Inverse Operator Research Articles

Theory on New Fractional Operators Using Normalization and Probability Tools

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag–Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proven. A conjecture on deconvolution is raised, which would permit completing the proposed theory.

Spent Time of Orbiting Eletron Generating the Quantized Energy En in the Hydrogen Like Atom

Despite the large literature on hydrogen-like atoms, one cannot find any mention as to how long an electron spends in the orbit to generate the quanized radiation energy En. Here, the Heisenberg representation of operators is exteded so as to be able to deal also with imverse oprators. This is necessary when dealing with Coulomb potentials. This then allows to write down the general hydrogen-like atom in a compact form. With the help of operator algebra that includes also the inverse operators, one arrives at the differential equation of the electron trajectory with respect to time.The time solution is achieved through established coupled integral equations. While already established quantized radiation electron energy En is proportional to 1/n2, the newely derived quantized electron circular time tn(e) is proportional to n3 with n=1,2,3,..,and tn(e) being of the order of 10−16 sec in magnitude.

Orthogonal Laurent Polynomials of Two Real Variables

ABSTRACTIn this paper, we consider an appropriate ordering of the Laurent monomials , that allows us to study sequences of orthogonal Laurent polynomials of the real variables and with respect to a positive Borel measure defined on such that . This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable and , and as a result we obtain five‐term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one‐variable case is also presented, along with some applications for future research.

On proportional hybrid operators in the discrete setting

In this article, we introduce a new nonlocal operator defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the ‐transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag‐Leffler sequences.

Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity

Classical General Relativity is a dynamical theory of spacetime metrics of Lorentzian signature. In particular the classical metric field is nowhere degenerate in spacetime. In its initial value formulation with respect to a Cauchy surface the induced metric is of Euclidian signature and nowhere degenerate on it. It is only under this assumption of non-degeneracy of the induced metric that one can derive the hypersurface deformation algebra between the initial value constraints which is absolutely transparent from the fact that the inverse of the induced metric is needed to close the algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface deformation algebra in quantum gravity has to address the issue of non-degeneracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop Quantum Gravity (LQG) most emphasis has been laid to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator topologies, dual spaces (habitats) and density weights have been proposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard constructions.

Homotopy Perturbation Method for Semi-Bounded Solution of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this paper, we introduce the standard Homotopy Perturbation Method (HPM) for obtaining semi-bounded solutions of the first kind of system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. We use the Gauss elimination technique to reduce the system of CSIEs to a diagonal triangle system of algebraic equations. We then apply the HPM to solve the resulting equations. By applying the theory of semi-bounded solutions of CSIEs, we can determine the inverse operators for the first kind of CSIEs. We demonstrate that the proposed method is exact for the system of characteristic SIEs, regardless of the choice of initial guesses (in the Holder class of functions). To illustrate the validity and accuracy of our proposed method, we supply and analyse three examples. We compare the results obtained using our method with those obtained using the Chebyshev collocation and Galerkin methods. Our method includes the ability to solve the complex-valued system of CSIEs. Based on the numerical results, we conclude that the HPM is more dominant than the other methods.

Particle swarm optimization method for parameter selecting in Tikhonov regularization method for solving inverse problems

In this article the Tikhonov method used as a regularization technique for solving the inverse problem liner operator equation of the first kind with noisy and noise-free data. We also provide the Tikhonov method's essential analysis for tackling inverse problems. Following that, we go over how to choose the Tikhonov regularization parameter by implementing the Particle Swarm Optimization (PSO) technique. The effectiveness of combining these two approaches particle swarm optimization and Tikhonov is demonstrated to be highly practicable.

Efficient least-squares reverse time migration with the approximate asymptotic inverse operator in place of the adjoint operator

Least-squares reverse-time migration (LSRTM) has emerged as a method to generate high-quality images by progressively decreasing the misfit between the observed data and the modeled data, in order to achieve an inverse effect of the modeling operator. In the construction of a inversion problem from a modeling operator, the standard procedure begins by seeking the adjoint operator. However, the adjoint operator is not the inverse operator, and thus it often needs numerous iterations to approach the inverse effect. Alternatively, we can seek the approximate asymptotic inverse operator as the migration operator. This allows us to achieve the inverse effect with fewer iterations. The approximate inverse operator lies in three aspects: the back-propagation wavefield, the bond betweeen the reflectivity function and the scattering function, and the weighting factor [Formula: see text], where [Formula: see text] is the incidence angle. The adjoint operator utilizes the observed data or data residual as sources to generate the back-propagation wavefield, whereas the approximate inverse operator treats these as boundary conditions. This subtle difference has implications for the amplitude perservation of the inverted images. By binding the reflectivity function and the scattering function through the factor, we can effectively employ a single modeling operator, based the scattering function, as the demigration operator for the reflectivity function. This strategy enables the simultaneous inversion for the scattering and reflectivity functions, effectively bypassing the challenges typically associated with direct modeling through the reflectivity function itself. Numerical examples indicate that 3 to 5 iterations may be adequate to produce high-quality images.

