Recurrence Scheme Research Articles

Simulations of X-ray focusing by zone plates in rotationally symmetric optical field utilizing the matrix-free Finite Difference Beam Propagation Method

We present the use of a finite difference method based on Crank-Nicholson scheme and recurrence scheme for computationally efficient simulation of the X-ray propagation through a zone plate. By introducing boundary and central conditions and by avoiding large matrix operations, the method achieves considerable speed, little memory occupation and low background noise. Accommodating refractive index profiles of arbitrary shape, it can be applied to assist optimizing X-ray zone plates and understanding focusing mechanism.

On computational aspects of high-order dual Hahn moments

In this paper, we present two new algorithms for the fast and stable computation of high-order discrete orthogonal dual Hahn polynomials (DHPs). These algorithms are essentially based on the proposed computation method of the initial values of DHPs following the order n and the variable s. For both algorithms, a single stable value is computed, fully independent of the gamma function that is the source of the numerical overflow, and then the rest of DHPs values are computed recursively via the proposed recurrence scheme. By analyzing the DHPs matrix, we propose a new method, which allows ensuring the numerical stability of high-order DHPs and dual Hahn moments (DHMs) until the last order. This method is based on the use of appropriate stability conditions. The results of simulations and comparisons carried out show on one hand that the second algorithm with the stability condition allows to compute DHPs up to the order n = 17,603 without propagation of numerical error. On the other hand, the performance of analyzing large-size signals and images by high-order DHMs computed by the proposed method significantly exceeds the existing methods in terms of numerical stability, accuracy of reconstruction and in terms of maximum size of the analyzed signals and images. After the acceptance of this paper, the proposed algorithms for high-order DHPs computation will be made publically available at https://github.com/AchrafDaoui/On-Computational-Aspects-of-High-Order-Dual-Hahn-Moments.

Solving Poisson’s equation for birth–death chains: Structure, instability, and accurate approximation

Poisson’s equation plays a fundamental role as a tool for performance evaluation and optimization of Markov chains. For continuous-time birth–death chains with possibly unbounded transition and cost rates as addressed herein, when analytical solutions are unavailable its numerical solution can in theory be obtained by a simple forward recurrence. Yet, this may suffer from numerical instability, which can hide the structure of exact solutions. This paper presents three main contributions: (1) it establishes a structural result (convexity of the relative cost function) under mild conditions on transition and cost rates, which is relevant for proving structural properties of optimal policies in Markov decision models; (2) it elucidates the root cause, extent and prevalence of instability in numerical solutions by standard forward recurrence; and (3) it presents a novel forward–backward recurrence scheme to compute accurate numerical solutions. The results are applied to the accurate evaluation of the bias and the asymptotic variance, and are illustrated in an example.

Taylor dispersion in Poiseuille flow in three-dimensional tubes of varying diameter.

Diffusion of particles carried by Poiseuille flow of the surrounding solvent in a three-dimensional (3D) tube of varying diameter is considered. We revisit our mapping technique [F. Slanina and P. Kalinay, Phys. Rev. E 100, 032606 (2019)2470-004510.1103/PhysRevE.100.032606], projecting the corresponding 3D advection-diffusion equation onto the longitudinal coordinate and generating an effective one-dimensional modified Fick-Jacobs (or Smoluchowski) equation. A different scaling of the transverse forces by a small auxiliary parameter ε is used here. It results in a recurrence scheme enabling us to derive the corrections of the effective diffusion coefficient and the averaged driving force up to higher orders in ε. The new scaling also preserves symmetries of the stationary solution in any order of ε. Finally we show that Reguera-Rubí's formula, widely applied for description of diffusion in corrugated tubes, can be systematically corrected by the strength of the flow Q; we give here the first two terms in the form of closed analytic formulas.

Integrable anharmonic potentials for stretching and bending molecular vibrations

A formalism is developed to enhance the concept of anharmonicity for a generalized unidimensional potential that includes the Morse, Pöschl–Teller, Rosen–Morse, Manning–Rosen, Eckart, Tietz and Wei Hua perturbed models as particular cases. To obtain an exact unperturbed solution of the Schrödinger equation for a family of unidimensional vibrational potentials, we apply a purely algebraic approach based on a factorization method according to a deductive Green’s treatment. This theory is shown to result in the construction of ladder operators, eigenvalues and eigenvectors in their general form. For a perturbed solution, we use a recurrence scheme that leads to the pertinent corrections in closed and clear form.

