This manuscript proposes a novel tangent-graph-based method that provides all distinctive topology paths, as a set of trajectories that can not be transformed into each other by gradual bending and stretching without colliding with obstacles. As the global optimal trajectory (limited by dynamic constraints) may not correspond to the shortest path (almost without dynamic constraints), it is important to provide all distinctive topology paths rather than just the globally shortest one. One of the application is provide initial paths for trajectory planning methods, such as time elastic band (TEB) and dynamic window approach (DWA). Considering that a mobile platform is always working in a dynamic changing environment and that update the whole graph is time consuming, a dynamic rectify method is proposed to update tangents according to local map provided by sensor reading. Finally, results under real grid maps are presented.

Path planning in 3D environment is a fundamental research area for robots and autonomous vehicles. Based on the principle ``the shortest path consists of tangents'', RimJump* is proposed as a tangent-based path planning method suitable for finding the shortest path (both off-ground and on-ground) in 3D space (e.g., octomap and point cloud) for mobile platform to follow. RimJump* searches the tangent graph in the form of a path tree and considers the geometrical properties of the locally shortest path. Therefore, the method can provide all of the locally shortest paths that connect the starting point and the target, including the globally shortest path. And the time cost of RimJump* is insensitive to map scale increases in comparison to methods that search the whole passable space rather than the surface of the obstacle, e.g., Dijkstra and A*. In the Results, RimJump* is compared with other methods in terms of path length and time cost.

In this review paper, we present a pellucid proof of how x tanh(x/µ) approximates |x|and is better than √x² +µ  when we are concernedwith accuracy.

We present smooth approximations to $ \vert{x} \vert $ using sigmoid functions. In particular $ x\,erf(x/\mu) $ is proved to be better smooth approximation for $ \vert{x} \vert $ than $ x\,tanh(x/\mu) $ and $ \sqrt{x^2 + \mu} $ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.

In this paper, we propose a methodology for analyzing the near-field coupling between two surface mount device (SMD) inductors using a 3-dimensional electromagnetic (3D-EM) model. To develop the 3D-EM model, we first extract the effective permeability of core magnetic material in the SMD using the loss tangent in the equivalent circuit model. Then the effective permeability is used in the magnetic material for the 3D-EM modeling of SMD inductor. The validity of the proposed 3D-EM model is confirmed by comparing the impedance and S-parameters obtained from both measured and EM-simulated values for the two near-field coupled SMDs. Finally, the near-field coupling effects between the two adjacent SMD inductors are visualized in terms of magnetic coupling path visualization (CPV) using the proposed 3D-EM model, which demonstrates its usefulness for near-field coupling analysis.

In this paper, an analytical form of the Unit Step Function (or Heaviside Step function) is presented. This important function constitutes a fundamental concept of Operational Calculus and is also involved in many other fields of applied and engineering mathematics. In particular, Heaviside Step Function is performed in a very simple manner by the use of a finite number of standard operations, as the summation of six inverse tangent functions. The novelty of this work when compared with other analytical expressions, is that the proposed exact formula contains two arbitrary single - valued continuous functions which satisfy only one restriction. In addition, the proposed representation is not exhibited in terms of miscellaneous special functions, e.g. Bessel functions, Beta function, Logistic function. Besides, it is neither the limit of a function, nor the limit of a sequence of functions with point – wise or uniform convergence. Hence, this formula may be much more practical, flexible and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

In this work, generalized geometry of second-order tangent groups and affine configuration complexes is proposed. Initially, geometry for higher weights n=4 and weights n=5 is presented through some interesting and suitable homomorphisms, finally, this geometry is extended and generalized for any weight n.

With the rapid advancements in remote sensing technology, the spectral information from hyperspectral remote sensing images has become increasingly rich, facilitating detailed spectral analysis of the Earth's surface objects. However, this abundance of spectral data poses significant challenges in data processing, such as the curse of dimensionality leading to the “Hughes” phenomenon, "strong correlation" due to high resolution, and "non-linear characteristics" caused by varied surface reflectance rates. Therefore, dimensionality reduction of hyperspectral data has become a crucial task. This paper, grounded in manifold theory and manifold learning techniques, and considering the non-linear structures and features in hyperspectral remote sensing data, elucidates the principles and processes of dimensionality reduction in hyperspectral remote sensing images using manifold learning, with a formalized expression of the process. This article introduces spectral information divergence (SID) into the nearest neighbor graph computation of manifold learning algorithms. The principles and computational processes of nearest neighbor graph algorithms based on Euclidean distance (ED), spectral angle mapping (SAM), and SID are studied, and a comparative analysis of the dimensionality reduction effects under these three metrics in hyperspectral data is conducted. Experiments on feature extraction under different metrics were performed using the publicly available Indian Pines hyperspectral dataset. The intrinsic features obtained post-dimensionality reduction were used as inputs for classification algorithms in ground objects classification experiments, with algorithm runtime, overall accuracy, and Kappa coefficient as evaluation metrics for dimensionality reduction quality. The results demonstrate that nearest neighbor graph computation based on SAM and SID outperforms traditional ED methods; SAM-based computation has the lowest time complexity, while SID-based manifold learning yields the highest accuracy in ground objects classification. Thus, manifold learning based on SAM and SID metrics proves to be an effective method for feature extraction in hyperspectral remote sensing data, underscoring the potential of manifold learning techniques in the dimensionality reduction of hyperspectral remote sensing images.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

A facile and environmentally benign method for a single-phase barium titanate synthesis in water vapor medium was studied to reveal the mechanism of phase transformation of the initial simple oxides mixture and estimate the capability of the product to be used as a raw material for low-frequency dielectric ceramics. The composition and structure of the reaction mixture treated in vapor at 130…150 °C as well as 230 °C for various time were investigated by means of XRD, SEM, HRTEM, EDX, and FTIR methods. The kinetics of the occurring phase transformation was described by Johnson-Mehl-Avrami-Erofeev equation. The reaction between the initial oxides was considered as a topochemical process with an apparent activation energy of 74…80 kJ mol^-1. A crucial role in this process belonged to the water vapor medium, which facilitated the generation of the reaction zone and its spreading inward the solid particles. The synthesized tetragonal barium titanate powder (mean particle size of 135 ± 24 nm) was sintered by a conventional technique at 1250 °C to obtain ceramics with grains of about 2 μm. Capacitance measurements performed a dielectric constant and loss tangent of ceramics, which reached 3879 and 6.7 10^-3, respectively, at 1 kHz and room temperature.