Spectral Problem Research Articles

Spectral Partition Problems with Volume and Inclusion Constraints

Spectral Partition Problems with Volume and Inclusion Constraints

Jade Identification Using Ultraviolet Spectroscopy Based on the SpectraViT Model Incorporating CNN and Transformer

Jade is a highly valuable and diverse gemstone, and its spectral characteristics can be used to identify its quality and type. We propose a jade ultraviolet (UV) spectrum recognition model based on deep learning, called SpectraViT, aiming to improve the accuracy and efficiency of jade identification. The algorithm combines residual modules to extract local features and transformers to capture global dependencies of jade’s UV spectrum, and finally classifying jade using fully connected layers. Experiments were conducted on a UV spectrum dataset containing four types of jade (natural diamond, cultivated diamond (CVD/HPHT), and moissanite). The results show that the algorithm can effectively identify different types of jade, achieving an accuracy of 99.24%, surpassing traditional algorithms based on Support Vector Machines (SVM) and Partial Least Squares Discriminant Analysis (PLS_DA), as well as other deep learning methods. This paper also provides a reference solution for other spectral analysis problems.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Spectral Unmixing of Pigments on Surface of Painted Artefacts Considering Spectral Variability

Abstract. Painted artefacts, such as murals and paintings, are the treasures of human civilization. Pigment is an important component of their surfaces. It is crucial to study the composition and proportion of pigments on the surface of painted artefacts for the field of heritage conservation. In this study, hyperspectral remote sensing was used to invert the abundance of mixed pigments by spectral unmixing. Spectral variability effects are often present in hyperspectral images. Hyperspectral images of cultural artefacts also suffer from spectral variability due to factors such as particle size and impurities within the pigment, and acquisition conditions. Therefore, spectral variability was incorporated into the Linear Mixed Model of the pigment spectral unmixing. The unmixing methods are classified into two categories based on whether or not spectral libraries are used. In the experiment, computer synthesized data and laboratory-made sample blocks of mixed pigments, which mixed with different ratios of Azurite and Malachite, were selected as validation data. Five commonly used algorithms for solving the spectral variability problem, namely MESMA, Fractional Sparse SU, ELMM, RUSAL, and BCM, are used for pigment unmixing and compared with FCLS, which does not consider spectral variability. The results show that ELMM has the highest unmixing accuracy of aRMSE and xRMSE compared to other methods. At the same time, it can be seen from the metric of SAM that ELMM is able to extract reliable endmember variability spectral, which is more suitable for solving the problem of spectral variability in pigment unmixing. Finally, we apply ELMM to real artefacts Yungang Grottoes mural paintings, and obtain a lower xRMSE than FCLS, which improves the unmixing accuracy.

Dirac points and inverse problems of quantum graphs associated with Archimedean tilings

Abstract One interesting phenomenon of graphene is the presence of the conical singularity
or Dirac points. Using the quantum graph model, we show that there exist three
classes of possible Dirac points for all of the periodic quantum graphs associated with
Archimedean tilings, when the potentials are identical and even. They occur at the
periodic eigenvalues, anti-periodic eigenvalues and other double eigenvalues of the dispersion
relations respectively. We also characterize their associated potentials. Moreover,
we show that there are no other possible Dirac points. Finally, we solve an
inverse spectral problem for the potential, given the knowledge of the pure point and
absolutely continuous spectra.

Physics-informed deep learning for structural dynamics under moving load

Physics-informed deep learning has emerged as a promising approach that incorporates physical constraints into the model, reduces the amount of data required, and demonstrates robustness and potential in dealing with limited datasets for a variety of studies. However, several key challenges still exist, with one being the spectral bias problem of deep learning in the simulation of functions with multi-frequency features. To overcome the challenge, this study proposes a novel physics-informed deep learning method, which integrates physics-informed neural network with Fourier transform so as to solve partial differential equations in the frequency domain, thus alleviating the problem of spectral bias of neural networks in the simulation of multi-frequency functions. In addition, the proposed method is used to focus on the forward simulation and parameter inverse identification issues in structural dynamics under moving loads. To illustrate the superiority of the method, the issues of dynamic response of simply supported beams under moving loads are presented as case studies, and the performance of the method in multiple cases is analysed and discussed. The research results demonstrate the feasibility and effectiveness of the method for structural dynamics simulation and parameter inverse identifications using limited datasets.

