Search Results (124)

In this paper, we introduce a new sequence, which approximates the Euler–Mascheroni constant $\gamma$ and converges faster to its limit, with the convergence rate $n^{-5}$. Also, for this constant, new inequalities are established. Our result, compared to other sequences with convergence rates $n^{-2}$, $n^{-3}$, or $n^{-4}$, improves some known results. Full article

We study a class of continued fraction transformations where the partial numerators are quadratic polynomials and the denominators are linear or constant. Using the Bauer–Muir transform, we establish two theorems that yield structurally distinct but equivalent continued fractions—one with rational coefficients and another with alternating forms. These transformations provide a unified framework for evaluating and simplifying continued fractions, including classical identities such as one of Euler, a recent result by Campbell and Chen, and several conjectures from the Ramanujan Machine involving $\pi$ and $\log 2.$ We conclude by discussing the potential extension of our methods to more general polynomial cases. Full article

Pharmacokinetic modelling is extensively used in understanding drug behavior, distribution and optimizing dosing regimens. This study presents a two-compartment pharmacokinetic model developed using three numerical approaches that includes the Euler method, fourth-order Runge–Kutta method, and Adams–Bashforth–Moulton method. The model incorporates key parameters including elimination, transfer rate constants, and compartment volumes. The numerical approaches are used to simulate the concentration of drug profiles, which are then compared to the exact solution. The results reveal that with an average error of 1.54%, the fourth-order Runge–Kutta technique provides optimized results compared to other methods when the overall average error is taken into account, which shows that the Runge–Kutta method is better in terms of accuracy and consistency for drug concentration estimates in the two-compartment model. This mathematical model may be used to optimize dosing procedures by providing a less complex method. Along with that, it also monitors therapeutic medication levels, which provides accurate analysis for drug distribution and elimination kinetics. Full article

Under cyclic moving load action, tensile-dominant structures are prone to crack initiation due to cumulative damage effects. The presence of cracks leads to structural stiffness degradation and nonlinear redistribution of dynamic characteristics, thereby compromising str18uctural integrity and service performance. The current research on the dynamic behavior of cracked structures predominantly focuses on transient analysis through high-fidelity finite element models. However, the existing methodologies encounter two critical limitations: computational inefficiency and a trade-off between model fidelity and practicality. Thus, this study presents an innovative analytical framework to investigate the dynamic response of cracked simply supported beams subjected to moving loads. The proposed methodology conceptualizes the cracked beam as a system composed of multiple interconnected sub-beams, each governed by the Euler–Bernoulli beam theory. At crack locations, massless rotational springs are employed to accurately capture the local flexibility induced by these defects. The transfer matrix method is utilized to derive explicit eigenfunctions for the cracked beam system, thereby facilitating the formulation of coupled vehicle–bridge vibration equations through modal superposition. Subsequently, dynamic response analysis is conducted using the Runge–Kutta numerical integration scheme. Extensive numerical simulations reveal the influence of critical parameters—particularly crack depth and location—on the coupled dynamic behavior of the structure subjected to moving loads. The results indicate that at a constant speed, neither crack depth nor position alters the shape of the beam’s vibration curve. The maximum deflection of beams with a 30% crack in the middle span increases by 14.96% compared to those without cracks. Furthermore, crack migration toward the mid-span results in increased mid-span displacement without changing vibration curve topology. For a constant crack depth ratio (γ_i = 0.3), the progressive migration of the crack position from 0.05 L to 0.5 L leads to a 26.4% increase in the mid-span displacement (from 5.3 mm to 6.7 mm). These findings highlight the efficacy of the proposed method in capturing the complex interactions between moving loads and cracked concrete structures, offering valuable insights for structural health monitoring and assessment. Full article

This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. $L^2$-norm stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstrate optimal convergence order in both spatial and temporal directions. The numerical results confirm that the proposed method achieves an accuracy of the order $O\left(\tau +h^{k+1}\right)$, where $\tau$ and h represent the time step and spatial mesh size, respectively. This work extends previous studies on one-dimensional problems to higher spatial dimensions, providing a robust framework for handling evolution equations with a weakly singular kernel. Full article

