General existence principles for Stieltjes differential equations with applications to mathematical biology

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.     

Characterization of semi-classical orthogonal polynomials on nonuniform lattices

We state and prove characterization theorem for semi-classical orthogonal polynomials on nonuniform lattices (quadratic lattices of a discrete or q-discrete variable). This theorem proves the equivalence between the four characterization properties, namely, the Pearson type equation for the linear functional, the strictly quasi-orthogonality of the derivatives, the structure relation, and the Riccati equation for the formal Stieltjes function. We give the classification of the semi-classical linear functional of class one on nonuniform lattice. Using the definition and the properties of the associated orthogonal polynomials, we prove that semi-classical orthogonal polynomials satisfy the second-order divided difference equation on nonuniform lattices.     

On generalized Stieltjes-Wigert and related orthogonal polynomials

The Charlier, Wall, and generalized Stieltjes-Wigert polynomials are characterized by a property involving the concept of kernel polynomials. This characterization leads to consideration of a certain functional equation satisfied by solutions of the associated Stieltjes moment problem. All distribution functions which satisfy this functional equation are found up to singular functions. This yields new distribution functions, both discrete and absolutely continuous, with respect to which generalized Stieltjes-Wigert polynomials are orthogonal.     

Computation of the eigenvalues of Fredholm–Stieltjes integral equations

The Rayleigh–Ritz and the inverse iteration methods are used in order to compute the eigenvalues of Fredholm–Stieltjes integral equations, i.e. Fredholm equations with respect to suitable Stieltjes-type measures. Some applications to the so-called ‘charged’ (in German ‘belastete’) integral equation, and particularly the problem of computing the eigenvalues of a string charged by a finite number of cursors are given.     

Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue–Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.     

The attractive Coulomb potential polynomials

TheJ matrix method in quantum mechanics developed by Heller, Reinhardt, and Yamani points to a set of orthogonal polynomials having a nonempty continuous spectrum in addition to an infinite discrete spectrum. Asymptotic methods are used to investigate the spectral properties of these polynomials. We also obtain generating functions for both numerator and denominator polynomials. The corresponding continued fraction is computed and the Stieltjes inversion formula is used to determine the distribution function.     

A 2-component Camassa-Holm Equation,Euler-Bernoulli Beam Problem,and Noncommutative Continued Fractions

A new approach to the Euler-Bernoulli beam based on an inhomogeneous matrix string problem is presented. Three ramifications of the approach are developed:

1. motivated by an analogy with the Camassa-Holm equation a class of isospectral deformations of the beam problem is formulated;
2. a reformulation of the matrix string problem in terms of a certain compact operator is used to obtain basic spectral properties of the inhomogeneous matrix string problem with Dirichlet boundary conditions;
3. the inverse problem is solved for the special case of a discrete Euler-Bernoulli beam. The solution involves a noncommutative generalization of Stieltjes’ continued fractions, leading to the inverse formulas expressed in terms of ratios of Hankel-like determinants.

© 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.     

Special functions as solutions to discrete Painlevé equations

We present results on special solutions of discrete Painlevé equations. These solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second-order (discrete) equations. These linear equations define the discrete analogues of special functions.     

Canonical spectral equation for empirical covariance matrices

We study asymptotic properties of normalized spectral functions of empirical covariance matrices in the case of a nonnormal population. It is shown that the Stieltjes transforms of such functions satisfy a socalled canonical spectral equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1176–1189, September, 1995.     

On the convergence of inexact Newton methods for discrete-time algebraic Riccati equations

In this paper, we present a convergence analysis of the inexact Newton method for solving Discrete-time algebraic Riccati equations (DAREs) for large and sparse systems. The inexact Newton method requires, at each iteration, the solution of a symmetric Stein matrix equation. These linear matrix equations are solved approximatively by the alternating directions implicit (ADI) or Smith?s methods. We give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact Newton sequences. We show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d-stability. The theoretical results developed in this paper are an extension to the discrete case of the analysis performed by Feitzinger et al. (2009) [8] for the continuous-time algebraic Riccati equations. In the last section, we give some numerical experiments.     

