Search Results (91)

This paper analyzes the stability of trajectory tracking in fixed-wing UAV swarms subject to time-delayed feedback control. A delay-dependent stability criterion is established using a combination of Routh–Hurwitz analysis and a transcendental characteristic equation method. The study identifies a critical delay threshold beyond which the tracking objective becomes unstable. The influence of delayed feedback on the system dynamics is analyzed, showing how time delays affect the swarm’s ability to maintain formation. Numerical simulations confirm the theoretical predictions and illustrate the loss of stability as the delay increases. The findings underline the importance of accounting for delays when evaluating control performance in UAV swarm coordination. Full article

We consider the melting of a one-dimensional domain ($x\ge 0$), initially at the melting temperature $u=0$, by fixing the boundary temperature to a value $u(0,t)=U_0>0$—the so called Stefan melting problem. The governing transient heat-conduction equation involves a time derivative and the spatial derivative of the temperature gradient. In the general case the order of the time derivative and the gradient can take values in the range $(0,1]$. In these problems it is known that the advance of the melt front $s\left(t\right)$ can be uniquely determined by a specified prefactor multiplying a power of time related to the order of the fractional derivatives in the governing equation. For given fractional orders the value of the prefactor is the unique solution to a transcendental equation formed in terms of special functions. Here, our main purpose is to provide efficient numerical schemes with low computational complexity to compute these prefactors. The values of the prefactors are obtained through a dimensionalization that allows the recovery of the solution for the quasi-stationary case when the Stefan number approaches zero. The mathematical analysis of this convergence is given and provides consistency to the numerical results obtained. Full article

The Dynamic Stiffness Matrix (DSM) of a structure is a frequency-dependent stiffness matrix relating the actions (forces and moments) and displacements (translations and rotations) when the structure vibrates at a given frequency. The DSM may be used to find the natural frequencies, modes, and structural response. For many structures, including skeletal frames of prismatic members, exact transcendental expressions for the DSM are readily available. This paper presents a mathematical proof of a linear determinantal relationship between the DSM of a skeletal frame when it is undamaged, cracked, and hinged at the crack location. The rotational stiffness or flexibility of the crack also appears as a linear term. This relationship gives, for the first time, an explicit equation to directly calculate the stiffness of the rotational spring representing a crack from measured natural frequencies for any potential crack location. Numerical examples demonstrate that computing the DSM of the intact and hinged structures gives an efficient solution method for the inverse problem of identifying crack location and severity. This paper also shows that an approximate DSM based on a finite element model can be used in the same way, making this procedure more versatile. Furthermore, new approximate expressions for the natural frequencies of structures with very small or very severe cracks are derived. An interesting relationship between the square of the bending moment in an undamaged beam and the determinant of the DSM of a hinged beam is also derived. This relationship, which can also be inferred from previous work, leads to a better understanding of the effect of crack location in specific vibration modes. Full article

New analytical solutions for the inverse kinematics problem of a 6R manipulator are proposed. Based on the assumption that the rotation axes of the last three links intersect at a common point, the problem is divided into orientation and transition problems. The position of the common point and the rotation angles of the first three links are determined using the equations of motion of the output link. A matrix equation for the rotation angles of the last three links is formulated. Solutions to the inverse kinematics problem are obtained for three models. In the first two, the rotation axis of the fourth link may not intersect the vertical rotation axis of the first. In the third model, the rotation axis of the fourth link intersects neither the vertical rotation axis of the first link nor the intersection point of the axes of the last two links. For all models, an analytical solution in closed form is obtained from the geometry of the mechanism. The solution for the third requires a preliminary search for the root of the transcendental equation for the rotation angle of the fourth link. Illustrative examples of calculations for a specific manipulator are given. Full article

The effective length factor (ELF) of bridge piers, a critical design parameter, is determined by solving the transcendental equation governing stability. Efficient and accurate solutions to these equations under various constraints are essential for automating bridge design software. In this paper, the bridge pier is simplified as an elastically restrained column based on the Timoshenko beam model, and the pier stability equation under general elastic constraints considering shear deformation is derived. By analyzing the distribution patterns of the solutions to the transcendental equations with and without considering shear deformation, a novel two-stage Adaptive Sequential Root Search Method based on bisection algorithm (ASRSBM2s) is proposed to calculate the ELF. In the first stage, the smallest positive root of the transcendental equation without considering shear deformation is first calculated, and the obtained positive root is used to restrict the solution domain of the transcendental equation considering shear deformation in the second stage. Compared with the results of the finite element method (FEM), the proposed algorithm can accurately determine the correct roots of the transcendental equation for various bridge scenarios, and the maximum relative error of the calculated ELF of bridge piers is below 2.5%. Full article

