Search Results (11)

We develop a fully gauge-invariant analysis of gravitational-wave polarizations in metric $f\left(R\right)$ gravity with a particular focus on the modified Starobinsky model $f\left(R\right)=R+\alpha R^2-2\mathsf{\Lambda }$, whose constant-curvature solution $R_d=4\mathsf{\Lambda }$ provides a natural de Sitter background for both early- and late-time cosmology. Linearizing the field equations around this background, we derive the Klein–Gordon equation for the curvature perturbation $\delta R$ and show that the scalar propagating mode acquires a mass $m_{\psi }^2=1/\left(6\alpha \right)$, highlighting how the same scalar degree of freedom governs inflationary dynamics at high curvature and the propagation of gravitational waves in the current accelerating Universe. Using the scalar–vector–tensor decomposition and a decomposition of the perturbed Ricci tensor, we obtain a set of fully gauge-invariant propagation equations that isolate the contributions of the scalar, vector, and tensor modes in the presence of matter. We find that the tensor sector retains the two transverse–traceless polarizations of General Relativity, while the scalar sector contains an additional massive scalar propagating degree of freedom, which manifests through breathing and longitudinal tidal responses depending on the wave regime and detector frame. Through the geodesic deviation equation—computed both in a local Minkowski patch and in fully covariant de Sitter form—we independently recover the same polarization content and identify its tidal signatures. The resulting framework connects the extra scalar polarization to cosmological observables: the massive scalar propagating mode sets the range of the fifth force, influences the time evolution of gravitational potentials, and affects the propagation and dispersion of gravitational waves on cosmological scales. This provides a unified, gauge-invariant link between gravitational-wave phenomenology and the cosmological implications of metric $f\left(R\right)$ gravity. Full article

This manuscript presents a comprehensive Lie symmetry analysis of the KdV-Burgers equation, a prototypical model for nonlinear wave dynamics incorporating dissipation and dispersion. We systematically derive its six-dimensional Lie algebra and construct an optimal system of one-dimensional subalgebras. This framework is used to perform a symmetry reduction, transforming the governing partial differential equation into a set of ordinary differential equations. A key contribution of this work is the identification and analysis of several non-trivial invariant solutions, including a new Galilean-boost-invariant solution related to an accelerating reference frame, which extends beyond standard traveling waves. Through a detailed physical interpretation supported by phase plane analysis and asymptotic methods, we elucidate how the mathematical symmetries directly manifest as fundamental physical behaviors. This reveals a clear classification of distinct wave regimes—from monotonic and oscillatory shocks to solitary wave trains governed by the interplay between nonlinearity, dissipation and dispersion. The numerical validation verify the accuracy and physical relevance of the derived invariant solutions, with errors less than $0.5\%$ in the Burgers limit and $3.2\%$ in the weak dissipation regime. Our work establishes a direct link between the model’s symmetry structure and its observable dynamics, providing a unified framework validated both analytically and through the examination of universal scaling laws. The results offer profound insights applicable to fields ranging from plasma physics and hydrodynamics to nonlinear acoustics. Full article

Research on surfaces generated by curves plays a central role in linking differential geometry with physical applications, especially following Hasimoto’s transformation and the development of Hasimoto-inspired surface models. In this work, we introduce a new class of such surfaces, referred to as RM Hasimoto surfaces, constructed by employing the rotation-minimizing (RM) Darboux frame along both timelike and spacelike curves in Minkowski 3-space ${\mathbb{E}}_1^3$. In contrast to the classical Hasimoto surfaces defined via the Frenet or standard Darboux frames, the RM approach eliminates torsional difficulties and reduces redundant rotational effects. This leads to more straightforward expressions for the first and second fundamental forms, as well as for the Gaussian and mean curvatures, and facilitates a clear classification of key parameter curves. Furthermore, we establish the associated evolution equations, analyze the resulting geometric invariants, and present explicit examples based on timelike and spacelike generating curves. The findings show that adopting the RM Darboux frame provides greater transparency in Lorentzian surface geometry, yielding sharper characterizations and offering new perspectives on relativistic vortex filaments, magnetic field structures, and soliton behavior. Thus, the RM framework opens a promising direction for both theoretical studies and practical applications of surface geometry in Minkowski space. Full article