This numerical study addressed the multi-objective optimization of a rocket-type Pulse Detonation Engine nozzle. The Pulse Detonation Engine consisted of a constant length, constant diameter cylindrical section plus a nozzle that could be either convergent, divergent, or convergent–divergent. The space of five design variables contained: equivalence ratio of the H2-Air mixture, convergent contraction ratio, divergent expansion ratio, dimensionless nozzle length, and convergent to divergent length ratio. The unsteady Euler-type numerical solver was quasi-one-dimensional with variable cross-sectional area. Chemistry was simulated by means of a one-step global reaction. The solver was used to generate three coarse five-dimensional data tensors that contained: specific impulse based on fuel, total impulse, and nozzle surface area, for each configuration. The tensors were decomposed using the High Order singular Value Decomposition technique. The eigenvectors of the decompositions were used to generate continuous descriptions of the data tensors. A genetic algorithm plus a Gradient Method optimization algorithm acted on the densified data tensors. Five different objective functions were considered that involved specific impulse based on fuel, total impulse, and nozzle surface area either separately or in doublets/triplets. The results obtained were discussed, both qualitatively and quantitatively, in terms of the different objective functions. Design guidelines were provided that could be of interest in the growing area of Pulse Detonation Engine engineering applications. Full article

The characteristic structure of the two-dimensional adjoint Euler equations is examined. The behavior is similar to that of the original Euler equations, but with the information traveling in the opposite direction. The compatibility conditions obeyed by the adjoint variables along characteristic lines are derived. It is also shown that adjoint variables can have discontinuities across characteristics, and the corresponding jump conditions are obtained. It is shown how this information can be used to obtain exact predictions for the adjoint variables, particularly for supersonic flows. The approach is illustrated by the analysis of supersonic flow past a double-wedge airfoil, for which an analytic adjoint solution is obtained in the near-wall region. The solution is zero downstream of the airfoil and piecewise constant around it except across the expansion fan, where the adjoint variables change smoothly while remaining constant along each Mach wave within the fan. Full article

The Euler difference approach has become a prevalent tool in the research of integral order differential equations. Nevertheless, a review of the literature reveals a dearth of studies examining fractional order models using the exponential Euler difference approach. The present study employs an exponential Euler difference approach to examine the properties of nonlocal discrete-time oscillators with Mittag–Leffler kernels and piecewise features, with the aim of providing insights into a continuous-time nonlocal nonlinear system. By employing impulsive equations of variations in constants with different forms in conjunction with the Gronwall inequality, a controller that is capable of instantaneously responding and stabilizing the nonlocal discrete-time oscillator is devised. This controller is realized through an associated algorithm. As a case study, the primary outcome is applied to a problem of impulsive stabilization in nonlocal discrete-time Chua’s oscillator. This article presents a stabilizing algorithm for piecewise nonlocal discrete-time oscillators developed using a novel impulsive approach. Full article

This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems involving infinity that are otherwise difficult to solve. Infinite numbers are a superset of complex numbers and can be either complex numbers or some quantification of infinity. The Riemann zeta function can be written as a sum of three rotational infinite numbers, each of which represents infinity. Using these infinite numbers and their properties, a correlation of the non-trivial zeros of the Riemann zeta function with each other is revealed and proven. In addition, an interesting relation between the Euler–Mascheroni constant (γ) and the non-trivial zeros of the Riemann zeta function is proven. Based on this analysis, complex series limits are calculated and important conclusions about the Riemann zeta function are drawn. It turns out that when we have non-trivial zeros of the Riemann zeta function, the corresponding Dirichlet series increases linearly, in contrast to the other cases where this series also includes a fluctuating term. The above theoretical results are fully verified using numerical computations. Furthermore, a new numerical method is presented for calculating the non-trivial zeros of the Riemann zeta function, which lie on the critical line. In summary, by using infinite numbers, aspects of the Riemann zeta function are explored and revealed from a different perspective; additionally, interesting mathematical relationships that are difficult or impossible to solve with other methods are easily analyzed and solved. Full article

An attitude and heading reference system (AHRS) based on the inertial measurement unit is crucial for various applications. In an AHRS, stationary alignments are performed to determine the initial orientation of the sensor frame with respect to the navigation frame. However, the stationary alignment accuracy is affected by sensor error factors. Therefore, several studies have attempted to analyze and minimize the effects of these errors. However, there have been no studies describing and analyzing the Euler angle errors for various sensor orientations. This paper presents the analytical formulation of the relationship between the sensor and the Euler angle errors based on accelerometer and magnetometer signals, regardless of alignment between the sensor and the navigation frames. We selected three-axis attitude determination (TRIAD) as the stationary alignment method and considered the scale, installation, and the offset errors, including noise and constant bias, as sensor error factors. The presented formulation describes the relationship between the sensor error factors and the Euler angle errors as a linear equation. To analyze the Euler angle errors, we performed both sensor-aligned and sensor-misaligned simulations in which the Euler angles were 0° and arbitrary, respectively. The results showed that the presented error formulation could describe the total Euler angle errors and the partial errors induced by each sensor error factor for both the sensor-aligned conditions and the arbitrary Euler angle configurations. Thus, the effects of each sensor error factor on the Euler angle errors can be analytically investigated using the presented formulations for random alignment. Full article