A discrete Schrödinger equation via optimal transport on graphs

In 1966, Edward Nelson presented an interesting derivation of the Schrödinger equation using Brownian motion. Recently, this derivation is linked to the theory of optimal transport, which shows that the Schrödinger equation is a Hamiltonian system on the probability density manifold equipped with the Wasserstein metric. In this paper, we consider similar matters on a finite graph. By using discrete optimal transport and its corresponding Nelson's approach, we derive a discrete Schrödinger equation on a finite graph. The proposed system is quite different from the commonly referred discretized Schrödinger equations. It is a system of nonlinear ordinary differential equations (ODEs) with many desirable properties. Several numerical examples are presented to illustrate the properties.     

Reduced Finite Element Discretizations of the Stokes and Navier-Stokes Equations

ABSTRACT

If finite element spaces for the velocity and pressure do not satisfy the Babu?ka-Brezzi condition, a stable conforming discretization of the Stokes or Navier-Stokes equations can be obtained by enriching the velocity space by suitable functions. Writing any function from the enriched space as a sum of a function from the original space and a function from the supplementary space, the discretization will contain a number of additional terms compared with a conforming discretization for the original pair of spaces. We show that not all these terms are necessary for the solvability of the discrete problem and for optimal convergence properties of the discrete solutions, which is useful for saving computer memory and for establishing a connection to stabilized methods.     

Indeterminacy Criteria for the Stieltjes Matrix Moment Problem

In this paper, we obtain criteria for the indeterminacy of the Stieltjes matrix moment problem. We obtain explicit formulas for Stieltjes parameters and study the multiplicative structure of the resolvent matrix. In the indeterminate case, we study the analytic properties of the resolvent matrix of the moment problem. We describe the set of all matrix functions associated with the indeterminate Stieltjes moment problem in terms of linear fractional transformations over Stieltjes pairs.     

Well-posedness of second order evolution equation on discrete time

We characterize the well-posedness for second order discrete evolution equations in unconditional martingale difference spaces by means of Fourier multipliers and R-boundedness properties of the resolvent operator which defines the equation. Applications to semilinear problems are given.     

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids

In this article, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long‐time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all time is established in a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality‐type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.     

Discrete May–Leonard Competition Models I

The local dynamics of two discrete-time models applicable to three competing plant species are shown to have properties similar to the May–Leonard (M-L) differential equation model. The dynamics of the two discrete models are shown to be similar. However, they are not dynamically consistent with the continuous M-L model. Unlike the continuous M-L model, the Hopf bifurcations of the discrete M-L models are not degenerate. The continuous M-L model is the limiting case of the discrete models.     

Convergence of Solutions of Discrete Reflected Backward SDE＇s and Simulations

The objective of this paper is to introduce elementary discrete reflected backward equations and to give a simple method to discretize in time a （continuous） reflected backward equation. A presentation of numerical simulations is also described.     

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.     

On the zeros of complex Van Vleck polynomials

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials, have been studied extensively in the case of real number parameters. In the complex case, relatively little is known. Numerical investigations of the location of the zeros of the Stieltjes and Van Vleck polynomials in special cases reveal intriguing patterns in the complex case, suggestive of a deeper structure. In this article we report on these investigations, with the main result being a proof of a theorem confirming that the zeros of the Van Vleck polynomials lie on special line segments in the case of the complex generalized Lamé equation having three free parameters. Furthermore, as a result of this proposition, we are able to obtain in this case a strengthening of a classical result of Heine on the number of possible Van Vleck polynomials associated with a given Stieltjes polynomial.     

Single-Server Queue with Markov-Dependent Inter-Arrival and Service Times

In this paper we study a single-server queue where the inter-arrival times and the service times depend on a common discrete time Markov chain. This model generalizes the well-known MAP/G/1 queue by allowing dependencies between inter-arrival and service times. The waiting time process is directly analyzed by solving Lindley's equation by transform methods. The Laplace–Stieltjes transforms (LST) of the steady-state waiting time and queue length distribution are both derived, and used to obtain recursive equations for the calculation of the moments. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the inter-arrival and service times. An erratum to this article is available at .