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Generalized indefinite strings provide a canonical model for self-adjoint operators with simple spectrum (other classical models are Jacobi matrices, Krein strings and 2 × 2 canonical systems). We prove a number of Szegő-type theorems for generalized indefinite strings and related spectral problems (including Krein strings, canonical systems and Dirac operators). More specifically, for several classes of coefficients (that can be regarded as Hilbert–Schmidt perturbations of model problems), we provide a complete characterization of the corresponding set of spectral measures. In particular, our results also apply to the isospectral Lax operator for the conservative Camassa–Holm flow and allow us to establish existence of global weak solutions with various step-like initial conditions of low regularity via the inverse spectral transform. [ABSTRACT FROM AUTHOR]

This paper deals with differential pencils possessing a term depending on the unknown function with a fixed argument. We deduce the so called main equation together with its fine structure for the spectral problem. Then, according to the boundary conditions and the position of argument, we describe two cases: degenerate and non-degenerate. For these two cases, the uniqueness of inverse spectral problem is studied and a constructive procedure for reconstructing the potentials along with necessary and sufficient conditions of the inverse problem solvability are obtained. [ABSTRACT FROM AUTHOR]

The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case. [ABSTRACT FROM AUTHOR]

This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra. [ABSTRACT FROM AUTHOR]

Rapidly developing machine learning methods have stimulated research interest in computationally reconstructing differential equations (DEs) from observational data, providing insight into the underlying mechanistic models. In this paper, we propose a new neural-ODE-based method that spectrally expands the spatial dependence of solutions to learn the spatiotemporal DEs they obey. Our spectral spatiotemporal DE learning method has the advantage of not explicitly relying on spatial discretization (e.g., meshes or grids), thus allowing reconstruction of DEs that may be defined on unbounded spatial domains and that may contain long-ranged, nonlocal spatial interactions. By combining spectral methods with the neural ODE framework, our proposed spectral DE method addresses the inverse-type problem of reconstructing spatiotemporal equations in unbounded domains. Even for bounded domain problems, our spectral approach is as accurate as some of the latest machine learning approaches for learning or numerically solving partial differential equations (PDEs). By developing a spectral framework for reconstructing both PDEs and partial integro-differential equations (PIDEs), we extend dynamical reconstruction approaches to a wider range of problems, including those in unbounded domains. [ABSTRACT FROM AUTHOR]

This paper presents an inverse problem of identifying two ionic parameters of a nonlocal reaction-diffusion system in cardiac electrophysiology modelling. We used a nonlocal FitzHugh-Nagumo monodomain model which describes the electrical activity in cardiac tissue with the diffusion rate assumed to depend on the total electrical potential in the heart. We established at first, the global Carleman estimate adapted to nonlocal diffusion to obtain our main result which is the uniqueness and the Lipschitz stability estimate for two ionic parameters (k , γ) . [ABSTRACT FROM AUTHOR]

In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integro-differential equations comprising of the Ψ-Caputo fractional derivative and the Ψ-Riemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko's attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, inverse problems are studied for Sturm–Liouville difference operators having $ \frac {p^{2}-1/4}{m^{2}} $ p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes. [ABSTRACT FROM AUTHOR]

We present a new expression for the Weyl-Titchmarsh matrix-valued function of a self-adjoint matrix-valued Schrödinger operator defined on the interval $ [0,b) $ [ 0 , b) , where $ 0 0 < b ≤ ∞. Let $ H_j=-\frac {d^2}{dx^2}I_m+Q_j $ H j = − d 2 d x 2 I m + Q j , j=1,2, be two self-adjoint Schrödinger operators in $ L^2((0,b))^{m\times m} $ L 2 ((0 , b)) m × m and $ Q_1=Q_2 $ Q 1 = Q 2 a.e. on the interval $ [0,a] $ [ 0 , a ] , where $ a\in (0,b) $ a ∈ (0 , b). It is assumed that the potentials $ Q_1 $ Q 1 and $ Q_2 $ Q 2 are sufficiently smooth in the right neighborhood of the point a, where the right-hand derivatives of $ Q_1=Q_2 $ Q 1 = Q 2 at a coincide up to a certain order. Let $ M_j(z) $ M j (z) be the Weyl-Titchmarsh functions of $ H_j=-\frac {{\rm d}^2}{{\rm d}x^2}I_m+Q_j $ H j = − d 2 d x 2 I m + Q j , j=1,2. As a specific application of this expression, we establish a high-energy asymptotic for the difference between $ M_1(z) $ M 1 (z) and $ M_2(z) $ M 2 (z). Besides, new proofs are given for the local Borg-Marchenko uniqueness theorem and the high-energy asymptotics of the Weyl-Titchmarsh functions. [ABSTRACT FROM AUTHOR]

