Spectral Problem Research Articles

НАХОЖДЕНИЕ СОБСТВЕННЫХ ФУНКЦИЙ ВОЗМУЩЕННЫХ САМОСОПРЯЖЕННЫХ ОПЕРАТОРОВ, ЗАДАННЫХ НА КОМПАКТНОМ ГРАФЕ

The substantiation of a numerical method for finding the eigenfunctions of perturbed self-adjoint operators given in a separable Hilbert space is considered. The method is adapted for problems given on finite connected oriented graphs with an arbitrary number of edges and vertices. The non-iterative numerical method is based on simple formulas using information about the spectral charac-teristics of an unperturbed operator and its perturbing operator. Approximation of the perturbed operator by a matrix is not required. As an example, the developed technique is applied to a perturbed spectral problem given on a three-edge graph of the "star" type. The results of a computational experiment conducted using a software package written in the Maple mathematical environment are presented.

Reliability-aware Restoration Framework for 4D Spectral Photoacoustic Data.

Spectral photoacoustic imaging (PAI) is a new technology that is able to provide 3D geometric structure associated with 1D wavelength-dependent absorption information of the interior of a target in a non-invasive manner. It has potentially broad applications in clinical and medical diagnosis. Unfortunately, the usability of spectral PAI is severely affected by a time-consuming data scanning process and complex noise. Therefore in this study, we propose a reliability-aware restoration framework to recover clean 4D data from incomplete and noisy observations. To the best of our knowledge, this is the first attempt for the 4D spectral PA data restoration problem that solves data completion and denoising simultaneously. We first present a sequence of analyses, including modeling of data reliability in the depth and spectral domains, developing an adaptive correlation graph, and analyzing local patch orientation. On the basis of these analyses, we explore global sparsity and local self-similarity for restoration. We demonstrated the effectiveness of our proposed approach through experiments on real data captured from patients, where our approach outperformed the state-of-the-art methods in both objective evaluation and subjective assessment.

On the Differential-Difference Sine-Gordon Equation with an Integral Type Source

ABSTRACT In this work, we study the integration of the differential-difference sine-Gordon equation with an integral type source. We deduce the time performance of the scattering data of the spectral problem which is associated with the discrete sine-Gordon equation. Using the inverse scattering method, we integrate the Cauchy problem for the differential-difference sine-Gordon equation with the integral type source in the class of the rapidly decreasing functions.

Distribution of eigenvalues of a perturbed differentiation operator on the interval

In this paper, we construct a characteristic determinant of the spectral problem of a first-order differential equation on an interval with an integral perturbation in the boundary value condition, which is an entire analytic function of the spectral parameter. Based on the formula for the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the perturbed spectral problem depending on the modulus of continuity of the subinteral function.

An Inverse Sturm–Liouville-Type Problem with Constant Delay and Non-Zero Initial Function

We suggest a new statement of the inverse spectral problem for Sturm–Liouville-type operators with constant delay. This inverse problem consists of recovering the coefficient (often referred to as potential) of the delayed term in the corresponding equation from the spectra of two boundary value problems with one common boundary condition. The previous studies, however, focus mostly on the case of zero initial function, i.e., they exploit the assumption that the potential vanishes on the corresponding subinterval. In the present paper, we waive that assumption in favor of a continuously matching initial function, which leads to the appearance of an additional term with a frozen argument in the equation. For the resulting new inverse problem, we pay special attention to the situation when one of the spectra is given only partially. Sufficient conditions and necessary conditions on the corresponding subspectrum for the unique determination of the potential are obtained, and a constructive procedure for solving the inverse problem is given. Moreover, we obtain the characterization of the spectra for the zero initial function and the Neumann common boundary condition, which is found to include an additional restriction as compared with the case of the Dirichlet common condition.

Weyl asymptotics for Poincaré–Steklov eigenvalues in a domain with Lipschitz boundary

We justify the Weyl asymptotic formula for the eigenvalues of the Poincaré–Steklov spectral problem for a domain bounded by a Lipschitz surface.