THE Q-ANALOGUES OF NONSINGULAR FRACTIONAL OPERATORS WITH MITTAG-LEFFLER AND EXPONENTIAL KERNELS

This paper is devoted to introducing a new [Formula: see text]-fractional calculus in the framework of Atangana–Baleanu ([Formula: see text]) and Caputo–Fabrizio ([Formula: see text]) operators. First, an appropriate [Formula: see text]-Mittag-Leffler function is defined, and then [Formula: see text]-analogues of fractional derivatives of Atangana–Baleanu–Riemann ([Formula: see text]) and Atangana–Baleanu–Caputo ([Formula: see text]) are derived. Next, the [Formula: see text]-analogues of proper fractional integrals in the [Formula: see text] sense are proved. Several important properties of these definitions are investigated by using the [Formula: see text]-Laplace transform. Additionally, a suitable [Formula: see text]-exponential function is defined, and the [Formula: see text]-analogues of [Formula: see text] fractional derivatives with their inverse operators are introduced. The higher-order extension of the [Formula: see text]-analogues of [Formula: see text] and [Formula: see text] fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of [Formula: see text]-[Formula: see text] calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.

Estimates of Eigenvalues and Approximation Numbers for a Class of Degenerate Third-Order Partial Differential Operators

In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time.

Microscopic derivation of the octupole magic numbers from symmetry considerations

The valence shells of medium mass and heavy nuclei consist of the normal and the intruder parity orbitals; therefore the Shell Model SU(3) symmetry of Elliott cannot have a straightforward application on them. The proxy-SU(3) can be applied instead, since it uses a unitary transformation, meant to act on the intruder orbitals to alter their parity and transform them to their proxy orbitals. The inverse unitary operator transforms the proxy orbitals back to the intruder ones. The highest weight proxy-SU(3) irreducible representations (irreps) allows one to determine the corresponding number of occupied intruder orbitals. In this way we obtain the so-called ‘octupole magic numbers’ 32, 56, 90, 134 and 194 without any parameter. Moreover, the proxy (unitary) mapping and its inverse transformation make the proxy space eligible for the calculation of observables associated with octupole deformation and the relevant treatment of mixed parity states. The implemented study validates the proxy-SU(3) approach with respect to the octupole deformation and suggests its full applicability in the corresponding mass regions.

Inverse covariance operators of multivariate nonstationary time series

Inverse covariance operators of multivariate nonstationary time series

Truncated Gauss-Newton full-waveform inversion of pure quasi-P waves in vertical transverse isotropic media

Full-waveform inversion (FWI) uses the full information of seismic data to obtain a quantitative estimation of subsurface physical parameters. Anisotropic FWI has the potential to recover high-resolution velocity and anisotropy parameter models, which are critical for imaging the long-offset and wide-azimuth data. We develop an acoustic anisotropic FWI method based on a simplified pure quasi P-wave (qP-wave) equation, which can be solved efficiently and is beneficial for the subsequent inversion. Using the inverse Hessian operator to precondition the functional gradients helps to reduce the parameter tradeoff in the multi-parameter inversion. To balance the accuracy and efficiency, we extend the truncated Gauss-Newton (TGN) method into FWI of pure qP-waves in vertical transverse isotropic (VTI) media. The inversion is performed in a nested way: a linear inner loop and a nonlinear outer loop. We derive the formulation of Hessian-vector products for pure qP-waves in VTI media based on the Lagrange multiplier method and compute the model update by solving a Gauss-Newton linear system via a matrix-free conjugate gradient method. A suitable preconditioner and the Eisenstat and Walker stopping criterion for the inner iterations are used to accelerate the convergence and avoid prohibitive computational cost. We test the proposed FWI method on several synthetic data sets. Inversion results reveal that the pure acoustic VTI FWI exhibits greater accuracy than the conventional pseudoacoustic VTI FWI. Additionally, the TGN method proves effective in mitigating the parameter crosstalk and increasing the accuracy of anisotropy parameters.

An Algorithm for Computing the n^th Root of Real Numbers using a General Method

Abstract: Based on an old article about extracting the ࢔ ࢎ࢚ of real numbers using a general method, we introduce an algorithm for computing such roots, using a different approach, through introducing an operator which we define as the inverse binomial operator .We illustrate the procedure by computing ࢔ ࢎ࢚ roots of some ,randomly, selected real numbers. To show the beauty of using such a new procedure, we compare our calculations with the calculations obtained, for the a certain example, via well-known iteration methods such as the bisection and Newton-Raphson methods; moreover a comparison is made for same purpose using continued fractions method. Finally, a conclusion is given to clarify the advantages of the inverse binomial operator.