In this paper, by employing the Euler–Maclaurin summation formula and real analysis techniques, an improved version of the parameterized Hardy–Hilbert inequality involving two partial sums is established. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed. Our results extend the classical Hardy–Hilbert inequality and improve certain existing results. Full article

Based on the Newton–Euler method, a multi-body coupling dynamics model of the separation process of underwater vehicles is established. The conditions of contact and detachment between the sub-vehicle and each group of elastic gaskets are analyzed in detail, and the elastic gasket constraint model is established to simulate the elastic contact and detachment process. Based on the Computational Fluid Dynamics (CFD) method, the hydrodynamic data of vehicles under different cases is calculated. In this context, a relatively accurate hydrodynamic database is established, where the hydrodynamic of the Unmanned Underwater Vehicle (UUV) is obtained through fitting, while those of the sub-vehicles are calculated using online interpolation. These provide conditions for realizing Fluid–Structure Interaction (FSI) calculation. Utilizing the FSI simulation method in the multi-body separation process, the separation dynamics of the multi-vehicle under the influence of elastic constraint parameters are analyzed. The simulation results show that the pitching attitude angles of the UUV and sub-vehicle in the separation process are negatively correlated with the change of elastic constraint stiffness, and the load is positively correlated with it, which are in opposite optimization directions. When the total stiffness of the elastic gaskets remains constant, changes in the number of elastic gaskets have a minimal impact on the UUV and sub-vehicle motion state during separation, but significantly affects the load fluctuations on the sub-vehicle, leading to structural vibration issues. The analysis method established in this paper is capable of quickly assessing the safety of underwater vehicle separation for different elastic gasket schemes, thereby facilitating the optimization of parameters. Full article

The mathematical constant $\pi$ and Euler numbers $E_n$ have long been of relevance in various branches of mathematics, particularly in number theory, combinatorics, numerical analysis, and mathematical physics. This review article introduces an exploration of their historical evolution, theoretical foundations, and recent advancements. We examine how $\pi$ and Euler numbers have facilitated advances in different scientific areas like quantum field theory and numerical algorithms. We introduce some emerging perspectives that highlight their interdisciplinary applications and potential future trajectories. Certainly, both numbers are of great relevance in current mathematical research, and their adaptability and ubiquity will continue to ensure their continuous appearance in future mathematical discoveries. The integration of $\pi$ and Euler numbers into advanced computational techniques and fields like artificial intelligence exemplifies their potential in driving mathematical innovation. Full article

Bottlenecks reduce both traffic safety and efficiency, resulting in congestion and collisions. The introduction of connected autonomous vehicles (CAVs) has had a significant impact on road networks and can improve traffic efficiency at bottlenecks. This paper proposes a microscopic traffic model to investigate CAV behavior at bottlenecks and examine the effect of cyberattacks. The model is developed using data collected from a roadside sensor node. It is implemented in MATLAB using the Euler scheme to simulate a platoon of vehicles on a circular road of length $1$ km. The performance is compared with the intelligent driver (ID) model. The results obtained indicate that the road capacity with the proposed model is $1.4$ times higher than with the ID model. Further, the proposed model results in nearly constant speeds with small variations, which is realistic. Conversely, the ID model produces large speed variations that are unrealistic. In addition, the proposed model results in less acceleration and deceleration, which leads to lower vehicle emissions and pollution. The efficiency is better than with the ID model due to CAV communication and coordination, so queues dissipate faster. The traffic flow with the proposed model increases as the density decreases, which is consistent with traffic dynamics. It is also shown that the proposed model can characterize CAV behavior under cyberattacks that cause disruptions in the data. Thus, it can be employed for traffic control and forecasting when bottleneck conditions exist and there is malicious behavior. Full article

In this paper, we consider the semilinear Dirichlet problem $\left({\mathcal{P}}_{\epsilon }\right):-\Delta u+V\left(x\right)u=u^{{\displaystyle \frac{n+2}{n-2}}-\epsilon }$, $u>0$ in $\Omega$, $u=0$ on ∂$\Omega$, where $\Omega$ is a bounded regular domain in ${\mathbb{R}}^n$, $n\ge 4$, $\epsilon$ is a small positive parameter, and V is a non-constant positive $C^2$-function on $\overline{\Omega }$. We construct interior peak solutions with isolated bubbles. This leads to a multiplicity result for $\left({\mathcal{P}}_{\epsilon }\right)$. The proof of our results relies on precise expansions of the gradient of the Euler–Lagrange functional associated with $\left({\mathcal{P}}_{\epsilon }\right)$, along with a suitable projection of the bubbles. This projection and its associated estimates are new and play a crucial role in tackling such types of problems. Full article