Predicting the evolution of dynamics from a given trajectory history of an unknown system is an important and challenging problem. This paper presents a model-free method of forecasting unknown chaotic systems through reconstructing vector fields from noisy measured data via an adaptation of optimal control methods. This technique is also applicable to partially observed systems using a Takens delay embedding approach. The algorithms are validated on the Lorenz system and the four-dimensional hyperchaotic Rössler system, and demonstrate successful predictions well beyond the Lyapunov timescale. It is found that for small datasets or datasets with large levels of noise, the prediction accuracy of partially observed systems approaches that of fully observed systems. The presented approach also allows the model-free assessment of local predictability on the attractor by evolving initial condition density through the reconstructed vector fields via estimation of the transfer operator. The method is compared to predictions made by an imperfect model which highlights the utility of model-free approaches when the only available models have significant model error. The capability of this method for reconstruction of continuous and global vector fields may be applied to model validation, forecasting of initial conditions not in the training set, and model-free filtering. [ABSTRACT FROM AUTHOR]

In this paper, we construct a counterexample to the uniqueness theorem by Barcilon (Geophys J Int 38(2):287–298, 1974), which is well-known in the field of inverse spectral problems. Our example shows that Barcilon’s three spectra do not uniquely specify the coefficients of the fourth-order differential equation. Our technique is based on the method of spectral mappings, which is a universal tool in the inverse spectral theory for higher-order differential operators. The example is obtained by a finite perturbation of the spectral data for the trivial problem with the zero coefficients. [ABSTRACT FROM AUTHOR]

We study the inverse spectral problems of recovering Dirac-type functional-differential operator with two constant delays a 1 and a 2 not less than one-third of the length the interval. It has been proved that the operator can be recovered uniquely from four spectra when 2 a 1 + a 2 2 is not less than the length of the interval, while it is not possible otherwise. [ABSTRACT FROM AUTHOR]

We study inverse spectral problems for Dirac-type functional-differential operators with two constant delays greater than two fifths the length of the interval, under Dirichlet boundary conditions. The inverse problem of recovering operators from four spectra has been studied. We consider cases when delays are greater or less than half the length of the interval. The main result of the paper refers to the proof that in both cases operators can be recovered uniquely from four spectra. [ABSTRACT FROM AUTHOR]

In this study, a diffusion operator is investigated on a star graph with nonhomogeneous edges. First, the behaviors of sufficiently large eigenvalues are learned, and then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points and to obtain a constructive solution to the inverse problems of this class. [ABSTRACT FROM AUTHOR]

We consider the inverse nodal problem for Sturm-Liouville (S-L) equation with frozen argument. Asymptotic behaviours of eigenfunctions, nodal parameters are represented in two cases and numerical algorithms are produced to solve the given problems. Subsequently, solution of inverse nodal problem is calculated by the second Chebyshev wavelet method (SCW), accuracy and effectiveness of the method are shown in some numerical examples. [ABSTRACT FROM AUTHOR]

The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an n-dimensional matrix eigenvalue problem is derived with a special matrix A:= [aij], that is, aij = 0 if i + j is odd. Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function q(x) in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method. [ABSTRACT FROM AUTHOR]

The reconstruction of potential function using nodal parameters is an inverse problem that has been studied in this work. An efficient and highly helpful transformation allowed for the extraction of a reconstruction formula for the problem’s potential function by a narrow collection of nodal data only. Additionally, the method’s efficacy was shown by a few numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, the half inverse problem and interior inverse problems for Dirac operators with discontinuity inside the interval (0, T) is considered. It is shown that (i) if the potential is given on (0 , (1 + α) T 4) , then one spectrum can uniquely determine the potential on the whole interval; (ii) one spectrum and a set of values of eigenfunctions at some internal points can also uniquely determine the potential on the whole interval. [ABSTRACT FROM AUTHOR]

