Search Results (779)

While the standard Schwarzschild metric is overwhelmingly employed in general relativity (GR) as the starting point for various spherical spacetime metric calculations, its isotropic (ISO) form is mentioned in more specialized contexts and its derivation is barely discussed in published GR literature. In this work, we review the isotropic metric, stressing that it stands out as a useful spherically symmetric metric to be employed also in traditional GR problems. We start by deriving the ISO metric through solving the vacuum field equations in Cartesian coordinates, thereby obtaining the Ricci tensor also in spherical coordinates. We then analytically calculate the Riemann tensor in Cartesian coordinates, proving its consistency with the Ricci tensor calculation for pedagogical reasons. Finally, from the Riemann tensor we exactly evaluate the Kretschmann scalar, which lacks metric singularities, a result consistent with the known singular behavior of the standard Schwarzschild metric. We conclude that the isotropic metric naturally emerges as a suitable candidate for modeling static neutron stars and regular black holes, thereby complementing the present attempts to understand these rapidly evolving research fields. Full article

Weak measurement enables the amplification of weak physical effects via post-selection and has become an important tool in precision optical metrology; however, conventional schemes based on mean-pointer shifts suffer from response saturation, limited linear range, and stringent stability requirements. Here, we propose and experimentally demonstrate a weak-measurement scheme based on spectral-interference trough shifts, where the zero-intensity points of the post-selected spectrum act as the measurement pointer, establishing an analytical mapping between the trough displacement and the target phase or time delay. Theoretical analysis shows that, under detector resolution limits, the measurement resolution depends solely on the frequency of extinction point and is independent of weak-value singular amplification or bias-phase modulation, thereby maintaining high sensitivity while avoiding pointer saturation. Experiments demonstrate that the trough-shift scheme achieves significantly better agreement between measured and theoretical sensitivities than biased weak measurement and provides a stable linear response without additional bias-compensation structures, reaching a minimum resolvable phase variation at the ${10}^{-7}$ level. Moreover, the approach intrinsically supports multi-period traceable measurements and exhibits strong robustness against intensity fluctuations and spectral distortions, offering a promising route toward high-sensitivity, large-dynamic-range, and stable weak measurement-based optical sensing. Full article

In this study, we investigate a class of mixed fractional partial integro-differential equations (FrPI-DE) involving symmetric singular kernels. The considered model problem involves Caputo fractional derivatives and integral operators that describe spatial interactions in a bounded domain. For the purpose of analysis, the original problem is reformulated in the form of a nonlinear Volterra–Fredholm integral equation (NV-FIE). The existence and uniqueness of the solution are established by the Banach fixed point theorem. To compute numerical solutions, a modified Toeplitz matrix method (TMM) is proposed to handle the singular kernel efficiently. The method transforms the integral equation to a system of nonlinear algebraic equations, which can be solved numerically. The convergence properties of the resulting numerical scheme are analyzed and illustrate the effectiveness of the method by providing numerical examples involving logarithmic, Cauchy-type, and weakly singular kernels. Numerical results indicate that the proposed method provides highly accurate approximations and exhibits stable convergence behavior for different parameter values. Furthermore, these results confirm the effectiveness and reliability of the proposed method for solving fractional integro-differential equations that include symmetric singular kernels. Full article

Electron fields (and more generally spinor fields) with a vortex structure in free space that allows them to have arbitrary integer orbital angular momentum along the direction of motion have been studied for some time. We point out that there are several ways to calculate the local velocity of the electron field, defined as the ratio of momentum density to energy density, and that all but one show a singular vorticity at the vortex line. That one, using the Dirac bilinear current with no derivatives, is the only one so far (to our knowledge) studied in the literature in this context and we further show how to understand an apparent conflict in the existing results. The momentum densities corresponding to the three possible velocity fields give different physical results, in particular regarding the electron induced quantum superkicks given to small electron-absorbing test objects. Full article