Christoffel transformations for (partial-)skew-orthogonal polynomials and applications

In this article, we consider the Christoffel transformations for skew-orthogonal polynomials and partial-skew-orthogonal polynomials. We demonstrate that the Christoffel transformations can act as spectral problems for discrete integrable hierarchies, and therefore we derive certain integrable hierarchies from these transformations. Some reductional cases are also considered.

Lump solutions for a (2+1)-dimensional generalized KP-type equation

A new ([Formula: see text])-dimensional generalized KP-type equation and its Lax pair are established. Based on gauge transformation between spectral problems, a Darboux transformation for the ([Formula: see text])-dimensional generalized KP-type equation is obtained. Some exact solutions are derived by utilizing the formula of Darboux transformation, including soliton solutions, trigonometric function solution, lump solutions, lump-soliton solution. In order to analyze the dynamical behavior of the solutions, the plots of solutions are depicted by Mathematical software.

Rethinking Pedagogical Use of Eye Trackers for Visual Problems with Eye Gaze Interpretation Tasks

Eye tracking technology enables the visualisation of a problem solver's eye movement while working on a problem. The eye movement of experts has been used to draw attention to expert problem solving processes in a bid to teach procedural skills to learners. Such affordances appear as eye movement modelling examples (EMME) in the literature. This work intends to further this line of work by suggesting how eye gaze data can not only guide attention but also scaffold learning through constructive engagement with the problem solving process of another human. Inferring the models’ problem solving process, be it that of an expert or novice, from their eye gaze display would require a learner to make interpretations that are rooted in the knowledge elements relevant to such problem solving. Such tasks, if designed properly, are expected to probe or foster a deeper understanding of a topic as their solutions would require not only following the expert gaze to learn a particular skill, but also interpreting the solution process as evident from the gaze pattern of an expert or even of a novice. This position paper presents a case for such tasks, which we call eye gaze interpretation (EGI) tasks. We start with the theoretical background of these tasks, followed by a conceptual example and representation to elucidate the concept of EGI tasks. Thereafter, we discuss design considerations and pedagogical affordances, using a domain-specific (chemistry) spectral graph problem. Finally, we explore the possibilities and constraints of EGI tasks in various fields that require visual representations for problem solving.

A multi-component super integrable Dirac hierarchy

We propose a method for generating higher-dimensional nonisospectral super integrable coupling hierarchies associated with a new type of higher-dimensional Lie superalgebra. As applications, we introduce a coupled and a multi-component super Dirac spectral problems respectively, and thus deduce a coupled generalized super integrable Dirac hierarchy and a multi-component generalized super integrable Dirac hierarchy. Additionally, we discuss the super Hamiltonian structures of the resulting higher-dimensional super Dirac hierarchies.

Black hole perturbation theory and multiple polylogarithms

We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with a positive cosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant is negative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail.

Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions

In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a π \pi -periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable x x ; and if the number π / 2 \pi /2 is a period (antiperiod) of the initial function, then the number π / 2 \pi /2 is a period (antiperiod) in the variable x x of the solution of the Cauchy problems for the Hirota equation.

A water probe for direct pH measurement of individual particles via micro-Raman spectroscopy

The acidity of atmospheric aerosols influences fundamental physicochemical processes that affect climate and human health. We recently developed a novel and facile water-probe-based method for directly measuring of the pH for micrometer-size droplets, providing a promising technique to better understand aerosol acidity in the atmosphere. The complex chemical composition of fine particles in the ambient air, however, poses certain challenges to using a water-probe for pH measurement, including interference from interactions between compositions and the influence of similar compositions on water structure. To explore the universality of our method, it was employed to measure the pH of ammonium, nitrate, carbonate, sulfate, and chloride particles. The pH of particles covering a broad range (0–14) were accurately determined, thereby demonstrating that our method can be generally applied, even to alkaline particles. Furthermore, a standard spectral library was developed by integrating the standard spectra of common hydrated ions extracted through the water-probe. The library can be employed to identify particle composition and overcome the spectral overlap problem resulting from similar effects. Using the spectral library, all ions were identified and their concentrations were determined, in turn allowing successful pH measurement of multicomponent (ammonium-sulfate-nitrate-chloride) particles. Insights into the synergistic effect of Cl–, NO3–, and NH4+ depletion obtained with our approach revealed the interplay between pH and volatile partitioning. Given the ubiquity of component partitioning and pH variation in particles, the water probe may provide a new perspective on the underlying mechanisms of aerosol aging and aerosol–cloud interaction.