To investigate the biomechanical properties of Camellia oleifera branches under two loading speeds within a specific diameter range, three-point bending tests were conducted using a universal material–testing machine. The tests were performed at loading speeds of 10 mm/min and 20 mm/min on branches with diameters ranging from 5 mm to 40 mm. This study aims to provide insights into the design of a manipulator gripper used in a vibrating harvester for Camellia oleifera fruit. Four main varieties of Camellia oleifera were tested to determine their elastic modulus. The nonlinear least squares method, based on the hyperbolic tangent function, was employed to fit the bending load–deflection curves of the branches. This process constructed multi-parameter transcendental equations involving elastic modulus, diameter, and loading speed. Results indicated that the branches of four Camellia oleifera varieties exhibited significant differences in their biomechanical properties, with their modulus of elasticity ranging from 459.01 MPa to 983.33 MPa. This suggests variability in the bending performance among different varieties. For instance, Huaxin branches demonstrated the highest rigidity, while Huashuo branches were softer in general. For the proposed empirical fitting equations, when the fitting parameter k is 168 ± 20 and the parameter c is 3.102 ± 0.421, the bending load–deflection relationship of the branches can be predicted more accurately. This study provides a theoretical basis for enhancing the efficiency of mechanized vibratory picking of Camellia oleifera and optimising the design of the gripper. Full article

To compute the maximum power point (MPP) from physical parameters of the single-diode model (SDM), it is necessary to solve a transcendental equation using numerical methods. This is computationally expensive and can lead to divergence problems. An alternative is to develop analytical approximations which can be accurate enough for engineering problems and simpler to use. Therefore, this paper presents approximations for computing the MPP of single-junction solar cells. Two special cases are considered: (i) SDM with only series resistance, and (ii) SDM with only shunt resistance. Power series closed-form expressions for the MPP are obtained using perturbation theory and the Lagrange inversion theorem. Validation of the formulas is performed using experimental data from six different technologies obtained from the NREL database and comparing the results with the numerical solution of the SDM and three approximations from the literature. The results show an absolute percentage error (APE) of less than 0.035% with respect to the real MPP measurements. In cases with limited computational resources, this value could be further improved by using a higher- or lower-order power-series approximation. Full article

Community lay leaders are critical in connecting professional services and general populations in communities. However, limited studies have explored the potential protective factors for psychological health among this group of people. In addition, based on the complex nature of spiritual health, the inconsistent findings of previous studies also suggested that different domains of spiritual health may shape psychological health differently in different contexts and among different socio-demographic groups. Therefore, we assessed the psychological health of Hong Kong community lay leaders after COVID-19 and examined the effects of different domains of spiritual health on psychological distress after controlling for age and gender. Cross-sectional data from 234 Hong Kong community lay leaders aged 18 to 84 were analyzed using structural equation modeling. The results showed that most Hong Kong community lay leaders reported moderate anxiety. In addition, personal and communal (one domain) and transcendental domains of spiritual health were negatively associated with depression, anxiety, and stress, and the environmental domain of spiritual health was positively associated with depression, anxiety, and stress. These findings imply the importance of considering both the positive and negative effects of spiritual health on psychological distress. Full article

In this paper, an ancient Babylonian algorithm for calculating the square root of 2 is unveiled, and the potential link between this primitive technique and an ancient Chinese method is explored. The iteration process is a symmetrical property, whereby the approximate root converges to the exact one through harmonious interactions between two approximate roots. Subsequently, the algorithm is extended in an ingenious manner to solve algebraic equations. To demonstrate the effectiveness of the modified algorithm, a transcendental equation that arises in MEMS systems is considered. Furthermore, the established algorithm is adeptly adapted to handle differential equations and fractal-fractional differential equations. Two illustrative examples are presented for consideration: the first is a nonlinear first-order differential equation, and the second is the renowned Duffing equation. The results demonstrate that this age-old Babylonian approach offers a novel and highly effective method for addressing contemporary problems with remarkable ease, presenting a promising solution to a diverse range of modern challenges. Full article

This article presents a buck converter in which the inductor has been replaced by a transmission line. This approach would be practically conceivable if the power supply and load had a greater spatial distance. Alternatively, the model derived in this way could also be regarded as an intermediate model in order to replace a power coil via discretization with a larger number of smaller coils and capacitors. In the time domain, this new converter can be described by a system of coupled partial and ordinary differential equations. In the frequency domain, a transcendental transfer function is obtained. For comparison with an equivalently parameterized conventional converter, Padé approximants are derived. A linear controller is designed for the converter topology under consideration. Full article