In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph. The trace formula of the operator is established by applying the residual technique in complex analysis. [ABSTRACT FROM AUTHOR]

We study the inverse spectral problems of recovering Sturm–Liouville differential operator with two constant delays a 1 {a_{1}} and a 2 {a_{2}} greater than one third of the interval. It has been proved that the operator can be recovered uniquely from four spectra under the condition 2 ⁢ a 1 + a 2 2 ≥ π {2a_{1}+\frac{a_{2}}{2}\geq\pi} , while it is not possible otherwise. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with an inverse problem with Robin boundary conditions, which states that, when the potential on [ 0 , 1 / 2 ] and the coefficient at the left end point are known a priori, a full spectrum uniquely determines its potential on the whole interval and the coefficient at the right end point. We shall give a new method for reconstructing the potential for this problem in terms of the Mittag-Leffler decomposition of entire functions associated with this problem. The new reconstructing method also deduces a necessary and sufficient condition for the existence issue. [ABSTRACT FROM AUTHOR]

We consider four common defects that appear in an infinite single-layer graphene lattice, i.e. single vacancy V 1 (5 − 9) , Stone-Wales SW (55 − 77) , and double vacancies V 2 (5 − 8 − 5) and V 2 (555 − 777). Investigating the honeycomb pattern of the graphene, we make use of the concept of shortest closed paths (periodic orbits) in the underlying topological structure. Using properties of the shortest closed paths of odd length we present and prove mathematically an algorithm that classifies which one of these defects occurs. [ABSTRACT FROM AUTHOR]

This paper presents classification and analysis of the mathematical models of the spread of COVID-19 in different groups of population such as family, school, office (3–100 people), town (100–5000 people), city, region (0.5–15 million people), country, continent, and the world. The classification covers major types of models (time-series, differential, imitation ones, neural networks models and their combinations). The time-series models are based on analysis of time series using filtration, regression and network methods. The differential models are those derived from systems of ordinary and stochastic differential equations as well as partial differential equations. The imitation models include cellular automata and agent-based models. The fourth group in the classification consists of combinations of nonlinear Markov chains and optimal control theory, derived by methods of the mean-field game theory. COVID-19 is a novel and complicated disease, and the parameters of most models are, as a rule, unknown and estimated by solving inverse problems. The paper contains an analysis of major algorithms of solving inverse problems: stochastic optimization, nature-inspired algorithms (genetic, differential evolution, particle swarm, etc.), assimilation methods, big-data analysis, and machine learning. [ABSTRACT FROM AUTHOR]

We consider self-adjoint Dirac-type systems with rectangular matrix potentials on the interval [0, b), where 0 < b ≤ ∞. We present a new proof of the local Borg–Marchenko uniqueness theorem. The high-energy asymptotics of the Weyl–Titchmarsh functions and the local Borg–Marchenko uniqueness theorem are derived for locally smooth potentials at the right endpoint. [ABSTRACT FROM AUTHOR]

We introduce Dirac-type operators with a global constant delay on a star graph consisting of m equal edges. For our introduced operators, we formulate an inverse spectral problem that is recovering the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except for a specific boundary vertex v 0 (called the root). For simplicity, we restrict ourselves to the constant delay not less than the edge length of the graph. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of extending Hamilton’s principle to the class of natural mechanical systems with random perturbing forces of white noise type. By the method of moment functions, we construct the functionals taking a stationary value on the solutions of a given stochastic equation of Lagrangian structure.

