity analysis. Then, a new numerical methodology for assessing dynamical integrity will be introduced, demonstrating its effectiveness through various examples. In the second part of the seminar, an alternative method that leverages critical slowing down near steady-state solutions to estimate dynamical integrity will be presented. Finally, a predictive technique for identifying regions with limited basin stability will be discussed.Related papers:
- Habib, G. (2021). Dynamical integrity assessment of stable equilibria: a new rapid iterative procedure. Nonlinear Dynamics, 106(3), 2073-2096.
- Szaksz, B., Stepan, G., & Habib, G. (2024). Dynamical integrity estimation in time delayed systems: a rapid iterative algorithm. Journal of Sound and Vibration, 571, 118045.
- Habib, G. (2023). Predicting saddle-node bifurcations using transient dynamics: a model-free approach. Nonlinear Dynamics, 111(22), 20579-20596.
7 November 2024
György Károlyi (Department of Nuclear Techniques, BME)
Dissipative dynamical systems, or, the transient charm of decay
In undriven dissipative systems all motion decays since dissipation continually decreases the available mechanical energy. Chaotic motion can only show up transiently. Traditional transient chaos is, however, caused by the presence of an infinity of unstable orbits. In the lack of these, chaos in undriven dissipative systems is of another type: it is termed doubly transient chaos as the strength of transient chaos is diminishing in time, and ceases asymptotically. To characterize the behavior of such systems, the snapshot view has been suggested, but it does not lead to a clear characterization of e.g. the fractality of the boundary between the basins of attraction. We suggest that a view based on equal energy levels might be a better choice.
17 October 2024
Beatrix Oroszi (Centre for Epidemiology and Surveillance, Semmelweis University)
Epidemiology of communicable diseases: dynamics of disease transmission and epidemic control from the perspective of epidemic modelling
In this interactive seminar, we will examine the epidemiological aspects that are integral to the development of models, as well as the epidemiological studies and other routine data collections that can provide critical input parameters. It is essential to gain an understanding of the limitations of such data collections, which will be illustrated by real-world examples. The conditions for the applicability of epidemiological models to public health decision making will then be discussed.
These issues will be explored through the lens of a realistic modelling example of a hypothetical but plausible pandemic emergency caused by a human-to-human respiratory virus. This will provide a context for understanding the theoretical underpinnings of epidemiology in mathematical modelling. At the end of the seminar, we will present our imaginary decision-makers with evidence-based answers to questions about epidemiological interventions in the example. We will recognise that, in certain circumstances, epidemiological models can inform specific decisions that have a significant impact on people's lives. We will therefore examine in detail the specific limitations of the epidemiological model and its outputs, and propose strategies to improve the quality of model outputs.
3 October 2024
Miklós Mincsovics (BME, Department of Analysis and Operations Research)
What makes a good student project?
This is the question we explore in the topic of differential equations and numerical methods. After some introductory thoughts, we will dive deeper into two possible projects: modeling addiction, and modeling the motion of a glider. We will investigate them primarily from an educational point of view, but we will also look at the results achieved so far as research.
19 September 2024
Csaba Farkas (Sapientia - Hungarian University of Transylvania, Department of Mathematics-Informatics)
Compact Sobolev embeddings on non-compact manifolds with applications
In this talk, we collect recent achievements obtained in the theory of geometric analysis, by exploiting elements from the calculus of variations, partial differential equations, Riemann-Finsler geometry and group theory. First, for a given complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259--268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). Then we give some applications for such results, first, we investigate the existence of multiple solutions for a low-dimensional Dirichlet eigenvalue problem with non-standard growth on a Randers space. Variational techniques, depending in particular on a minimax theorem and the previous compact embeddings, are employed to establish the existence of multiple group invariant solutions. After that, we deal with a quasilinear elliptic equation involving a critical Sobolev exponent on non-compact Randers spaces. Under very general assumptions on the perturbation, we prove the existence of a non-trivial solution. The approach is based on the direct methods of the calculus of variations. We end this talk with the multiplicity of singular fourth-order Schrödinger equation on Hadamard manifolds.
Presentations in 2023/24, 2019/20, 2018/19, 2017/18, 2016/17