In classical explicit Runge–Kutta methods for solving initial value problems (IVPs) of the form $y^{\prime }\left(x\right)=f(x,y),y\left(x_0\right)=y_0$, the first stage is typically given by evaluating the right-hand side at the initial point, i.e., $f_1=f(x_0,y_0)$. However, this approach becomes inefficient or even ill-posed when the $f(x_0,y_0)$ exhibits a singularity at $x_0$, as is common in many physically motivated problems such as the Lane–Emden equation or Thomas–Fermi model. To address this issue, we propose an alternative approach that was originally introduced by Oliver for low-order methods. In this formulation, the first stage is shifted away from the singular point and is instead evaluated at a shifted location: $f_1=f(x_0+c_1\tau ,y_0)$, where $\tau$ is the step size and $c_1\ne 0$ is a nonzero coefficient. This allows the method to bypass the singularity while preserving consistency with the IVP. We derive the corresponding order conditions for algebraic order six and construct an eight-stage scheme satisfying these constraints. The resulting method demonstrates significantly improved efficiency when applied to problems with initial-point singularities, outperforming classical Runge–Kutta pairs of orders $6\left(5\right)$ and even $8\left(7\right)$. Full article

A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential $V^{\left(ICO\right)}\left(x\right)=\mathrm{i}{x}^3$, which is by itself not acceptable as a meaningful quantum model due to a combination of its non-Hermiticity, unboundedness, and most of all the Riesz-basis non-diagonalizability of the Hamiltonian, known as its intrinsic exceptional point (IEP) feature. For the purposes of a perturbation-theory-based simulation of the emergence of such a singular system, a simplified (though not too strictly related) toy-model Hamiltonian is proposed. It combines an $N-$point discretization of the real line of coordinates with an ad hoc interaction in a two-parametric N-by-N-matrix Hamiltonian $H=H^{\left(N\right)}(A,B)$. After such a simplification, one can still encounter a somewhat weaker form of non-diagonalizability at the conventional Kato’s exceptional-point (EP) limit of parameters $(A,B)\to (A^{\left(EP\right)},B^{\left(EP\right)})$. The IEP-non-diagonalizability phenomenon itself appears mimicked by the less enigmatic EP degeneracy of the discrete toy model, especially at large $N\gg 1$. What we gain is that, in contrast to the IEP case, the regularization of the simplified toy model in vicinity to the black conventional EP becomes feasible. Full article

Position-Based Visual Servoing (PBVS) and Image-Based Visual Servoing (IBVS) struggle to balance end effector pose accuracy and robustness in apple picking. They are also prone to target loss and control singularities. A progressive Hybrid Automatic Switching Visual Servoing (HAVS) method is proposed and applied to an apple-picking robotic system. HAVS integrates PBVS and IBVS to coordinate control of the manipulator end effector pose. A depth-based switching function is designed. When target depth is below an optimal threshold, the controller switches to PBVS for precise final positioning. This reduces target loss and control singularities. An adaptive proportional-derivative (PD) controller with fuzzy gain scheduling updates the control gains online to enhance responsiveness and stability. The hardware consists of a six-axis manipulator, a depth camera, and a mobile base. You Only Look Once version 5 (YOLOv5) performs apple detection and generates control commands. Indoors, success rate was 96%, which was 4 and 10 percentage points higher than PBVS only and IBVS only. Average picking time was 12.5 s, 0.3 s, and 1.1 s shorter. Outdoors, success rate was 87.5%, average time was 13.2 s, and damage rate was 4.2%. This method provides a reference implementation for visual servo control in agricultural picking robots. Full article