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present and distribute a parallel finite-element toolbox written in the free software FreeFEM for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFEM with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFEM with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox's results with known theoretical and numerical findings that have been reported in the literature. Program summaryProgram Title: FFEM_BdG_ddm_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/w9hg964wpb.1Licensing provisions: GPLv3Programming language:FreeFEM (v 4.12) free software (www.freefem.org)Nature of problem: Among the plethora of configurations that may exist in Gross-Pitaevskii (GP) equations modeling one or two-component Bose-Einstein condensates, only the ones that are deemed spectrally stable (or even, in some cases, weakly unstable) have high probability to be observed in realistic ultracold atoms experiments. To investigate the spectral stability of solutions requires the numerical study of the linearization of GP equations, the latter commonly known as the Bogoliubov-de Gennes (BdG) spectral problem. The present software offers an efficient and reliable tool for the computation of eigenvalues (or modes) of the BdG problem for a given two- or three-dimensional GP configuration. Then, the spectral stability (or instability) can be inferred from its spectrum, thus predicting (or not) its observability in experiments.Solution method: The present toolbox in FreeFEM consists of the following steps. At first, the GP equations in two (2D) and three (3D) spatial dimensions are discretized by using P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. For a given configuration of interest, mesh adaptivity in FreeFEM is deployed in order to reduce the size of the problem, thus reducing the toolbox's execution time. Then, stationary states of the GP equations are obtained by a Newton method whose backbone involves the use of a reliable and efficient linear solver judiciously selected from the PETSc1 library. Upon identifying stationary configurations, to trace branches of such solutions a parameter continuation method over the chemical potential in the GP equations (effectively controlling the number of atoms in a BEC) is employed with step-size adaptivity of the continuation parameter. Finally, the computation of the stability of branches of solutions (i.e. the BdG spectrum), is carried out by accurately solving, at each point in the parameter space, the underlying eigenvalue problem by using the SLEPc2 library. Three-dimensional computations are made affordable in the present toolbox by using the domain decomposition method (DDM). In the course of the computation, the toolbox stores not only the solutions but also the eigenvalues and respective eigenvectors emanating from the solution to the BdG problem. We offer examples for computing stationary configurations and their BdG spectrum in one- and two-component GP equations.Running time: From minutes to hours depending on the mesh resolution and space dimension.

The novel triangular spectral indices for characterizing winter wheat drought

Agricultural drought threatens food security and agricultural sustainable development. There have been numerous spectral indices from remote sensing images developed for monitoring crop drought. However, most present spectral indices are focusing on crop growth and Land Surface Temperature (LST), and the crop canopy water content are in less consideration simultaneously. Additionally, the Normalized Difference Vegetation Index (NDVI) is used for characterizing crop growth in almost all spectral drought indices, with the spectral saturation problem of NDVI for closed crop canopy. When vegetation cover is high, NDVI values tend to saturate, which makes them insensitive to further changes in crop health. Therefore, the NDVI saturation phenomenon may lead to an underestimation of the extent of crop drought, as it is not effective in identifying subtle changes in crops under high-density vegetation conditions. Hence, we propose three novel triangular spectral indices for characterizing winter wheat drought using three features including LAI, Land Surface Water Index (LSWI) and LST. For validating the proposed spectral indices, we compared the agreement between these indices with measured Relative Soil Moisture (RSM) and Volumetric Water Content (VWC) of soil in agricultural meteorological station and present popular indices including Crop Water Stress Index (CWSI), Temperature-Vegetation Drought Index (TVDI), and Vegetation Health Index (VHI). The results revealed that our proposed indices including Euclidean distance Crop Health Index (ECHI), Difference Crop Health Index (DCHI) and Perpendicular Water Stress Index (PWSI) outperformed the popular CWSI, TVDI and VHI, with stronger correlations with measured RSM and VWC in agricultural meteorological station. Secondly, there are spatial consistencies for characterizing winter wheat drought between proposed ECHI, DCHI and PWSI with popular CWSI, TVDI and VHI. In addition, our proposed ECHI, DCHI and PWSI have achieved good performance of drought monitoring both in irrigated and rainfed croplands. All these results suggest that our proposed indices have great potential in crop drought monitoring.

Generalized Multiscale Finite Element Method for discrete network (graph) models

In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.

Multi-geometric discrete spectral problem with several pairs of zeros for Sasa–Satsuma equation

Sasa–Satsuma equation is proposed to model the propagation and interaction of the sub-picosecond or femtosecond pulses in a monomode optical fiber. Different from several integrable equations in the Ablowitz–Kaup–Newell–Segur system, the higher-order zeros of Riemann–Hilbert problem for the Sasa–Satsuma appear in quadruples. A new approach to study the multi-geometric discrete spectral problem with several pairs of zeros for the Sasa–Satsuma equation is proposed. Thus, the complete soliton solutions corresponding to the higher-order zeros with arbitrary geometric and algebraic multiplicities are derived. Moreover, the inelastic interactions between or among the solitons corresponding to the higher-order non-elementary zeros exhibit the shape-changing phenomena.

