Mathematical Colloquium

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.

We present several methods for constructing elliptic curves with a given torsion group and high (in some case highest known) rank over Q and Q(t). One motivation for studying such curves comes from their application in Lenstra\\\'s elliptic curve factorization method. For some of the torsion groups, the constructions over Q(t) use elliptic curves induced by rational Diophantine triples, in particular, triples contained in parametric families of rational Diophantine sextuples. In finding curves over Q with a higher rank, we search for suitable specializations by using Mestre-Nagao sums and then try to compute the rank by available software. Let us mention that similar methods were used recently by Elkies and Klagsbrun in finding the curve of record rank 29. We will also compare our results in the construction of infinite families of elliptic curves with large rank and given torsion with the recent heuristics by Park, Poonen, Voight and Wood which predicts the upper bounds for the rank of such families of elliptic curves.
This is a joint work with Juan Carlos Peral and partly with Matija Kazalicki.

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.

Heterogeneous robot cooperation, robot manipulation of various objects, networked control systems in smart cities, path planning for heterogeneous robots in dynamic environments are examples of systems where adaptive optimal control can be successfully employed. In this talk, problems and results of these algorithms will be presented which enabled these systems to adapt to changing environments while maintaining the sub-optimal performance. Current research activities related to these topics will be presented.

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.

Computer criminal acts are criminal acts which is happening almost daily and therefore it is necessary to know how to treat them national criminal legislation. Of course, it treats them like other criminal acts and punishes them as prescribed by the Criminal Code. In order to computer criminal act (and other criminal acts prescribed by the Criminal Code) to have been realized, must be fulfilled certain assumptions, and when it be fulfilled, criminal proceedings may begin.