This paper deals with several equations of mathematical physics written in explicit form with their solutions. In Theorem 1, an oblique derivative problem for the string equation is studied. More precisely, the initial-boundary value problem for the string equation is investigated. The corresponding vector field on the boundary is non-vanishing and does not have a characteristic direction, but can be tangential to some part of the boundary, and it is allowed to change sign. A classical solution exists with suitable compatibility conditions at the corner points. The picture changes significantly in the case of the wave equation with several (say two: 2D) space variables in a circular cylinder. The initial-boundary value problem turns out to be underdetermined with an infinite-dimensional kernel if the boundary vector field is orthogonal to the time axis. By prescribing extra conditions on the generatrices of the cylinder where the vector field is tangential to the cylinder, we obtain a unique classical solution. In Theorem 2, we consider the Cauchy problem in the interior of the parabola of the Lorentzian-type eikonal equation and find its unique classical solution in $\{0\le x_2\le 1/2\}\cap \{x_2\ge {\displaystyle \frac{x_1^2}{2}}\}$. Propagation of singularities for the D and 3 D hyperbolic (Klein–Gordon) equations in ${\mathbf{R}}^4$, ${\mathbf{R}}^8$ is studied in Theorem 3. In the double characteristic points, the wave front propagates either along the surface of the characteristic cone, or in the solid cone starting from $(t_0,x_0)$. Full article

We show that recent numerical findings of universal scaling relations in systems of noisy, aligning self-propelled particles by Rüdiger Kürstencan robustly be explained by perturbation theory and known results for the Mathieu equation with purely imaginary parameter. In particular, we highlight the significance of a cascade of exceptional points that leads to non-trivial fractional scaling exponents in the singular-perturbation limit of high activity. Crucially, these features are rooted in the Fokker–Planck operator corresponding to free self-propulsion. This can be viewed as a dynamical phase transition in the dynamics of noisy active matter. We also predict that these scaling relations depend on the symmetry of the alignment interactions and discuss the relevance of this structure in the free propagation for self-alignment and cohesion-type interactions. Full article

To meet the requirements of effectiveness and strength in actual engineering, based on the dynamic fracture characteristics, the dynamic propagation of orthogonal anisotropic interface cracks in piezoelectric bimaterials was analyzed. By performing Laplace transformation and Fourier transformation on the governing equations, the problem was transformed into a singular integral equation. Using the Chebyshev point method and Laplace inversion, the stress and electric displacement intensity factors at the crack tip of the orthogonal anisotropic interface were obtained. The results show that the crack length affects the dimensionless function. The longer the crack, the larger the dimensionless function. Under certain conditions, the smaller the elastic parameters, the smaller the dimensionless dynamic stress intensity factor. At the same time, the impact time also affects the dynamic crack propagation. With the passage of time, the dimensionless function first increases, then reaches a peak, and finally oscillates and converges to the static value. On this basis, the response surface method was used for analysis and prediction. The R² value of the random forest model is 0.9886, which indicates that the model has high predictive accuracy. When the optimal values of A ($d_1/a$), B ($c_{p}t/a$) and C ($c_{44}^{(2)}/c_{44}^{(1)}$) are 0.4045, 1.6797 and 1.9035 respectively, the stress intensity reaches its maximum value of 1.3375. Full article

The present work is focused on a predator–prey model with the Holling type II interaction function, which is influenced by diffusion, advection and nonlinear reaction effects. Firstly, we show that the solutions of this dynamical model are bounded and unique. Secondly we use the Lyapunov function and then show that the equilibrium points are globally stable. Thirdly, we obtain the solution profile when the diffusion coefficient is small. For this purpose we introduce self-similar structures to convert the nonlinear partial differential equations into nonlinear ordinary differential equations and then use the singular perturbation technique to solve these equations. Fourthly, we use the Hamiltonian and Lighthill’s technique to obtain upper stationary solutions for a small coefficient of the advection term. Lastly, we consider a large diffusion coefficient and obtain the asymptotic profiles of nonstationary solutions with the help of nonlinear point scaling. Full article

