arXiv subjects

Aleš Návrat

Publications and source records attributed to Aleš Návrat.

7 recordsLinked to original sources

arXiv2025

Understanding the Geometry of Faulted Power Systems under High Penetration of Inverter-Based Resources via Ellipse Fitting and Geometric Algebra

Power systems with high penetration of inverter-based resources (IBR) present significant challenges for conventional protection schemes, with traditional distance protection methods failing to detect line-to-line faults during asymmetric conditions. This paper presents a methodology for electrical fault detection and classification using ellipse fitting and geometric algebra applied to voltage and current space curves. The approach characterizes electrical faults by fitting ellipses to voltage vector data, enabling fault detection with only a quarter-cycle. The method employs bivector components for line-to-ground fault classification, while ellipse parameters identify line-to-line and three-phase faults. The geometric representation preserves voltage or current curve shapes in three-dimensional space, overcoming Clarke transform limitations when zero-sequence components are present. Validation using simulations and laboratory experiments demonstrates accurate fault identification and magnitude estimation, providing enhanced power system protection capabilities.

eess.SY↗

arXiv2024

Sub-Riemannian optimal synthesis for Carnot groups with the structure of a path geometry

This paper explicitly constructs the complete set of optimal sub-Riemannian geodesics starting from a point for certain Carnot groups of step two. These are groups of dimension 2n+1 equipped with a left-invariant distribution of dimension n+1 such that at each point, there is a unique direction defining a nontrivial Lie bracket. A suitable explicit expression of geodesics, together with symmetries of the structure, allows us to identify the cut time and the cut locus by applying the so-called extended Hadamard technique.

math.DG↗

arXiv2023

Quantization of Two- and Three-player Cooperative Games Based on QRA

In this paper, a novel quantization scheme for cooperative games is proposed. The considered circuit is inspired by the Eisert-Wilkens-Lewenstein protocol modified to represent cooperation between players and extended to $3$-qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. To distribute players' payoffs after the measurement, the expected value of the Shapley value with respect to quantum probabilities is employed. We study how entanglement, representing the level of pre-agreement between players, affects the final distribution of utility. The paper also demonstrates how all necessary calculations can be automatized using the Quantum Register Algebra and GAALOP software.

math.QA↗

arXiv2022

Quantum Register Algebra: the mathematical language for quantum computing

We introduce Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present GAALOP (Geometric Algebra Algorithms Optimizer) implementation of our approach. Using the QRA basis vectors definitions given in Section 4 and the framework based on the de Witt basis presented in Section 5, we are able to fully describe and compute with QRA in GAALOP using the geometric product. We illustrate the intuitiveness of this computation by presenting the QRA form for the well known SWAP operation on a two qubit register.

quant-ph↗

arXiv2022

On symmetries of a sub--Riemannian structure with growth vector $(4,7)$

We study symmetries of specific left--invariant sub--Riemannian structure with filtration $(4,7)$ and their impact on sub--Riemannian geodesics of corresponding control problem. We show that there are two very different types of geodesics, they either do not intersect the fixed point set of symmetries or are contained in this set for all times. We use the symmetry reduction to study properties of geodesics.

math.DG↗

arXiv2015

A nonstandard operator for almost Grassmannian geometries with a torsion

A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geometry. It is obtained by translating a Grassmannian analogue of the Laplace operator to a certain tractor bundle with a specific tractor connection, which is not normal in general.

math.DG↗

arXiv2014

Nonstandard operators in Grassmannian geometry

We present a construction of curved analogues of the nonstandard operators on Grassmannians parallel to the construction of the Paneitz operator via the curved Casimir operator, but technically more demanding. In particular, the construction breaks down in the presence of torsion. In the second part, we prove that the nonstandard operators are not strongly invariant.

math.DG↗