Matematički kolokvij

Despite the impressive progress in the practical use of neural networks, our mathematical understanding of their inner mechanisms and why they generalize so well remains incomplete. Beyond classical results such as the universal approximation theorem by Hornik, Stinchcombe, and White (1989), a key question is which functions can be represented exactly by neural networks, as explored by Hertrich, Basu, Di Summa, and Skutella (2022). This talk focuses on the effect of sparsity within the computational graph of maxout neural networks, introducing an additional parameter for assessing expressivity that complements traditional depth-width trade-off considerations. Our work relies on a duality between positively homogeneous continuous piecewise-linear functions and a class of virtual polytopes --- formal differences of convex polytopes. At any neuron, rather than focusing on the functions that can be computed, we consider the polytopes that can emerge. Tracking how the dimensions of these polytopes evolve from layer to layer offers new insights into neural network expressivity. One key insight is that sparsity has a decisive restrictive effect on the expressivity of neural networks --- one that cannot be fully compensated by increasing a networks\\\' width.

Na ovom predavanju prikazat ćemo nekoliko metoda za konstruiranje eliptičkih krivulja sa zadanom torzijskom grupom i velikim (u nekim slučajevima i rekordnim) rangom nad Q i Q(t). Jedna motivacija za proučavanje takvih krivulja dolazi od njihove primjene u Lenstrinoj metodi faktorizacije pomoću eliptičkih krivulja. Za neke od torzijskih grupa, konstrukcije nad Q(t) koriste eliptičke krivulje inducirane racionalnim Diofantovim trojkama, posebno trojke sadržane u parametarskim familijama racionalnih Diofantovih šestorki. U pronalaženju krivulja iznad Q s još većim rangom, tražimo odgovarajuće specijalizacije koristeći Mestre-Nagaove sume te zatim pokušavamo izračunati rang pomoću dostupnog specijaliziranog softvera. Spomenimo da su slične metode koristili Elkies i Klagsbrun u nedavnom nalaženju krivulje rekordnog ranga 29. Također ćemo usporediti naše rezultate u konstrukciji beskonačnih familija eliptičkih krivulja velikog ranga i zadane torzije s heuristikom iz članka Park-Poonen-Voight-Wooda koja predviđa gornje granice za rang takvih familija eliptičkih krivulja.
Rezultati predstavljaju zajednički rad s Juanom Carlosom Peralom te dijelom s Matijom Kazalickim.

Balanced truncation is a classical approach to model reduction that has long been the ”gold standard” for high fidelity reduced order modeling for large scale linear dynamical systems. I will discuss a novel data-driven reformulation of this approach that does not require intrusive access to internal system dynamics, that is, knowledge of an original system realization is unnecessary. Instead, observations are accumulated of the system response - either sampling the transfer function evaluated at complex driving frequencies, or sampling (in time) the system’s impulse response. Notably, we do not require access to state-space trajectory snapshots as would be found in POD-type methods, nor do we approximate any system Grammians. A variety of numerical experiments are provided that verify the effectiveness of this approach.

Kooperativni heterogeni roboti, robotska manipulacija različitim objektima, umreženi sustavi upravljanja, planiranje putanja heterogenih robota u dinamičnim okruženjima primjeri su sustava u kojima se adaptivno optimalno upravljanje uspješno koristi. U sklopu ovog predavanja predstavit će se problemi i rezultati algoritama koji omogućuju sustavima prilagodbu u promjenjivim okruženjima uz zadržavanje suboptimalnosti. Također će se prezentirati trenutne istraživačke aktivnosti povezane s navedenim temama.

In this talk we will present a survey of the main topics and results from the recently published research monograph "Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions" by the coauthors Michel L. Lapidus, Goran Radunović and Darko Žubrinić. In the monograph, a new theory of fractal zeta functions and complex dimensions valid for arbitrary subsets of Euclidean spaces has been developed. This theory provides a significant extension of the existing theory of zeta functions for fractal strings, i.e., "fractal" subsets of the real line. Two new classes of fractal zeta functions,
namely, the distance and the tube zeta function, are introduced and their properties are investigated. In fact, for a given subset, its complex dimensions are defined as poles or even more general singularities of the corresponding fractal zeta function. The complex dimensions themselves, although defined analytically, have a deep connection with the fractal geometry of the given set.

Računalna kaznena djela su kaznena djela koja se događaju gotovo svakodnevno i zbog toga valja znati kako ih tretira nacionalno kazneno zakonodavstvo. Naravno, tretira ih kao i ostala kaznena djela, a kažnjava ih onako kako je to propisano Kaznenim zakonom. Da bi se računalno kazneno djelo (i ostala kaznena djela propisana Kaznenim zakonom) smatralo ostvarenim, moraju biti ispunjene određene pretpostavke, a kada su one ispunjene, kazneni postupak može početi.