The localized excitation on the Weierstrass elliptic function periodic background for the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation

In this paper, the linear spectral problem associated with the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (gvcKP) equation with the Weierstrass function as the external potential is investigated based on the Lamé function, from which some new localized nonlinear wave solutions on the Weierstrass elliptic ℘-function periodic background are obtained by the Darboux transformation. The degenerate solutions on the ℘-function periodic background for the gvcKP equation can be derived by taking the limits of the half-periods ω 1, ω 2 of ℘(x), whose evolution and corresponding dynamics are also discussed. The findings show that nonlinear waves on the ℘-function periodic background behave as different types of nonlinear waves in different spaces, including periodic waves, vortex solitons and interaction solutions, aiding in elucidating some physical phenomena in the related fields, such as the physical ocean and nonlinear optics.

Algebraic solitons in the massive Thirring model.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

[formula omitted]-dressing approach and [formula omitted]-soliton solutions of the general reverse-space nonlocal nonlinear Schrödinger equation

Using the ∂̄-dressing method, we study the general reverse-space nonlocal nonlinear Schrödinger (nNLS) equation. Beginning with a 3 × 3 matrix ∂̄-problem, the associated spatial and time spectral problems are obtained through two linear constraint equations. Furthermore, the gauge equivalence between the Heisenberg chain equation and the general reverse-space nNLS equation is established. By employing a recursive operator, a hierarchy for the general reverse-space nNLS equation is proposed. Moreover, by selecting a suitable spectral transformation matrix, the N-soliton solutions of the general reverse-space nNLS equation are calculated, yielding the explicit one-soliton and two-soliton solutions.

Finite element analysis for the Navier-Lamé eigenvalue problem

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k≥1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.

Solitons with varying amplitudes and/or varying velocities in a variable-coefficient mixed spectral generalized Sasa–Satsuma equation revealed through the Riemann–Hilbert method

The integrable Sasa–Satsuma (SS) equation, a higher-order Nonlinear Schrödinger (NLS)-type model, has important applications in the field of optics. In this study, we present a generalized SS equation with variable coefficients derived from a mixed spectral problem, abbreviated as vcmsgSSE, and construct its N-soliton solution using the Riemann–Hilbert (RH) method. First, we provide the Lax pair associated with the vcmsgSSE and perform a spectral analysis on it, which allows us to derive a solvable RH problem. Then, by solving this RH problem we construct an explicit expression for N-soliton solution of the vcmsgSSE. Finally, under a special reduction, we obtain the specific one-soliton solution and two-soliton solution of the vcmsgSSE, and use them to illustrate graphically the N-soliton solution with varying amplitude and/or varying velocity corresponds to the physical background of non-uniform medium assumed by the vcmsgSSE.

Optical solitons and their propagations in a time-varying coefficient modified nonlinear Schrödinger equation with non-zero boundary conditions via Riemann–Hilbert method

In this article, a time-varying coefficient modified nonlinear Schrödinger equation (tvcmNLSE) with non-zero boundary conditions (NZBCs) at infinity, phase term of which contains time-varying coefficients, is proposed to describe the nonlinear propagation behavior of optical pulses in non-uniform optical fibre. Meanwhile, the Riemann–Hilbert method (RHM) for the tvcmNLSE with NZBCs is presented for the first time. Specifically, a special transformation is constructed to convert the tvcmNLSE with NZBCs into the case with constant NZBCs, and the associated asymptotic spectral problem (ASP) is obtained. By introducing a suitable two-piece Riemann surface to map the original spectral parameter into a single-valued one, and analyzing the ASP, we construct the corresponding Jost solutions and spectral matrix, and sequentially analyze their analytical, symmetric, and asymptotic properties. Then, a generalized Riemann–Hilbert problem (RHP) is established that relates to the converted tvcmNLSE with constant NZBCs. Using the reconstruction formula, we further derive N-soliton solution with NZBCs in the absence of reflection potential, and analyze the dynamic characteristics of single and double solitons. Most notably, by modulating discrete spectral points (DSPs) and time-varying coefficients, we discover some previously unreported new waveforms, such as parabolic and periodic solitons with NZBCs. These waveforms have important implications in both theory and practice. In addition, the influences of different time-varying coefficients, NZBCs, and DSPs on soliton solutions are also discussed and analyzed. This study shows that the selection of time-varying coefficients, distribution of DSPs, and constraints from NZBCs all have significant impacts on the properties of soliton solutions.

RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

On spectral eigenmatrix problem for the planar self-affine measures with three digits

On spectral eigenmatrix problem for the planar self-affine measures with three digits