Search Results (11)

With the continuous growth of the global population, cities worldwide face the challenge of limited surface land area, making the utilization of underground space increasingly important. The structural stability of underground tunnels is a critical component of underground space safety, influenced by the distribution of the surrounding composite strata and hydrogeological environment. To better analyze the structural stability of underground tunnels, this study proposes a method for estimating the distribution of composite strata that considers the surrounding hydrogeological conditions. The method uses a hydrogeological analysis of the tunnel area to determine the spatial estimation range and unit scale to meet the actual project requirements and then uses the geostatistical kriging method to obtain a distance-weighted interpolation algorithm for the impact area. First, the spatial data are used to obtain the statistical characteristics. Second, the statistical data are interpolated, multifractal theory is used to compensate for the kriging method of sliding weighted average defects, and the local singularity of the regionalized variables is measured. Finally, the mean results of 100 simulations are compared with the empirical results for the tunnel. The interpolation results reveal that this method can be used to quickly obtain good interpolation results. Full article

Recently we studied a collocation–quadrature method in weighted ${\mathbf{L}}^2$ spaces as well as in the space of continuous functions for a Volterra-like integral equation of the form $u\left(x\right)-{\int }_{\alpha x}^{1}h(x-\alpha y)u\left(y\right)dy=f\left(x\right),0<x<1,$ where $h\left(x\right)$ (with a possible singularity at $x=0$) and $f\left(x\right)$ are given (in general complex-valued) functions, and $\alpha \in (0,1)$ is a fixed parameter. Here, we want to investigate the same method for the case when $\alpha =1.$ More precisely, we consider (in general weakly singular) Volterra integral equations of the form $u\left(x\right)-{\int }_0^{x}h(x,y){(x-y)}^{\kappa }u\left(y\right)dy=f\left(x\right),0<x<1,$ where $\kappa >-1$, and $h:D⟶\mathbb{C}$ is a continuous function, $D=\left\{(x,y)\in {\mathbb{R}}^2:0<y<x<1\right\}.$ The passage from $0<\alpha <1$ to $\alpha =1$ and the consideration of more general kernel functions $h(x,y)$ make the studies more involved. Moreover, we enhance the family of interpolation operators defining the approximating operators, and, finally, we ask if, in comparison to collocation–quadrature methods, the application of the Nyström method together with the theory of collectively compact operator sequences is possible. Full article

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov spaces. The spectral theory of families of boundary-interpolating operators is also expanded, offering precise eigenvalue estimates and functional calculus applications. Furthermore, we explore Bessel operators under non-standard measures, such as fractal and weighted geometries, uncovering new analytical phenomena. Key implications include advanced insights into singular integrals, heat kernel behavior, and the boundedness of Riesz transforms, with potential applications in fractal geometry, constrained wave propagation, and mathematical physics. Full article

We study an interpolation problem (objects singular at a prescribed finite set) for curves instead of hypersurfaces. We study singular curves in projective and multiprojective spaces. We construct curves that are singular (or with maximal dimension Zariski tangent space) at each point of a prescribed finite set, while the curves have low degree or low “complexity” (e.g., they are complete intersections of hypersurfaces of low degree). We discuss six open problems on the existence and structure of the base locus of the set of all hypersurfaces of a given degree and singular at a prescribed number of general points. The tools come from algebraic geometry, and some of the results are only existence ones or only asymptotic ones (but with as explicit as possible bounds). Some of the existence results are almost constructive, i.e., in our framework, random parameters should give a solution, or otherwise, take other random parameters. Full article

Due to the assumption of acceleration variation in traditional step-by-step integration methods such as Newmark, sufficiently small time steps are required to ensure numerical stability and accuracy in dynamic systems. In contrast, the state-space approach, based on piecewise interpolation of discrete load functions, does not rely on predetermined acceleration assumptions and has demonstrated high efficiency in terms of stability and accuracy. The original state-space method requires the calculation of the inverse of the structural mass in the transition matrix. However, when a lumped mass matrix is used, this computation renders the entire mass matrix singular, resulting in an invalid solution expression. To address this issue, this study proposes an improved state-space approach for the transient analysis of large-scale structural systems with local nonlinearities. In this approach, a nonlinear force corrector is introduced as an external force term applied to the linear elastic system to account for the nonlinear behavior of locally yielding components. Consequently, the original nonlinear dynamic system can be transformed into an equivalent linear elastic transient system. Furthermore, based on the lumped mass matrix, a first-order ordinary differential state-space equation for such an equivalent linear elastic transient system is derived. Simulation results from three transient system examples show that the state-space approach outperforms the Newmark method in terms of accuracy and stability for dynamic systems characterized by high frequency and low damping. The prediction results show that the state-space approach appears to be insignificantly affected by the choice of the consistent or lumped mass matrix. The numerical results show that the root-mean-square errors between the consistent and lumped matrices in the top displacement time histories of a 15-storey plane frame under various seismic intensities are all less than 1%, and in the base reaction time histories responses the discrepancies are only about 0.5%, indicating that the use of lumped mass matrices is quite reliable. When many nodes or degrees of freedom have no assigned mass, the dimensionality of the state-space equation can be significantly reduced using the lumped mass approach. Therefore, the simulation of large-scale systems can be simplified by employing the improved state-space approach with lumped mass matrices, yielding results nearly identical to those obtained using traditional methods. In conclusion, the improved state-space approach has great potential for the simulation of transient behavior in large-scale systems with local nonlinearities. Full article

