Infinite Linear Systems Research Articles

Estimation of Carleman operator from a univariate time series.

Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems.

Tight infinite matrices

We give a simple proof of a recent result of Gollin and Joó (Matching variables to equations in infinite linear equation systems. Linear Algebra Appl. 2023;660:40–46. doi: 10.1016/j.laa.2022.12.002): if a possibly infinite system of homogeneous linear equations A x → = 0 → , where A = ( a i , j ) is an I × J matrix, has only the trivial solution, then there exists an injection ϕ : J → I , such that a ϕ ( j ) , j ≠ 0 for all j ∈ J .

SPECTRAL DYNAMIC STIFFNESS METHOD FOR THE FLUTTER PROBLEM OF COMBINED PLATES

At present time, the spectral dynamic stiffness method is being actively developed as an alternative to the finite element method for vibration and stability problems of composite structures from beams, rods, plates and shells. This approach, based on exact solutions of governing differential equations, makes it possible to more effectively study the problem in the medium and high frequency ranges, and gives analytical expressions for natural modes. It is proposed to use the advantages of this method to study the problems of dynamic stability and flutter of an orthotropic composite plate in a supersonic gas flow. Using the linear approximation of piston theory, solution of the problem is investigated according to the Galerkin method on the basis of the eigenforms of a composite plate in vacuum. According to this approach the boundary value problem is reduced to a homogeneous infinite linear algebraic system of equations with coefficients are depending from physical-mechanical and geometrical parameters of the problem. The frequency parameter is included in the system linearly, that allows us to reduce eigenproblem for infinite system to the problem of determining the eigenvalues and vectors of a matrix. The convergence of the Galerkin method depending on the number of basis functions is studied numerically. It is shown that the first 16 eigenforms provide the good convergence of the method. Examples of numerical implementation are given, obtained solution allow us to study the dependence of the critical velocity from the properties of the material and geometry of the combined plate.

Diffraction of a symmetric TM mode at an open-ended deeply corrugated waveguide with a small period

We investigate the diffraction of a symmetric TM mode at an open-ended corrugated cylindrical waveguide with a flange. This mode can be generated, in particular, by a charged particle bunch moving along the waveguide axis. We analyze the so-called “longwave” range when the wavelengths and the waveguide radius are much greater than the corrugation period. In this case, the corrugated waveguide wall can be replaced with a smooth one on which the so-called equivalent boundary conditions (EBC) are fulfilled. Here, we also assume that the wall is deeply corrugated, i.e. the corrugation depth is of the same order as an inverse wavenumber. The approach applied uses the solution of the corresponding Wiener–Hopf–Fock equation. This solution is in turn utilized to construct an infinite linear system for reflection coefficients which can be solved numerically using the reduction technique. Integral representations for the components of the electromagnetic field are obtained. Typical radiation patterns are presented and analyzed.

Matching variables to equations in infinite linear equation systems

A fundamental result in linear algebra states that if a homogenous linear equation system has only the trivial solution, then there are at most as many variables as equations. We prove the following generalisation of this phenomenon. If a possibly infinite homogenous linear equation system with finitely many variables in each equation has only the trivial solution, then there exists an injection from the variables to the equations that maps each variable to an equation in which it appears.

Scattering characteristics through multiple regions of the wave-bearing trifurcated waveguide

ABSTRACT Present communication addresses the scattering of the dominant acoustic wave mode propagating from a semi-rigid waveguide. The underlying waveguide is tuned because of trifurcated inlet regions whereas outer surfaces are flexural. The matching of the scattering duct modes (orthogonal and non-orthogonal) is accomplished by employing continuity of pressure and velocity across the interface. The reflection and transmission coefficients are calculated using mode matching methods. The obtained infinite linear algebraic systems are truncated and solved numerically. The expressions for energy functionals in different duct regions are obtained and sketched subject to different parameters of interest. Through mathematical and graphical illustrations, the obtained solution is authenticated with the law of energy conservation and verification of geometric conditions.