Generation of Coulomb matrix elements for the 2D quantum harmonic oscillator

The Fock-Darwin states are the natural basis functions for a system of interacting electrons trapped inside a 2D quantum dot. Interaction effects at the mean field level or more elaborate quantum many body descriptions rely on an accurate evaluation of Coulomb matrix elements. In this work, we derive a highly efficient recurrence scheme to compute these elements in Fock-Darwin basis. The algorithm is best implemented on symbolic calculus platforms, preventing the appearance of rounding-off errors. The high speed achieved allows us to calculate all matrix elements in a basis set of several hundred states in very reasonable times (a few hours on a standard computer). We also make use of symmetry to reduce the number of elements to be computed. Finally, we check the reliability of floating-point evaluation for existing analytical expressions. We confirm the stability of a two-folded nonalternating sum for elements involving only the lowest energy levels and put on display the daunting limitations of the expressions commonly used to compute the elements in the general case.

Recursion for the Smallest Eigenvalue Density of $$\beta $$ β -Wishart–Laguerre Ensemble

The statistics of the smallest eigenvalue of Wishart-Laguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost significance in understanding the spectral behavior of Wishart-Laguerre ensembles and, among other things, unveil the underlying universality aspects in the asymptotic limits. However, obtaining exact and explicit expressions by expanding determinants or Pfaffians becomes impractical if large dimension matrices are involved. For the real matrices ($\beta=1$) Edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices. Very recently, an analogous recurrence scheme has been obtained for the complex matrices ($\beta=2$). In the present work we extend this to $\beta$-Wishart-Laguerre ensembles for the case when exponent $\alpha$ in the associated Laguerre weight function, $\lambda^\alpha e^{-\beta\lambda/2}$, is a non-negative integer, while $\beta$ is positive real. This also gives access to the smallest eigenvalue density of fixed trace $\beta$-Wishart-Laguerre ensemble, as well as moments for both cases. Moreover, comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly. Exact evaluations for large values of $n$ (the matrix dimension) and $\alpha$ also enable us to compare with Tracy-Widom density and large deviation results of Katzav and Castillo. We also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the Wigner-Smith matrix and are relevant to the problem of quantum chaotic scattering.

Recurrence scheme for FDTD-compatible discrete Green's function derived based on properties of Gauss hypergeometric function

ABSTRACTIn this paper, the formulation of one-dimensional FDTD (Finite-difference time-domain)-compatible discrete Green's function (DGF) is derived based on the Gauss hypergeometric function (GHF). The properties of GHF make it possible to derive the recurrence scheme only in the time domain for the DGF generation. Furthermore, this recurrence scheme is valid for any stable time-step size and can be implemented using standard numerical precision of computations. The proposed derivation is obtained without processing in any symbolic mathematics software and relies on the application of known properties of GHF. The difference between the developed recurrence scheme and the direct FDTD simulation is approximately at the level of numerical noise, which confirms the correctness of our derivation. The results obtained should be useful for the development of computational techniques employing FDTD and the diakoptic approach.

Development of a method for the creation of 3d advertising printing products

The analysis of characteristic features of 3D advertising design is carried out. It was found that since the input information concerning the task of structuring the stages of creation of 3D advertising printing products is qualitative, an expert approach should be used for its solution. As experts, the technologists of the leading Ukrainian publishing and printing companies were presented. Using the expert approach on the basis of these features, the structuring of development stages of 3D advertising printing products was formed. The list and content of the parameters of such structuring are conditioned by specific types of 3D advertising. To take into account the separate categories and properties of 3D advertising, a technology for choosing the options for implementing 3D advertising installations was developed. This technology is based on the use of simulation tools. The choice of acceptable alternatives is based on the use of the proposed basic recurrence scheme. The list of estimating parameters that influence the efficiency of advertising installations and allow evaluating the quality of the logo created by calligraphy means is proposed. A mathematical model for creating a projection of an advertising image is developed. This model provides features of graphic processing of illustrations such as distortion of the original 3D advertising image horizontally, distortion of the original 3D advertising image vertically, converting the pixel coordinate in the desired direction, calculating the size of the 3D advertising image. Based on the developed mathematical model, a program was created that allows forming the distorted appearance of any image, which can be applied to the surface. The method of creating 3D advertising was implemented in the BPWin software product. Software implementation allows optimizing the 3D advertising development process