We present a novel application of transcendental equations for nonlinear distance optimization in hyperbolic space. Through asymptotic approximations using Fourier and Taylor series expansions, we obtain approximations for the transcendental equations with non-zero real values on the boundary λ. The series expansion of the logarithmic form of our equations around two arbitrary points P₁ and P₂ can be used to find values close to definite coordinates on λ. Applying principles from the Poincaré hyperbolic disk—a non-Euclidean space with constant negative curvature—we construct optimization methods following λ of our transcendental equations. Full article

The main contribution of this paper is to present a simple algorithm that theoretically and numerically assesses the switching angles of an inverter operated with the SPWM technique. This technique is the most widely used for eliminating harmonics in DC-AC converters for powering motors, renewable energy applications, household appliances, etc. Unlike conventional implementations of the SPWM technique based on the analog or digital comparison of a sinusoidal signal with a triangular signal, this paper mathematically performs this comparison. It proposes a simple solution to solve the transcendental equations arising from the mathematical analysis numerically. The technique is validated by calculating the total harmonic distortion (THD) of the generated signal theoretically and numerically, and the results indicate that the calculated angles produce the same distribution of harmonics calculated analytically and numerically. The algorithm is limited to single-phase inverters with unipolar SPWM. Full article

The low-speed turning maneuverability of vehicles is closely related to a well-known offtracking phenomenon which occurs when the rear wheels of a turning vehicle deviate towards the inside of a horizontal curve. Although numerous mathematical models and computer programs for vehicle swept path analysis have been developed in the past, only a few of them can calculate the maximum transient offtracking of a turning vehicle, yet with limited accuracy. The authors were motivated by this fact to find a new mathematical solution for maximum transient offtracking calculation of a single-unit vehicle when negotiating circular curves. In the first stage, a transcendental equation defining vehicle maximum transient offtracking position is derived and numerically solved by Python 3.10.12. In the second stage, the polynomial regression model predicting accurate numerical solutions of the transcendental equation with the desired level of accuracy was developed and tested. The new calculation method is simple enough to simply take the vehicle datum length, circular curve radii, and turn angle, while instantly producing the maximum transient offtracking value, without the need to draw any of the vehicle movement trajectories. Full article

The analytical generalization for N periodic potential wells coupled to a probe rectangular-like potential and a zero potential is extremely important in the study of one-dimensional periodic potentials in solid state physics, e.g., in the calculation of transport, optical, and magnetic properties. These findings raise the possibility of calculating equations for the generalization of N arbitrary potentials related to any potential $V\left(x\right)$ using special functions as a solution. In this work, a novel analytical generalization of the transcendental energy equation, group velocity, and effective mass for N-coupled potentials to a probe one-dimensional potential $V=V\left(x\right)$ was proposed. Initially, two well-known linear periodic potentials $V=V\left(x\right)$ were employed to obtain analytical solutions for rectangular-like and Dirac-delta potentials. Python libraries were used to easily represent the equations for one or two rectangular-like potentials coupled with an arbitrary potential, highlighting the transcendental energy, group velocity, and effective mass. The results showed that the group velocity behavior changed its orientation due to the sign of the potential, whereas the width of the potential $V\left(x\right)$ strongly influenced the group velocity behavior. The effective mass was also modified by the potential shapes, and their combinations, both effective mass and group velocity, exhibited similar physical behaviors to those found in ordinary rectangular-like potentials. Full article

RNA transcripts play a crucial role as witnesses of gene expression health. Identifying disruptive short sequences in RNA transcription and regulation is essential for potentially treating diseases. Let us delve into the mathematical intricacies of these sequences. We have previously devised a mathematical approach for defining a “healthy” sequence. This sequence is characterized by having at most four distinct nucleotides (denoted as $nt\le 4$). It serves as the generator of a group denoted as $f_p$. The desired properties of this sequence are as follows: $f_p$ should be close to a free group of rank $nt-1$, it must be aperiodic, and $f_p$ should not have isolated singularities within its $S{L}_2\left(\mathbb{C}\right)$ character variety (specifically within the corresponding Groebner basis). Now, let us explore the concept of singularities. There are cubic surfaces associated with the character variety of a four-punctured sphere denoted as $S_2^4$. When we encounter these singularities, we find ourselves dealing with some algebraic solutions of a dynamical second-order differential (and transcendental) equation known as the Painlevé VI Equation. In certain cases, $S_2^4$ degenerates, in the sense that two punctures collapse, resulting in a “wild” dynamics governed by the Painlevé equations of an index lower than VI. In our paper, we provide examples of these fascinating mathematical structures within the context of miRNAs. Specifically, we find a clear relationship between decorated character varieties of Painlevé equations and the character variety calculated from the seed of oncomirs. These findings should find many applications including cancer research and the investigation of neurodegenative diseases. Full article