The graph theory-based approach to the three-body problem is introduced. Vectors of linear and angular momenta of the particles form the vertices of the graph. Scalar products of the vectors of the linear and angular momenta define the colors of the links connecting the vertices. The bi-colored, complete graph emerges. This graph is called the “momenta graph”. According to the Ramsey theorem, this graph contains at least one mono-chromatic triangle. This is true even for chaotic motion of three bodies; thus, illustrating the idea supplied by the Ramsey theory, total chaos is impossible. Coloring of the graph is independent on the rotation of frames; however, it is sensitive to Galilean transformations. The coloring of the momenta graph remains the same for general linear transformations of vectors with a positive-definite matrix. For a given motion, changing the order of the vertices does not change the number and distribution of monochromatic triangles. Symmetry of the momenta graph is addressed. The symmetry group remains the same for general linear transformation of vectors of the linear and angular momenta with a positive-definite matrix. Conditions defining conservation of the coloring of the momenta graph are addressed. The notion of the stereographic momenta graph is introduced. Shannon entropy of the momenta graph is calculated. The particular configurations of bodies are addressed, including the Lagrange configuration and the figure eight-shaped motion. The suggested approach is generalized for the quantum field theory with the Pauli–Lubanski pseudo-vector. The suggested coloring procedure is the Lorenz invariant. Full article

This paper presents a learning-assisted approach for state estimation of quadruped robots using observations of proprioceptive sensors, including multiple inertial measurement units (IMUs). Specifically, one body IMU and four additional IMUs attached to each calf link of the robot are used for sensing the dynamics of the body and legs, in addition to joint encoders. The extended Kalman filter (KF) is employed to fuse sensor data to estimate the robot’s states in the world frame and enhance the convergence of the extended KF (EKF). To circumvent the requirements for the measurements from the motion capture (mocap) system or other vision systems, the right-invariant EKF (RI-EKF) is extended to employ the foot IMU measurements for enhanced state estimation, and a learning-based approach is presented to estimate the vision system measurements for the EKF. One-dimensional convolutional neural networks (CNN) are leveraged to estimate required measurements using only the available proprioception data. Experiments on real data from a quadruped robot demonstrate that proprioception can be sufficient for state estimation. The proposed learning-assisted approach, which does not rely on data from vision systems, achieves competitive accuracy compared to EKF using mocap measurements and lower estimation errors than RI-EKF using multi-IMU measurements. Full article

Unmanned aerial vehicles (UAVs) offer an efficient solution for power grid maintenance, but collision avoidance during return flights is challenged by crossing power lines, especially for small drones with limited computational resources. Conventional visual systems struggle to detect thin, intricate power lines, which are often overlooked or misinterpreted. While deep learning methods have improved static power line detection in images, they still struggle with dynamic scenarios where collision risks are not detected in real time. Inspired by the hypothesis that the Lobula Giant Movement Detector (LGMD) distinguishes sparse and incoherent motion in the background by detecting continuous and clustered motion contours of the looming object, we propose a Scale-Invariant Looming Detector (SILD). SILD detects motion by preprocessing video frames, enhances motion regions using attention masks, and simulates biological arousal to recognize looming threats while suppressing noise. It also predicts impending collisions during high-speed flight and overcomes the limitations of motion vision to ensure consistent sensitivity to looming objects at different scales. We compare SILD with existing static power line detection techniques, including the Hough transform and D-LinkNet with a dilated convolution-based encoder–decoder architecture. Our results show that SILD strikes an effective balance between detection accuracy and real-time processing efficiency. It is well suited for UAV-based power line detection, where high precision and low-latency performance are essential. Furthermore, we evaluated the performance of the model under various conditions and successfully deployed it on a UAV-embedded board for collision avoidance testing at power lines. This approach provides a novel perspective for UAV obstacle avoidance in power line scenarios. Full article

We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles ${\overrightarrow{p}}_i$ serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are ${\overrightarrow{p}}_{cmi}.$ If ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))\ge 0$ is true, the vectors of momenta of the particles numbered i and j are connected with a red link; if ${(\overrightarrow{p}}_{cmi}(t)·{\overrightarrow{p}}_{cmj}(t))<0$ takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed. Full article