No-neighbours recurrence schemes for space-time Green’s functions on a 3D simple cubic lattice

Application of multivariate creative telescoping to a finite triple sum representation of the discrete space-time Green’s function for an arbitrary numeric (non-symbolic) lattice point on a 3D simple cubic lattice produces a fast, no-neighbours, seventh-order, eighteenth-degree, discrete-time recurrence scheme.For arbitrary numeric lattice points outside the diagonal symmetry planes, the seven numeric eighteenth-degree polynomial coefficients of the recurrence scheme are products of polynomials with integer coefficients that are linear in the recurrence index n, and two polynomials of degree four, and five polynomials of degree twelve that are irreducible over the field of integers. Owing to the symmetry of the scalar Green’s function upon interchanging any of the lattice point coordinates, the twelfth degree polynomials with integer coefficients may each be expanded in terms of 102 elementary symmetric polynomials in symbolic lattice point coordinates. The recurrence schemes determined by the telescoper for 102 distinct numeric lattice points can be used to form linear systems of equations. These are solved for the coefficients of the elementary symmetric polynomials required to construct the symbolic polynomial coefficients of the generic 3D recurrence scheme.Given its compact and straightforward 2D counterpart, this 3D recurrence scheme is far more intricate than expected, and is most efficiently presented through tables of coefficients. However, the scheme and the resulting lattice Green’s function sequences also exhibit more features. The complexity reduces for lattice points on diagonal symmetry planes, yielding a fast no-neighbours, fifth-order, twelfth-degree, discrete-time recurrence scheme. An illustrative example reveals unexpected phenomena, e.g. a late-time, high-frequency interplay of resonances that appears anomalous but can be fully explained, and the possible occurrence of removable recurrence scheme singularities. These effects are studied in detail in separate papers.

Investigation of the Acoustic Interaction of Shells with Liquid

We propose a mathematical model of acoustic interaction of the shells of revolution with liquid under axisymmetric loads. The model is based on the linear relations of Timoshenko–Mindlin shells and the acoustic approximation for the liquid. We formulate the initial boundary-value problem and the corresponding variational problem of interaction between two media. For the solution of the variational problem of acoustic interaction between the shell of revolution and the liquid, we develop a projectionmesh scheme, where the Galerkin semidiscretization is used together with approximations of the finiteelement method in the space variables and the one-step recurrence scheme of integration with respect to time. The stress-strain state of the shell subjected to the action of normal hydrostatic pressure is analyzed. The results of numerical analysis of deflections obtained by using the proposed method are compared with the available analytic solutions.

Smallest eigenvalue density for regular or fixed-trace complex Wishart–Laguerre ensemble and entanglement in coupled kicked tops

The statistical behaviour of the smallest eigenvalue has important implications for systems which can be modeled using a Wishart–Laguerre ensemble, the regular one or the fixed trace one. For example, the density of the smallest eigenvalue of the Wishart–Laguerre ensemble plays a crucial role in characterizing multiple channel telecommunication systems. Similarly, in the quantum entanglement problem, the smallest eigenvalue of the fixed trace ensemble carries information regarding the nature of entanglement.For real Wishart–Laguerre matrices, there exists an elegant recurrence scheme suggested by Edelman to directly obtain the exact expression for the smallest eigenvalue density. In the case of complex Wishart–Laguerre matrices, for finding exact and explicit expressions for the smallest eigenvalue density, existing results based on determinants become impractical when the determinants involve large-size matrices. In this work, we derive a recurrence scheme for the complex case which is analogous to that of Edelman’s for the real case. This is used to obtain exact results for the smallest eigenvalue density for both the regular, and the fixed trace complex Wishart–Laguerre ensembles. We validate our analytical results using Monte Carlo simulations. We also study scaled Wishart–Laguerre ensemble and investigate its efficacy in approximating the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace ensemble to investigate the behaviour of the smallest eigenvalue in the paradigmatic system of coupled kicked tops.