A Finite-Dimensional Integrable System Related to the Kadometsev–Petviashvili Equation

In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem Lφ=(∂2−v∂−λu)φ=λφx. By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao’s method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived.

Rate-Splitting Multiple Access for Quantized Multiuser MIMO Communications

This paper investigates the sum spectral efficiency maximization problem in downlink multiuser multiple-input multiple-output systems with low-resolution quantizers at an access point (AP) and users. We consider rate-splitting multiple access (RSMA) to enhance spectral efficiency by offering opportunities to boost achievable degree-of-freedom. Optimizing RSMA precoders, however, is highly challenging due to the minimum rate constraint when determining the common rate. The quantization errors coupled with the precoders make the problem more complicated. In this paper, we develop a novel RSMA precoding algorithm incorporating quantization errors for maximizing the sum spectral efficiency. To this end, we first obtain an approximate spectral efficiency in a smooth function. Subsequently, we derive the first-order optimality condition in the form of the nonlinear eigenvalue problem (NEP). We propose a computationally efficient algorithm to find the principal eigenvector of the NEP as a sub-optimal solution. We also extend the weighted minimum mean square error-based RSMA precoding to the considered quantization system. Simulation results validate the proposed methods. The key benefit of using RSMA over spatial division multiple access (SDMA) comes from the ability of the common stream to balance between the channel gain and quantization error in multiuser MIMO systems with different quantization resolutions.

-dressing method for the complex modified KdV equation

The dressing method based on the 2 × 2 matrix -problem is generalized to study the complex modified KdV equation (cmKdV). Through two linear constraint equations, the spatial and time spectral problems related to the cmKdV equation are derived. The gauge equivalence between the cmKdV equation and the Heisenberg chain equation is obtained. Using a recursive operator, a hierarchy of cmKdV with source is proposed. On the basis of the -equation, the N-solition solutions of the cmKdV equation are obtained by selecting the appropriate spectral transformation matrix. Furthermore, we get explicit one-soliton and two-soliton solutions.

A posteriori analysis for a mixed formulation of the Stokes spectral problem

A posteriori analysis for a mixed formulation of the Stokes spectral problem

A method for calculating the radiation characteristics of nonequilibrium laboratory plasmas

One of the fundamentally important issues in the characterization of a dense high-temperature plasma is the description of its self-radiation aimed at determining the radiation losses and their spectral composition. In this paper, a new method is proposed for calculating the radiation characteristics of a high-temperature nonequilibrium plasma. The method is based on a collisional radiative model, which makes it possible to calculate the charge-state distribution in the plasma and the excited-state distribution of the plasma ions. The transfer equations are solved separately for the radiation with a continuous spectrum, combining the bremsstrahlung and recombination radiation spectra, and for the spectral line radiation. A feature of the method is that, when calculating the mean line absorption coefficients, dimensionless weighting coefficients are used, which make it possible to take into account the contribution of the strongest lines. The use of weighting coefficients has the result that even a one-group approximation yields good agreement with the numerical solution of the complete spectral problem.

A new numerical approach for solving shape optimization fourth-order spectral problems among convex domains

A new numerical approach for solving shape optimization fourth-order spectral problems among convex domains

On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function px, which belongs to the space L20,1. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when px≡0) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions px, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L20,1, is dense within the space L20,1. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.