Related papers: Exploring Steinitz-Rademacher polyhedra: A challen…
We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle $\mathcal{R}$, and outputs an explicit system of piecewise linear motion planners for…
We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant,…
There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer.…
The aim of this paper is to use the framework of incidence geometry to develop a theory that permits to model both the inner and outer automorphisms of a group G simultaneously. More precisely, to any group G, we attempt to associate an…
We develop the Reidemeister theory of iterations of a group endomorphism $\varphi$ and study the asymptotic behavior of the sequence of the Reidemeister numbers of iterations $\{R(\varphi^k)\}$, the essential periodic $[\varphi]$-orbits and…
Recent advances in Large Language Models have led to Large Reasoning Models, which produce step-by-step reasoning traces. These traces offer insight into how models think and their goals, improving explainability and helping users follow…
Recent large-scale reasoning models have achieved state-of-the-art performance on challenging mathematical benchmarks, yet the internal mechanisms underlying their success remain poorly understood. In this work, we introduce the notion of a…
This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…
We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but…
Across various scientific and engineering domains, a growing interest in flexible and deployable structures is becoming evident. These structures facilitate seamless transitions between distinct states of shape and find broad applicability…
A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique…
Understanding the notion of a model is not always easy in logic courses. Hence, tools such as Euler diagrams are frequently applied as informal illustrations of set-theoretical models. We formally investigate Euler diagrams as an…
Results of number of geometric operations (often used in technical practise, as e.g. the operation of blending) are in many cases surfaces described implicitly. Then it is a challenging task to recognize the type of the obtained surface,…
We present detailed simulations addressing recent electronic interference experiments, where a metallic gate is used to locally modify the Fermi wave-length of the charge carriers. Our numerical calculations are based on a solution of the…
In additive combinatorics, Erd\"{o}s-Szemer\'{e}di Conjecture is an important conjecture. It can be applied to many fields, such as number theory, harmonic analysis, incidence geometry, and so on. Additionally, its statement is quite easy…
The statistics of meander and related problems are studied as particular realizations of compact polymer chain foldings. This paper presents a general discussion of these topics, with a particular emphasis on three points: (i) the use of a…
We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of lambda theories. Relying on the notion of easy intersection type theory we successfully build a filter model in which the interpretation…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
The promise of machine learning has been explored in a variety of scientific disciplines in the last few years, however, its application on first-principles based computationally expensive tools is still in nascent stage. Even with the…
Logical reasoning is central to complex human activities, such as thinking, debating, and planning; it is also a central component of many AI systems as well. In this paper, we investigate the extent to which encoder-only transformer…