Nonlinear Recurrence Relations Research Articles

A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation

We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation which determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.

Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach

In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform (AT) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The AT converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results.

Casting light on shadow Somos sequences

AbstractRecently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the $\mathbb{C}$ -algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.

Implementation of Elliptic Net Scalar Multiplication Computation for NIST P-192 Curve using Python

Elliptic curve cryptography is one of the most efficient public-key cryptosystems compared to the Rivest-Shamir-Addleman scheme. One of the methods to compute elliptic curve scalar multiplication is division polynomials which utilize the non-linear recurrence relation also known as the elliptic net. Previous studies were related to elliptic curve scalar multiplication via elliptic net which implemented the use of C++, GP/PARI or Java programming. This study aims to implement the Python programming in the SageMath compiler for computing elliptic net scalar multiplication based on the National Institute of Standards and Technology curve of the type P-192. The implementation could be seen at three main processes which are the scalar decomposition, the blocks of elliptic net generation namely the initial and final blocks, as well as scalar multiplication computation. The base point and prime field of the short Weierstrass curve with a large scalar are the parameters used in the process. The results showed that the implemented programming was easier while working with a large prime number field, vulnerable to errors and can be easily verified. The implemented programming can also be used to compute scalar multiplication on other standardizations of elliptic curve cryptography such as Brainpool 384-bit prime field or NUMS 256-bit prime field curves.

Uniform hydrostatic stresses inside a coated non-parabolic inhomogeneity in the vicinity of a concentrated couple

We establish the existence of an internal uniform hydrostatic state of stress inside a coated non-parabolic open elastic inhomogeneity when the surrounding matrix is subjected to a concentrated couple and uniform remote in-plane stresses. The proof of existence is accomplished through analytic continuation and the introduction of a specialized form of the conformal mapping function containing an infinite number of first-order poles. The complex constants appearing in the mapping function can be uniquely determined from a resulting set of non-linear recurrence relations for given material, geometric and loading parameters. The convergence criterion for the series in the mapping function is rigorously established. A simple condition on the remote loading in the matrix is found for given elastic constants of the inhomogeneity and the matrix in order to ensure that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. The analysis indicates that the internal uniform hydrostatic stress field, the constant mean stress in the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are all independent of the specific open shapes of the two interfaces and are also independent of the existence of the nearby concentrated couple. However, the non-parabolic open shapes of the two interfaces are significantly influenced by the concentrated couple. We also accomplish the design of an uncoated non-parabolic harmonic elastic inhomogeneity in the presence of an arbitrary number of concentrated couples applied in the matrix.

Using nonlinear difference equations to study Quicksort algorithms

Using nonlinear difference equations, combined with symbolic computations, we make a detailed study of the running times of numerous variants of the celebrated Quicksort algorithms, where we consider the variants of single-pivot and multi-pivot Quicksort algorithms as discrete probability problems. With nonlinear difference equations, recurrence relations and experimental mathematics techniques, explicit expressions for expectations, variances and even higher moments of their numbers of comparisons and swaps can be obtained. For some variants, Monte Carlo experiments are performed, the numerical results are demonstrated and the scaled limiting distribution is also discussed.

Linear fractional transformations and nonlinear leaping convergents of some continued fractions

For $$\alpha _0 = \left[ a_0, a_1, \ldots \right] $$ an infinite continued fraction and $$\sigma $$ a linear fractional transformation, we study the continued fraction expansion of $$\sigma (\alpha _0)$$ and its convergents. We provide the continued fraction expansion of $$\sigma (\alpha _0)$$ for four general families of continued fractions and when $$\left| \det \sigma \right| = 2$$. We also find nonlinear recurrence relations among the convergents of $$\sigma (\alpha _0)$$ which allow us to highlight relations between convergents of $$\alpha _0$$ and $$\sigma (\alpha _0)$$. Finally, we apply our results to some special and well-studied continued fractions, like Hurwitzian and Tasoevian ones, giving a first study about leaping convergents having steps provided by nonlinear functions.

Applications of derivative and difference operators on some sequences

In this study, depending on the upper and the lower indices of the hyperharmonic number h(r), nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.

Quasirenormalizable Quantum Field Theories

Leading logarithms (LLs) in massless non-renormalizable effective field theories (EFTs) can be computed with the help of non-linear recurrence relations. These recurrence relations follow from the fundamental requirements of unitarity, analyticity and crossing symmetry of scattering amplitudes and generalize the renormalization group technique for the case of non-renormalizable EFTs. We review the existing exact solutions of non-linear recurrence relations relevant for field theoretical applications. We introduce the new class of quantum field theories (quasi-renormalizable field theories) in which the resummation of LLs for $2 \to 2$ scattering amplitudes gives rise to a possibly infinite number of the Landau poles.

Free subgroups of free products and combinatorial hypermaps

We derive a generating series for the number of free subgroups of finite index in Δ+=Zp∗Zq by using a connection between free subgroups of Δ+ and certain hypermaps (also known as ribbon graphs or “fat” graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy.We also study the generating series for conjugacy classes of free subgroups of finite index in Δ+, which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.