Post-therapeutic relapse, poor survival rates and increasing incidence justify the search for novel therapeutic targets and strategies in cutaneous malignant melanoma (CMM). Within this context, a potential oncogenic role for TrkA in CMM is suggested by reports of NTRK1 amplification, enhanced TrkA expression and intracellular TrkA activation associated with poor prognosis. TrkA, however, exhibits tumour-suppressing properties in melanoma cell lines and has recently been reported not to be associated with CMM progression. To better understand these contradictions, we present the first analysis of potential oncogenic alternative TrkA mRNA splicing, associated with TrkA immunoreactivity, in CMMs, and compare the behaviour of fully spliced TrkA and the alternative TrkAIII splice variant in BRAF(V600E)-mutated A375 melanoma cells. Alternative TrkA splicing in CMMs was associated with unfolded protein response (UPR) activation. Of the several alternative TrkA mRNA splice variants detected, TrkAIII was the only variant with an open reading frame and, therefore, oncogenic potential. TrkAIII expression was more frequent in metastatic CMMs, predominated over fully spliced TrkA mRNA expression in ≈50% and was invariably linked to intracellular phosphorylated TrkA immunoreactivity. Phosphorylated TrkA species resembling TrkAIII were also detected in metastatic CMM extracts. In A375 cells, reductive stress induced UPR activation and promoted TrkAIII expression and, in transient transfectants, promoted TrkAIII and Akt phosphorylation, enhancing resistance to reductive stress-induced death, which was prevented by lestaurtinib and entrectinib. In contrast, fully spliced TrkA was dysfunctional in A375 cells. The data identify fully spliced TrkA dysfunction as a novel mechanism for reducing melanoma suppression, support a causal relationship between reductive stress, UPR activation, alternative TrkAIII splicing and TrkAIII activation and characterise a targetable oncogenic pro-survival role for TrkAIII in CMM. Full article

The information contained in an n-dimensional (nD) density matrix ρ is parametrized and interpreted in terms of its asymmetry properties through the introduction of a family of components of purity that are invariant with respect to arbitrary rotations of the nD Cartesian reference frame and that are composed of two categories of meaningful parameters of different physical nature: the indices of population asymmetry and the intrinsic coherences. It is found that the components of purity coincide, up to respective simple coefficients, with the intrinsic Stokes parameters, which are also introduced in this work, and that determine two complementary sources of purity, namely the population asymmetry and the correlation asymmetry, whose weighted square average equals the overall degree of purity of ρ. A discriminating decomposition of ρ as a convex sum of three density matrices, viz. the pure, the fully random (maximally mixed) and the discriminating component, is introduced, which allows for the definition of the degree of nonregularity of ρ as the distance from ρ to a density matrix of a system composed of a pure component and a set of 2D, 3D,… and nD maximally mixed components. The chiral properties of a state ρ are analyzed and characterized from its intimate link to the degree of correlation asymmetry. The results presented constitute a generalization to nD systems of those established and exploited for polarization density matrices in a series of previous works. Full article

Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\). Full article

This paper describes a novel sensor system to estimate the motion of a stereo camera. Local invariant image features are matched between pairs of frames and linked into image trajectories at video rate, providing the so-called visual odometry, i.e., motion estimates from visual input alone. Our proposal conducts two matching sessions: the first one between sets of features associated to the images of the stereo pairs and the second one between sets of features associated to consecutive frames. With respect to previously proposed approaches, the main novelty of this proposal is that both matching algorithms are conducted by means of a fast matching algorithm which combines absolute and relative feature constraints. Finding the largest-valued set of mutually consistent matches is equivalent to finding the maximum-weighted clique on a graph. The stereo matching allows to represent the scene view as a graph which emerge from the features of the accepted clique. On the other hand, the frame-to-frame matching defines a graph whose vertices are features in 3D space. The efficiency of the approach is increased by minimizing the geometric and algebraic errors to estimate the final displacement of the stereo camera between consecutive acquired frames. The proposed approach has been tested for mobile robotics navigation purposes in real environments and using different features. Experimental results demonstrate the performance of the proposal, which could be applied in both industrial and service robot fields. Full article