A quadrotor unmanned aerial vehicle (UAV) must achieve desired flight missions despite internal uncertainties and external disturbances. This paper proposes an adaptive trajectory tracking control method that attenuates unknown uncertainties and disturbances. Although the quadrotor is underactuated, a fully actuated controller is designed using backstepping control. To avoid repeated derivatives of control inputs, a dynamic surface method introduces a filter and auxiliary controller. Lyapunov criteria guide adaptive laws for tuning controller gain and filters. A low-power observer is integrated for state estimation. Additionally, a disturbance observer is developed and combined with the control scheme to handle unknown disturbances. Simulations on a DJI F450 quadrotor demonstrate that the proposed control algorithm offers strong trajectory-tracking performance and system stability under multiple uncertainties and external disturbances during flight.

We present a construction of left braces of right nilpotency class at most two based on suitable actions of an abelian group on itself with an invariance condition. This construction allows us to recover the construction of a free right nilpotent one-generated left brace of class two.

Let $B^{H}$ be a d-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f\,:\,[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process $B^{H}+f$. It aims to highlight how each component f, E and F is involved in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$. When F is a singleton and f is a general measurable drift, some new estimates are obtained for the last probability by means of suitable Hausdorff measure and capacity of the graph $Gr_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f)\vert_E$ (the restriction of $B^H+f$ to the subset $E\subset (0,\infty)$).

The transport industry of Ukraine is an integral part of its economy. According to the National Transport Strategy of Ukraine, a critical strategic goal is to enhance transport safety. Currently, there is a gap in mobile devices capable of automatically measuring slopes and evenness of both runways and road surfaces in two coordinates. This paper addresses the creation of new methods for assessing longitudinal and transverse slopes using micromechanical systems. The study highlights international experiences, presents practical applications and proposes strategies for overcoming implementation challenges. A detailed roadmap for deployment and further improvements is provided.

This paper analyses the performance of the Australian and New Zealand Satellite-Based Augmentation System (Aus-NZ SBAS) test-bed to evaluate its use in civil aviation applications with a focus on dual-frequency multi-constellation (DFMC) signals. The Aus-NZ SBAS test-bed performance metrics were determined using kinematic data recorded in flight across a variety of environments and operational conditions. A total of 14 tests adding up to 32 h of flight were evaluated. Flight test data were processed in both the L1 SBAS and DFMC SBAS modes supported by the test-bed broadcasts. The performance results are reviewed regarding accuracy, availability and integrity metrics and compared with the requirement thresholds defined by the International Civil Aviation Organisation (ICAO) for Precision Approach (PA) flight operations. The experimentation performed does not allow continuity assessment as specified in the standard due to a long-term statistical requirement and inherent limitations imposed by the reference station network. Analysis of flight test results shows that DFMC SBAS provides several performance improvements over single-frequency SBAS, tightening both horizontal and vertical protection levels and resulting in greater service availability during the approach.

Given a self-morphism $\phi$ on a projective variety defined over a number field k, we prove two results which bound the largest iterate of $\phi$ whose evaluation at P is quasi-integral with respect to a divisor D, uniformly across P defined over a field of bounded degree over k. The first result applies when the pullback of D by some iterate of $\phi$ breaks up into enough irreducible components which are numerical multiples of each other. The proof uses Le’s algebraic-point version of a result of Ji–Yan–Yu, which is based on Schmidt subspace theorem. The second result applies more generally but relies on a deep conjecture by Vojta for algebraic points. The second result is an extension of a recent result of Matsuzawa, based on the theory of asymptotic multiplicity. Both results are generalisations of Hsia–Silverman, which treated the case of morphisms on ${\mathbb{P}}^1$.

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for $k=3$. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.