Phase transition in quantum mechanics is interpreted as an evolution, at the end of which, typically, a parameter-dependent and Hermitizable Hamiltonian $H\left(g\right)$ loses its observability. In the language of mathematics, such a “quantum catastrophe” occurs at an exceptional point of order N (EPN). Although the Hamiltonian $H\left(g\right)$ itself becomes unphysical in the limit of $g\to g^{EPN}$, it is shown that it can play the role of an unperturbed operator in an innovative perturbation-approximation analysis of the vicinity of the EPN singularity. As long as such an analysis is elementary at $N\le 3$ and purely numerical at $N\ge 5$, we pick up $N=4$ and demonstrate that for an arbitrary quantum system, the specific (i.e., already sufficiently phenomenologically rich) EP4 degeneracy becomes accessible via a unitary evolution process. This process is shown realizable inside a parametric domain ${\mathcal{D}}_{\mathrm{physical}}$, the boundaries of which are determined, near $g^{EP4}$, non-numerically. Possible relevance of such a mathematical result in the context of non-Hermitian photonics is emphasized. Full article

Off-axis reflective telescopes are prone to component misalignment due to external environmental factors and mechanical vibrations. This misalignment introduces low-order aberrations, which severely degrade imaging quality. Thus, active misalignment correction is crucial for maintaining the imaging performance of off-axis reflective telescopes. Current computer-aided alignment technologies for optical systems mostly rely on wavefront sensors to acquire aberrations at multiple fixed fields of view (FOVs) or even the full FOV. This significantly increases system complexity and hinders practical engineering applications. To address this issue, this study first conducts sensitivity analysis of misaligned degrees of freedom (DOFs) using a mode truncation algorithm based on singular value decomposition (SVD). A compensation strategy is proposed to avoid the aberration coupling effect. Furthermore, two novel misalignment aberration compensation methods for off-axis reflective telescopes are presented. These methods require only a single focal spot image and eliminate the need for aberration detection and iterative calculations. One method directly solves component misalignment errors using a convolutional neural network (CNN) based on the system’s point spread function (PSF). To further improve compensation performance, an improved method fusing spot images and Zernike coefficients is proposed. In practical misalignment correction, both methods input a single acquired focal spot image into a well-trained model to obtain the misalignment compensation amount. Simulation experiments demonstrate that the improved method, which uses Zernike polynomial coefficients as an intermediate feature bridge, effectively establishes the mapping relationship between spot images and misalignment amounts. It achieves higher solution accuracy and better aberration compensation effect compared to the direct CNN method. This verifies the necessity of extracting Zernike polynomial coefficient features from spot images. Comparative experiments with the traditional sensitivity matrix method show that the two proposed methods outperform the sensitivity matrix method in aberration compensation accuracy over a large misalignment range. Comprehensive simulation results confirm the feasibility and effectiveness of the proposed methods. They overcome the limitations of existing methods, such as complex structure, high cost, and low efficiency, to a certain extent. Full article

This paper examines the existence and stability of an m-cyclic coupled system of higher-order fractional differential equations with non-singular kernels. Sufficient conditions for the existence and stability of solutions are obtained using fixed-point techniques. Two numerical examples involving coupled and triply coupled systems are presented to validate the theoretical results, and simulations of the triply coupled case illustrate the influence of different fractional orders on the system dynamics. Full article

We conduct a numerical study of integral algebraic equations (IAEs) with singular points, which pose significant challenges for standard computational methods. The presence of singular points often renders classical discretization schemes unstable and inaccurate. This work explores the reformulation of such problems using a least squares framework to restore numerical stability. By recasting the singular IAE as a minimization problem, the least squares method effectively handles the non-integrability and ill-conditioning inherent in direct approaches. We provide a numerical analysis of the proposed scheme and present results from several test cases, demonstrating its superior performance in terms of the convergence rate and solution quality compared to conventional methods. Our findings establish the least squares method as a viable and effective tool for solving singular IAEs. Full article