The primary goal of this work is to solve the problem of drug diffusion through a thin membrane using a differential quadrature approach with drastically different shape functions, such as Lagrange interpolation and discrete singular convolution (the delta Lagrange kernel and the regularized Shannon kernel). A nonlinear partial differential equation with two time- and space-dependent variables governs the system. To reduce the two independent variables by one, the partial differential equation is transformed into an ordinary differential equation using a one-parameter group transformation. With the aid of the iterative technique, the differential quadrature methods change this equation into an algebraic equation. Then, using a MATLAB program, a code is created that solves this equation for each shape function. To ensure the validity, efficiency, and accuracy of the developed techniques, the computed results are compared to previous numerical and analytical solutions. In addition, the L∞ error is applied. As a consequence of the numerical outcomes, the differential quadrature method, which is primarily based on a discrete singular convolution shape function, is an effective numerical method that can be used to solve the problem of drug diffusion through a thin membrane, guaranteeing a higher accuracy, faster convergence, and greater reliability than other techniques. Full article

Dual NURBS interpolation has been proven an essential technique for high-speed precision machining of complex surfaces. The solution of rotation angles and their derivatives is the basis of kinematic transformation and feedrate optimization in dual NURBS interpolation. The characteristics of the rotation motion of five-axis machine tools with different structures are analyzed. A generic model of dual heads of the vertical five-axis machine tool is established to unify the solution of rotation angles. Then, a generic method for solving the rotation angles and derivatives based on the vector inner product is proposed, and the solution space is analyzed. A singularity handling is given to avoid abrupt rotation angles based on the higher derivatives of the tool orientation vector. The proposed method obtained smooth rotation angles at the singularity points in the cardioid dual NURBS interpolation experiment. It reduced the machining time by 43.3% compared with the simple inverse trigonometric method based on kinematic transformation. Experiment results demonstrate that the proposed method is feasible and effective, and has significant theoretical and practical value for optimizing five-axis CNC machining. Full article

This paper appertains the presentation of a Clenshaw–Curtis rule to evaluate highly oscillatory Fredholm integro-differential equations (FIDEs) with Cauchy and weak singularities. To calculate the singular integral, the unknown function approximated by an interpolation polynomial is rewritten as a Taylor series expansion. A system of linear equations of FIDEs obtained by using equally spaced points as collocation points is solved to obtain the unknown function. The proposed method attains higher accuracy rates, which are proven by error analysis and some numerical examples as well. Full article

In this paper, we provide a detailed exposition of a generalized multivariate Birkhoff interpolation scheme $(Z,S,E)$ and introduce the notions of invariant interpolation space and singular interpolation space. We prove that the space ${\mathcal{P}}_S$, which is spanned by the monomial sequence S, is invariant or singular if the incidence matrix E satisfies some conditions. The advantage of our results lie in the fact that we can deduce whether ${\mathcal{P}}_S$ is always proper or not for all choices of the given node set Z, just from the property of the incidence matrix E, with very low computational complexity. Full article

We attempt to study three significant tests of general relativity in higher dimensions, both in commutative and non-commutative spaces. In the context of non-commutative geometry, we will consider a solution of Einstein’s equation in higher dimensions, with a source given by a static, spherically symmetric Gaussian distribution of mass. The resulting metric would describe a regular or curvature singularity free black hole in higher dimensions. The metric should smoothly interpolate between Schwarzschild geometry at large distance, and de-Sitter spacetime at short distance. We will consider gravitational redshift, lensing, and time delay in each sector. It will be shown that, compared to the four-dimensional spacetime, there can be significant modifications due to the presence of extra dimensions and the non-commutative corrected black holes. Finally, we shall attempt to obtain a lower bound on the size of the extra dimensions and on the mass needed to form a black hole in different dimensions. Full article

Sparse linear estimation of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale partial differential equations (PDE) dynamical systems with inherent scale separation that impact the system dimensionality. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches require the knowledge of the underlying low-dimensional space spanning the manifold in which the system resides. In this paper, we adopt an approach that learns basis from singular value decomposition (SVD) of training data to recover sparse information. This results in a set of four design parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of design parameters implicitly determines the choice of algorithm as either $l_2$ minimization reconstruction or sparsity promoting $l_1$ minimization reconstruction. In this work, we systematically explore the implications of these design parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement, particularly interpolation points from the discrete empirical interpolation method (DEIM), provide the best balance of computational complexity and accurate reconstruction. Full article