Mode-matching analysis of flexural trifurcated waveguide with porosity effects

The strong potential of porous media to absorb and change an incident sound wave interacting with a fluid that fills the pores of the material has sparked interest in their acoustic properties. Therefore, the current study describes the energy distribution and absorption mechanism of a dominating acoustics wave mode traveling out of the mouth of a semi-rigid waveguide. The underlying structure is tuned due to a trifurcated inlet region with flexible outer boundaries. In the outlet region of the duct, porosity effects for different porous materials are encountered. The continuity of pressure and velocity through the interface is used to match the scattering orthogonal and non-orthogonal duct modes. Mode matching techniques are used to compute the reflection and transmission coefficients. The resulting infinite linear algebraic systems are truncated and numerically solved. Of particular interest, the effects of considering different porous materials are observed and analyzed thoroughly. The obtained solution is confirmed to satisfy all geometrical conditions as well as the law of conservation of energy by mathematical and graphical representation.

Cramer's rule for implicit linear differential equations over a non-Archimedean ring

We consider a linear nonhomogeneous $m$-th order differential equation in a ring of formal power series with coefficients from some field of characteristic zero. This equation has infinite many solutions in this ring -- one for each initial condition of the corresponding Cauchy problem. These solutions can be found using classical methods of differential equation theory. Let us suppose the coefficients of the equation and the coefficients of nonhomogeneity belong to some integral domain $K$. We are looking for a solution in the form of a formal power series with coefficients from this integral domain. The methods of classical theory do not allow us to find out whether there exists an initial condition that corresponds to the solution of the coefficients from $K$ and do not allow find this initial condition. To solve this problem, we use the method proposed by U. Broggi. This method allows to find a formal solution of the linear nonhomogeneous differential equation in the form of some special series. In previous articles, sufficient conditions for the existence and uniqueness of a solution were found for a certain class of rings $K$ with a non-Archimedean valuation. If these conditions hold, the formal power series obtained using the Broggi’s method is considered. Its coefficients are the sums of series that converge in the non-Archimedean topology considered. It is shown that this series is the solution from $K[[x]]$ of our equation. Note that this equation over a ring of formal power series can be considered as an infinite linear system of equations with respect to the coefficients of unknown formal power series. In this article it is proved that this system can be solved by some analogue of Cramer's method, in which the determinants of infinite matrices are found as limits of some finite determinants in the non-Archimedean topology.

Dynamic responses of a plate with two circular cavities subjected to SH waves

This paper focuses on the scattering of SH waves by two circular cavities in an elastic plate. In the course of this study, wave function unfolding combined with the complex function, the mirror method, and the multi-polar coordinate translation technique are used to derive the incident and scattered conductive waves that fit the free conditions of stress on the upper and lower surface of the band domain. Then according to the stress free conditions of the edge of cylindrical hole, the infinite linear algebraic equations system is obtained. The unknown coefficients are finally calculated by intercepting the finite terms. The results show that when two cylindrical cavities of the same size are present within the plate, the presence of cavity 2, in most cases, leads to an increase in the concentration of kinetic stress along the edges of cavity 1; When two cylindrical cavity sizes are different, at medium and high frequency incidence, the enlargement of cavity 2 will lead to an increase in the kinetic stress concentration of cavity 1.

Diffraction at the Open End of Dielectric-Lined Circular Waveguide

A rigorous approach for solving canonical circular open-ended dielectric-lined waveguide diffraction problems is presented. This is continuation of our recent paper [1] where a simpler case of uniform dielectric filling has been considered. Here we deal with the case of an open-ended circular waveguide with layered dielectric filling which is closer to potential applications. The presented method uses the solution of corresponding Wiener-Hopf-Fock equation and leads to an infinite linear system for reflection coefficients (S-parameters) of the waveguide, the latter can be efficiently solved numerically using the reducing technique. As a specific example directly applicable to beam-driven radiation sources based on dielectric-lined capillaries, diffraction of a slow TM symmetrical mode at the open end of the described waveguide is considered. A series of such modes forms the wakefield (Cherenkov radiation field) generated by a charged particle bunch during its passage along the vacuum channel axis. Calculated S-parameters were compared with those obtained from COMSOL simulation and an excellent agreement was shown. This method is expected to be very convenient for analytical investigation of various electromagnetic interactions of Terahertz (THz) waves (both free and guided) and charged particle bunches with slow-wave structures prospective in context of modern beam-driven THz emitters, THz accererators and THz-based bunch manipulation and bunch diagnostic systems.

Drazin-Star and Star-Drazin Inverses of Bounded Finite Potent Operators on Hilbert Spaces

The aim of this work is to extend to bounded finite potent endomorphisms on arbitrary Hilbert spaces the notions of the Drazin-Star and the Star-Drazin of matrices that have been recently introduced by D. Mosić. The existence, structure and main properties of these operators are given. In particular, we obtain new properties of the Drazin-Star and the Star-Drazin of a finite complex matrix. Moreover, the explicit solutions of some infinite linear systems on Hilbert spaces from the Drazin-Star inverse of a bounded finite potent endomorphism are studied.

Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

Batch sojourn time in the processor sharing queue with geometric batch size

In this paper, we analyze the sojourn time of an entire batch in a processor sharing queue, where geometrically distributed batches arrive according to a Poisson process and individual jobs require exponentially distributed service times. By conditioning on the number of jobs in the queue and the number of jobs in a tagged batch, we establish recurrence relations for conditional sojourn times, which subsequently allow us to derive a partial differential equation for an associated bivariate generating function. This equation involves an unknown generating function, whose series expansion can be computed by solving an infinite lower triangular linear system. Once this unknown function is determined, we determine the Laplace transform and the mean value of the sojourn time of a batch in the system.

Core-Nilpotent Endomorphisms of Infinite-Dimensional Vector Spaces

The aim of this work is to develop a general theory of core-nilpotent endomorphisms of arbitrary vector spaces, such that endomorphisms of finite-dimensional vector spaces and finite potent endomorphisms of infinite-dimensional vector spaces are particular cases of the CN-endomorphisms studied in this theory. For these CN-endomorphisms, we introduce an index that generalizes the index of a finite square matrix and we prove the existence of the Drazin inverse and reflexive generalized inverses. In particular, we characterize all endomorphisms that have Drazin inverse on arbitrary vector spaces. Moreover, we offer a method to study infinite linear systems associated with CN-endomorphisms.

Anti-plane dynamic analysis of a quarter-space piezoelectric material with an elliptic notch in the corner

The aim of this article is to investigate the scattering of anti-plane shear waves by elliptic notches in a quarter-space piezoelectric material. The wave field and electric field expressions in this region are derived using the wave function expansion method. An infinite linear system of equations is established by applying the boundary conditions of no traction and no potential. The results obtained are accurate due to the good convergence of the wave function. The displacement amplitude and dynamic stress in the vicinity of the elliptic arc notch under different characteristic parameters and different wave numbers are computed numerically.

Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.

Diffraction at the Open-Ended Dielectric-Loaded Circular Waveguide: Rigorous Approach

An elegant and convenient rigorous approach for solving circular open-ended dielectric-loaded waveguide diffraction problems is presented. It uses the solution of corresponding Wiener-Hopf-Fock equation and leads to an infinite linear system for reflection coefficients (S-parameters) of the waveguide, the latter can be efficiently solved numerically using the reducing technique. As a specific example directly applicable to beam-driven radiation sources based on dielectric-lined capillaries, diffraction of a slow TM symmetrical mode at the open end of a circular waveguide with uniform dielectric filling is considered. A series of such modes forms the wakefield (Cherenkov radiation field) generated by a charged particle bunch during its passage along the waveguide axis. Calculated S-parameters were compared with those obtained from COMSOL simulation and an excellent agreement is shown. This method is expected to be very convenient for analytical investigation of various electromagnetic interactions of Terahertz (THz) waves (both free and guided) and charged particle bunches with slow-wave structures prospective in context of modern beam-driven THz emitters, THz accererators and THz-based bunch manipulation and bunch diagnostic systems.

The Multi-step Transformation Control and Exponentially Stability for Linear Discrete-time Systems with Additional Control Delay

Background: Few results about the controller of linear discrete-time system with control delay are reported. Methods: By means of the multi-step transformation with memory, the linear discrete-time systems with additional control delay can be transformed to the equivalent linear discrete-time systems without control delay, and the dimension of the transformed system is not increased. By designing the optimal controller of the finite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained. At the same time, by designing the optimal controller of the infinite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained as well. Results: The corresponding finite horizon optimal controller has proved to render the closed-loop system exponentially stable. And the corresponding infinite horizon optimal controller has proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally, two examples are used to verify the theoretical results of this paper. Conclusion: The controller design and the exponentially stability of discrete-time linear system with additional state delay will be investigated in the future.

Moore-Penrose inverse of some linear maps on infinite-dimensional vector spaces

The aim of this work is to characterize linear maps of infinite-dimensional inner product spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix $A\in \text{Mat}_{n\times m} ({\mathbb C})$. Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, the least norm solution of an infinite linear system from the Moore-Penrose inverse offered is studied.