In this study, singular Sturm–Liouville operators on a star graph with edges are investigated. First, the behavior of sufficiently large eigenvalues is learned. Then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points. Lastly, a constructive solution to the inverse problems of this class is obtained. [ABSTRACT FROM AUTHOR]

A third order self-adjoint differential operator with periodic boundary conditions and an one-dimensional perturbation has been considered. For this operator, we first show that the spectrum consists of simple eigenvalues and finitely many eigenvalues of multiplicity two. Then the expressions of eigenfunctions and resolvent are described. Finally, the inverse problems for recovering all the components of the one-dimensional perturbation are solved. In particular, we prove the Ambarzumyan-type theorem and show that the even or odd potential can be reconstructed by three spectra. [ABSTRACT FROM AUTHOR]

In the work, we transform the discontinuous periodic Sturm–Liouville problems into the new problems by rotating. We present the uniqueness theorem and the reconstruction algorithm of discontinuous periodic Sturm–Liouville operator by studying the inverse problems for the new problems. [ABSTRACT FROM AUTHOR]

The connection between numerical methods for solving differential equations and machine learning has been revealed recently. Differential equations have been proposed as continuous analogues of deep neural networks, and then used in handling certain tasks, such as image recognition, where the training of a model includes learning the parameters of systems of ODEs from certain points along their trajectories. Treating this inverse problem of determining the parameters of a dynamical system that minimize the difference between data and trajectory by a gradient-based optimization method presents the solution of the adjoint equation as the continuous analogue of backpropagation that yields the appropriate gradients. The paper explores an abstract approach that can be used to construct a family of loss functions with the aim of fitting the solution of an initial value problem to a set of discrete or continuous measurements. It is shown, that an extension of the adjoint equation can be used to derive the gradient of the loss function as a continuous analogue of backpropagation in machine learning. Numerical evidence is presented that under reasonably controlled circumstances the gradients obtained this way can be used in a gradient descent to fit the solution of an initial value problem to a set of continuous noisy measurements, and a set of discrete noisy measurements that are recorded at uncertain times. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the parameter-dependent conformable Sturm–Liouville problem (PDCSLP) in which its transmission conditions are arbitrary finite numbers at an interior point in [ 0 , π ] . Also, we prove the uniqueness theorems for inverse second order of conformable differential operators by applying three spectra with jumps and eigen-parameter-dependent boundary conditions. To this end, we investigate the PDCSLP in three intervals [ 0 , π ] , [0, p], and [ p , π ] where p ∈ (0 , π) is an interior point. [ABSTRACT FROM AUTHOR]

In this manuscript, we examine a fractional inverse problem of order 0 < ρ < 1 in a Banach space, including the Atangana–Baleanu fractional derivative in the Caputo sense. We use an overdetermined condition on a mild solution to identify the parameter. The major strategies for determining the outcome are a direct approach using the Volterra integral equation for sufficiently regular data. For less regular data, an optimal control approach uses Euler–Lagrange (EL) equations for the fractional order control problem (FOCP) and a numerical approach for solving FOCP. At last, a numerical example is provided in the support of our results. [ABSTRACT FROM AUTHOR]

We prove local solvability and stability of the inverse Robin–Regge problem in the general case, taking eigenvalue multiplicities into account. We develop the new approach based on the reduction of this inverse problem to the recovery of the Sturm–Liouville potential from the Cauchy data [ABSTRACT FROM AUTHOR]

In this paper, we for the first time get constructive solution for the inverse Sturm–Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse spectral problem for the third-order differential equation with distribution coefficient. The inverse problem consists in the recovery of the differential expression coefficients from the spectral data of two boundary value problems with separated boundary conditions. For this inverse problem, we solve the most fundamental question of the inverse spectral theory about the necessary and sufficient conditions of solvability. In addition, we prove the local solvability and stability of the inverse problem. Furthermore, we obtain very simple sufficient conditions of solvability in the self-adjoint case. The main results are proved by a constructive method that reduces the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. In the future, our results can be generalized to various classes of higher-order differential operators with integrable or distribution coefficients. [ABSTRACT FROM AUTHOR]

In the last years, there has been considerable interest in inverse spectral problems for functional-differential operators with frozen argument. Until now, however, no aspects of their stability have been studied. One of the difficulties here is caused by the non-standard characterization of the spectra of such operators. In the present paper, we eliminate this gap and introduce a natural metric that allows us to obtain the Lipschitz stability on each ball of finite radius. Along with the previous results, this brings a final missing piece to the well-posedness of the inverse problem under consideration. [ABSTRACT FROM AUTHOR]