Engineering Approach to Calculating QoS of Server with Self-Similar Incoming Traffic Based on Recursive Scalable Poisson Model

To date the prospects for using the accumulated over many years mathematical and software for the modeling of telecommunications with the Poisson input flow are under a big question. The matter is that a new fractal queuing theory is already on the threshhold. This article formulates and solves the problem of application of a queuing system model with a Poisson incoming flow for the purposes of server modeling described by QS with self-similar incoming traffic of the "fractal Brownian motion" type (according to Norros). Based on the results of the morphological analysis, the Norros model was decomposed into Poisson components connected by a scalable recurrence scheme. The variance of the number of packets in the server, raised to the power determined by the Hurst parameter acts as the similarity coefficient of fractal and Poisson QSs. The method for rescaling Poisson solutions into fractal solutions was constructed on the basis of the similarity coefficient. According to this method in order to find the fractal delay of access, the Poisson delay should be multiplied by the similarity coefficient, and to estimate the probability of packet loss, it is necessary to extract a root of degree equal to the similarity coefficient from classical exponential losses. The scope of the re-scaling method focuses on the pre-project stages of creating telecommunications, where there is no need for high accuracy of simulation results.

Decomposition of quantum gates

A recurrence scheme is presented to decompose an n-qubit unitary gate to the product of no more than N(N - 1)/2 single qubit gates with small number of controls, where N = 2n. Detailed description of the recurrence steps and formulas for the number of k-controlled single qubit gates in the decomposition are given. Comparison of the result to a previous scheme is presented, and future research directions are discussed.

Asymptotics in Random Recursive Circuits*

A random recursive circuit is a combinatorial structure with a stochastic growth rule. These models are profusely used in computer science as data structures and in other fields, such as biostatistics, to model the spread of epidemics. The evolution of this random structure can be described by means of a generalized Polya urn model where different colors of balls represent different kind of nodes of the circuit. We prove that the process that represents the proportion of balls of each color fits a Robbins–Monro stochastic recurrence scheme and, then, sufficient conditions are established in order to obtain almost sure convergence and a central limit theorem for this process.

Random site dilution properties of frustrated magnets on a hierarchical lattice

We present a method to analyze the magnetic properties of frustrated Ising spin models on specific hierarchical lattices with random dilution. Disorder is induced by dilution and geometrical frustration rather than randomness in the internal couplings of the original Hamiltonian. The two-dimensional model presented here possesses a macroscopic entropy at zero temperature in the large size limit, very close to the Pauling estimate for spin-ice on the pyrochlore lattice, and a crossover towards a paramagnetic phase. The disorder due to dilution is taken into account by considering a replicated version of the recursion equations between partition functions at different lattice sizes. An analysis to first order in replica number allows a systematic reorganization of the disorder configurations, leading to a recurrence scheme. This method is numerically implemented to evaluate thermodynamical quantities such as specific heat and susceptibility in an external field.

A general symplectic method for the response analysis of infinitely periodic structures subjected to random excitations

A general symplectic method for the random response analysis of infinitely periodic structures subjected to stationary/non-stationary random excitations is developed using symplectic mathematics in conjunction with variable separation and the pseudo-excitation method (PEM). Starting from the equation of motion for a single loaded substructure, symplectic analysis is firstly used to eliminate the dependent degrees of the freedom through condensation. A Fourier expansion of the condensed equation of motion is then applied to separate the variables of time and wave number, thus enabling the necessary recurrence scheme to be developed. The random response is finally determined by implementing PEM. The proposed method is justified by comparison with results available in the literature and is then applied to a more complicated time-dependent coupled system.