Inverse matrix estimations by iterative methods with weight functions and their stability analysis

In this paper, we construct a parametric family of iterative methods to compute the inverse of a nonsingular matrix. This class is free of inverse operators. We prove the third-order of convergence under some conditions involving the parameter of the family. Moreover, a dynamical analysis is made for the first time to a matrix iterative method, finding intervals of stability, that include but are wider than those found in the convergence analysis. Numerical tests on large random matrices confirm the results found.

Adomian Decomposition Method With Inverse Differential Operator and Orthogonal Polynomials for Nonlinear Models

A proficient Adomian decomposition method is proposed amidst the presence of inverse differential operator and orthogonal polynomials for solving nonlinear differential models. The method is indeed a reformation of the standard Adomian method thereby improving the rapidity of the solution's convergence rate. A generalized recurrent scheme for a general nonlinear model was derived and further utilized to solve certain nonlinear test models. Lastly, numerical results are reported in comparative tables, demonstrating absolute error differences between the exact and approximate solutions with regards to various employed orthogonal polynomials.

Analysis and simulation of the effect of large optical range difference of common path coherent-dispersion spectrometer on the detection of exoplanet radial velocities

The Exoplanet Explorer common-path coherent-dispersion spectrometer (CODES) utilizes a unique combination of an asymmetric common path Sagnac interferometer and a low to medium resolution spectrometer. The ideal optical range difference (OPD) interval for CODES is OPD ∈ {15.06 mm, 19.45 mm}; however, the OPD of CODES is 64.3 mm to achieve better detection accuracy. Though they will increase the accuracy of detection, large OPDs outside of the ideal interval will also reduce the contrast of the interference fringes, making phase changes more hazy. This may also significantly affect the radial velocity inversion and reduce CODES's instrumental accuracy. This study creates an inverse tone mapping operator based on the photographic model and designs an inverse tone mapping algorithm called CODESCE. The outcomes of the experiments demonstrate that the tone mapping algorithm CODESCE in this work is appropriate for enhancing the contrast of interference fringe images with high OPD, and it can enhance the contrast of interference fringes by three orders of magnitude when OPD = 64.3 mm; the processed interference fringes are located in the range of interference fringe curves of the optimal OPD. By comparison with other current approaches, the suggested algorithm yields superior processing outcomes.

What Observables Are Needed for Precision Data-Enabled Learning of Inverse Operators?

Abstract The advent of easy access to large amount of data has sparked interest in directly developing the relationships between input and output of dynamic systems. A challenge is that in addition to the applied input and the measured output, the dynamics can also depend on hidden states that are not directly measured. In general, it is unclear what type of data, such as past input and or past output is needed, to learn inverse operators (that predict the input needed to track a desired output for control purposes) with a desired precision. The main contribution of this work is to show that, irrespective of the selected model, removing the hidden-state dependence and achieving a desired precision of inverse operators require (i) a sufficiently-long past history of the output and (ii) sufficiently-precise estimates of the output's instantaneous time derivatives that are necessary and sufficient for linear systems, and under some conditions, for nonlinear systems. This insight, about the required observables (output history and derivative) for removing the hidden-state dependence and achieving precision, is used to develop a data-enabled algorithm to learn the inverse operator for multi-input multi-output square systems. Simulation examples are used to illustrate that neural nets (with universal approximation property) can learn the inverse operator with sufficient precision only if the required observables, identified in this work, are included in training.

СОБСТВЕННЫЕ ЧАСТОТЫ И ФОРМЫ КОЛЕБАНИЙ ОБОЛОЧЕК ВРАЩЕНИЯ.

A method is proposed for constructing a solution to the problem of natural oscillations of shells of revolution described by surfaces of rotation of the second order. The initial equations of shell theory are arranged in a sequence that allows one to calculate the unknowns in an iterative process. The system of the first two momentless equations and the first two equations of infinitesimal bending in the proposition that their inverse operators can be constructed, can be reduced to a single equation for the vector of tangential deformations. The possibility of such a representation is due to the research of V.Z. Vlasov shells of revolution, the meridians of which are described by second-order curves. It is shown that the results of calculating the natural frequencies of vibrations of loose shells given in the well-known reference book on strength, stability and vibrations are incorrect.

Deep Neural Network-Oriented Indicator Method for Inverse Scattering Problems Using Partial Data

We consider the inverse scattering problem to reconstruct an obstacle using partial far-field data due to one incident wave. A simple indicator function, which is negative inside the obstacle and positive outside of it, is constructed and then learned using a deep neural network (DNN). The method is easy to implement and effective as demonstrated by numerical examples. Rather than developing sophisticated network structures for the classical inverse operators, we reformulate the inverse problem as a suitable operator such that standard DNNs can learn it well. The idea of the DNN-oriented indicator method can be generalized to treat other partial data inverse problems.