In this article, we study the wellposedness of the inverse problem for Sturm-Liouville equation with coulomb potential. We will consider two different inverse problem for Sturm-Liouville equation with coulomb potential. We will prove that ıf the spectral characteristics of this problems are close to each other, then the difference between their potential functions is sufficiently small. [ABSTRACT FROM AUTHOR]

We investigate a simple objective for nonlinear instrumental variable (IV) regression based on a kernelized conditional moment restriction known as a maximum moment restriction (MMR). The MMR objective is formulated by maximizing the interaction between the residual and the instruments belonging to a unit ball in a reproducing kernel Hilbert space. First, it allows us to simplify the IV regression as an empirical risk minimization problem, where the risk function depends on the reproducing kernel on the instrument and can be estimated by a U-statistic or V-statistic. Second, on the basis this simplification, we are able to provide consistency and asymptotic normality results in both parametric and nonparametric settings. Finally, we provide easy-to-use IV regression algorithms with an efficient hyperparameter selection procedure. We demonstrate the effectiveness of our algorithms using experiments on both synthetic and real-world data.

In this paper we study a partial inverse spectral problem for non-self-adjoint Sturm–Liouville operators with a constant delay and show that subspectra of two boundary value problems with one common boundary condition are sufficient to determine the complex potential. We developed the Horváth's method in [M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 2001, 10, 4155–4171] for the self-adjoint Sturm–Liouville operator without delay into the non-self-adjoint Sturm–Liouville differential operator with a constant delay. [ABSTRACT FROM AUTHOR]

We study inverse spectral problems for radial Schrödinger operators in L2(0, 1). It is well known that for a radial Schrödinger operator, two spectra for the different boundary conditions can uniquely determine the potential. However, if the spectra corresponding to the radial Schrödinger operators with the two potential functions miss a finite number of eigenvalues, what is the relationship between the two potential functions? Inspired by Hochstadt (1973)'s work, which handled the Sturm-Liouville operator with the potential q ∈ L1(0, 1), we give a corresponding result for radial Schrödinger operators with a larger class of potentials than L1(0, 1). When q ∈ L1(0, 1), we also consider the case where the spectra corresponding to the radial Schrödinger operators with the two potential functions miss an infinite number of eigenvalues and the eigenvalues are close in some sense. [ABSTRACT FROM AUTHOR]

In this paper, the reconstruction of a differential operator from incomplete spectral characteristics is concerned. We prove that coefficients of a second-order differential equation with the eigen-parameter appearing at the right-end boundary condition can be uniquely determined from incomplete nodal data. The new results in this paper are unlike to the known results for the classical Sturm-Liouville operator in [21]. [ABSTRACT FROM AUTHOR]

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves global delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a locally nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce globally nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area for further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability. [ABSTRACT FROM AUTHOR]

This article is concerned with the inverse eigenvalue problem for rank one perturbations of the Sturm-Liouville operator. I obtain the relationship between the spectra of the Sturm-Liouville operator and its rank one perturbations, and from the spectra I reconstruct the perturbed operators, provided that the potential is known a priori.

In the Sturm-Liouville boundary value problem the Dirichlet boundary condition at the point zero has not yet been studied in sufficient details. In this paper the author gives the necessary and sufficient conditions for two sequences { μ n } n = 0 ∞ and { a n } n = 0 ∞ to be correspondingly the eigenvalues and norming constants of the Sturm-Liouville problem with Dirichlet condition at zero. In particular, the constructive solution of inverse problem is given. [ABSTRACT FROM AUTHOR]

We firstly study two partial inverse problems for a second-order differential equation with a special boundary condition and recover this operator from the known spectral data. Then we investigate the missing eigenvalue problem for this operator. [ABSTRACT FROM AUTHOR]

Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input–output equations and, in particular, characteristic sets of differential ideals. The precise relation between identifiability and input–output identifiability is subtle. The goal of this note is to clarify this relation. The main results are: identifiability implies input–output identifiability; these notions coincide if the model does not have rational first integrals; the field of input–output identifiable functions is generated by the coefficients of a "minimal" characteristic set of the corresponding differential ideal. We expect that some of these facts may be known to the experts in the area, but we are not aware of any articles in which these facts are stated precisely and rigorously proved. [ABSTRACT FROM AUTHOR]