Recursive evaluation of space-time lattice Green's functions

Up to a multiplicative constant, the lattice Green's function (LGF) as defined in condensed matter physics and lattice statistical mechanics is equivalent to the Z-domain counterpart of the finite-difference time-domain Green's function (GF) on a lattice. Expansion of a well-known integral representation for the LGF on a ν-dimensional hyper-cubic lattice in powers of Z−1 and application of the Chu–Vandermonde identity results in ν − 1 nested finite-sum representations for discrete space-time GFs. Due to severe numerical cancellations, these nested finite sums are of little practical use. For ν = 2, the finite sum may be evaluated in closed form in terms of a generalized hypergeometric function. For special lattice points, that representation simplifies considerably, while on the other hand the finite-difference stencil may be used to derive single-lattice-point second-order recurrence schemes for generating 2D discrete space-time GF time sequences on the fly. For arbitrary symbolic lattice points, Zeilberger's algorithm produces a third-order recurrence operator with polynomial coefficients of the sixth degree. The corresponding recurrence scheme constitutes the most efficient numerical method for the majority of lattice points, in spite of the fact that for explicit numeric lattice points the associated third-order recurrence operator is not the minimum recurrence operator. As regards the asymptotic bounds for the possible solutions to the recurrence scheme, Perron's theorem precludes factorial or exponential growth. Along horizontal lattices directions, rapid initial growth does occur, but poses no problems in augmented dynamic-range fixed precision arithmetic. By analysing long-distance wave propagation along a horizontal lattice direction, we have concluded that the chirp-up oscillations of the discrete space-time GF are the root cause of grid dispersion anisotropy. With each factor of ten increase in the lattice distance, one would have to roughly double the pulse width of the source signature to keep pulse distortion at bay. The GF time sequences can also be used for an efficient computation of discrete space-frequency LGFs, especially if one employs Aitken's δ2 process for the acceleration of the convergence of the consecutive partial sums.

Nonlinear viscoelastic analysis of orthotropic beams using a general third-order theory

The displacement based finite element model of a general third-order beam theory is developed to study the quasi-static behavior of viscoelastic rectangular orthotropic beams. The mechanical properties are considered to be linear viscoelastic in nature with a scope to undergo von Kármán nonlinear geometric deformations. A differential constitutive law is developed for an orthotropic linear viscoelastic beam under the assumptions of plane-stress. The fully discretized finite element equations are obtained by approximating the convolution integrals using a trapezoidal rule. A two-point recurrence scheme is developed that necessitates storage of data from the previous time step only, and not from the entire deformation history. Full integration is used to evaluate all the stiffness terms using spectral/hp lagrange polynomials. The Newton iterative scheme is employed to enhance the rate of convergence of the nonlinear finite element equations. Numerical examples are presented to study the viscoelastic phenomena like creep, cyclic creep and recovery for thick and thin beams using classical mechanical analogues like generalized n-parameter Kelvin-Voigt solids and Maxwell solids.

Spontaneous remission in thyroid cancer patients after biochemical incomplete response to initial therapy

To validate the American Thyroid Association (ATA) initial risk of recurrence scheme and the Memorial Sloan Kettering Cancer Center (MSKCC) response to therapy re-stratification approach in a large cohort of patients with differentiated thyroid cancer (DTC) treated outside of the United States. Retrospective chart review. Five hundred and six patients with DTC followed for a median of 10 years after total thyroidectomy and RAI remnant ablation at a major cancer centre in Brazil. Final clinical outcomes were assessed based on American Joint Cancer Committee (AJCC)/Union Internationale Contre le Cancer (UICC) staging, ATA risk stratification and response to therapy assessment (excellent, acceptable, biochemical incomplete and structural incomplete). The AJCC/UICC staging system did not adequately stratify patients with regard to the risk of recurrence/persistent disease. However, the ATA system demonstrated a 13% risk of recurrent/persistent disease in low-risk patients, 36% in intermediate risk patients, and 68% in high-risk patients. Furthermore, an excellent response to therapy decreased the risk of recurrent/persistent disease to 1·4%. At the time of final follow-up, 34% of the biochemical incomplete response patients had been re-classified as having no evidence of disease (NED) without having received any additional therapy beyond continue levothyroxine suppression. Conversely, even after additional therapies, only 9% of the patients with an incomplete structural response were eventually re-classified as NED. These data validate the ATA risk classification as an excellent initial predictor of recurrent/persistent disease and confirm the clinical utility of the MSKCC dynamic risk assessment system in a cohort of patients evaluated and treated outside the United States.