Internal uniform hydrostatic stresses in a three-phase non-elliptical inclusion subjected to a nearby concentrated couple

We apply conformal mapping techniques with analytic continuation to study the existence of a uniform hydrostatic stress field inside a non-elliptical inclusion bonded to an infinite matrix via a finite thickness interphase layer when the matrix is simultaneously subjected to a concentrated couple as well as uniform remote in-plane stresses. We show that the desired internal uniform hydrostatic stress field is possible for given material and geometric parameters provided a certain constraint is placed on the remote loading. Subsequently, when the single loading parameter, five material parameters and three geometric parameters are prescribed, all of the unknown complex coefficients appearing in the series representing the corresponding conformal mapping function can be uniquely determined from a set of nonlinear recurrence relations. We find that the internal uniform hydrostatic stress field, the constant mean stress in the interphase layer and the hoop stress along the inner interface on the interphase layer side are all unaffected by the existence of the concentrated couple whereas the non-elliptical shape of the (three-phase) inclusion is attributed solely to the influence of the nearby concentrated couple.

A Nonlinear Extension of Fibonacci Sequence

A new extension of Fibonacci sequence which yields a nonlinear second order recurrence relation is defined. Some identities and congruence properties for the new sequence are obtained.

The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

The diagonal spin–spin correlations of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation . The later recurrence relations are an example of discrete Garnier equations which, in turn, are extensions to a ‘discrete Painlevé V’ system.

Mitigation of premixed flame-sustained thermoacoustic oscillations using an electrical heater

To stabilize unstable combustion systems in gas turbines, aero-engines, rocket motors, boilers and furnaces, there is a need to develop a simple but applicable mitigation/control strategy. The present work considers using an electrical heater to mitigate limit cycle thermoacoustic oscillations sustained by a premixed flame. For this, nonlinear recurrence relation analysis of the thermoacoustic system with the premixed flame and an electrical heater confined is conducted first. It is shown that the heater placed at a proper location results in significant damping on the thermoacoustic oscillations. Experimental measurements are then performed. It shows that the heater can dampen thermoacoustic oscillations, depending strongly on (1) its location and (2) its output electrical power. The minimum electrical power Qmins is determined and compared for four different axial locations. The optimum position xopts at which the heater is most effective in damping combustion-excited limit cycle oscillations is shown to be at 0.72⩽xs/L⩽0.75 for the eigenmode with frequency ω1/2π≈240Hz. This finding agrees well with those from the energy exchange analyses via linearized flame and heater models. It is also experimentally shown that sound pressure level is reduced by 70dB, as the heater is placed at xopts and its output power is increased to 218.2W. Furthermore, the thermoacoustic mode frequencies are found to be increased with the limit cycle oscillating being dampened. The successful experimental demonstration opens up another possible way to mitigate/control thermoacoustic oscillations using an electrical heater.

Periodic Solutions of Certain Differential Equations with Piecewise Constant Argument

Existence criteria are derived for the eventually periodic solutions of a class of differential equations with piecewise constant argument whose solutions at consecutive integers satisfy nonlinear recurrence relations. The proof characterizes the initial values of periodic solutions in terms of the coefficients of the resulting difference equations. Sufficient conditions for the unboundedness, boundedness, and symmetry of general solutions also follow from the corresponding properties of the difference equations.

Nonlinear recurrences related to Chebyshev polynomials

We use two nonlinear recurrence relations to define the same sequence of polynomials, a sequence resembling the Chebyshev polynomials of the first kind. Among other properties, we obtain results on their irreducibility and zero distribution. We then study the \(2\times 2\) Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval \((-1,1)\), while those of the other half lie on the unit circle. Some further extensions and generalizations of these results are indicated.

COMPLETE PERIODICITY ANALYSIS FOR A DISCONTINUOUS RECURRENCE EQUATION

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang bang control is studied. A complete analysis of its periodic behavior is given. In particular, we show that each solution is periodic and its prime period can be determined by two of its consecutive terms. By means of our periodicity analysis, we may then solve the steady state problem which to our knowledge is not solved by other means.

Reduction of order of cluster-type recurrence relations

Certain nonlinear recurrence relations (of the real line) can be studied within the framework of cluster algebra theory. For this type of relations we develop the tools of Poisson and pre-symplectic structures compatible with a cluster algebra, in order to understand how these structures enable to reduce the recurrence relation to one of lower order. Several examples are worked in detail.

Higher order analogues of Tracy–Widom distributions via the Lax method

We study the distribution of the largest eigenvalue in Hermitian one-matrix models when the spectral density acquires an extra number of k − 1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy nonlinear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990s. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painlevé XXXIV hierarchy. These are the higher order analogues of the Tracy–Widom distribution. Using known Bäcklund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painlevé II hierarchy.

Complete set of periodic solutions of a discontinuous recurrence equation

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang‐bang control is studied. We show that each solution is periodic and its prime period can be determined by three of its consecutive terms. By means of this periodicity property, we may then solve the steady state problem which to our knowledge is not solved by other means.