Abstract In this study we are concerned with the properties of the sequence of coefficients ( β n ) n ⩾ 0 in the recurrence relation satisfied by the sequence of monic symmetric polynomials, orthogonal with respect to the symmetric sextic Freud weight ω ( x ; τ , t ) = exp ( − x 6 + τ x 4 + t x 2 ) , x ∈ R , with real parameters τ and t . It is known that the recurrence coefficients β n for the symmetric sextic Freud weight satisfy a fourth-order nonlinear discrete equation, which is a special case of the second member of the discrete Painlevé I hierarchy, often known as the ‘string equation’. The recurrence coefficients have been studied in the context of Hermitian one-matrix models and random symmetric matrix ensembles with researchers in the 1990s observing ‘chaotic, pseudo-oscillatory’ behaviour. More recently, this ‘chaotic phase’ was described as a dispersive shockwave in a hydrodynamic chain. Our emphasis is a comprehensive study of the behaviour of the recurrence coefficients as the parameters τ and t vary. Extensive computational analysis is carried out, using Maple, for critical parameter ranges, and graphical plots are presented to illustrate the behaviour of the recurrence coefficients as well as the complexity of the associated Volterra lattice hierarchy. The corresponding symmetric sextic Freud polynomials are shown to satisfy a second-order differential equation with rational coefficients. The moments of the weight are examined in detail, including their integral representations, differential equations, and recursive structure. Closed-form expressions for moments are obtained in several special cases in terms of generalised hypergeometric functions and modified Bessel functions, and asymptotic expansions for the recurrence coefficients are given. The results highlight the rich algebraic and analytic structures underlying the Freud weight and its connections to integrable systems.

Asymptotic properties of a special solution to the (3,4) string equation

The electric guitar is a musical instrument based on string vibration, that is well known by physics. This work presents the development of an experimental setup capable of verifying the vibrating string equation for electric guitar strings, by building an electric guitar prototype as well as a digital dynamometer prototype to measure guitar string tension. With the developed apparatus, it was possible to verify the relationship between vibrating string frequency and physical quantities such as length, tension, linear density and number of the harmonic associated with the string. Beyond the verification of the vibrating string equation, it was possible to characterize the guitar strings, as well as measure the differences between brands and types of strings. The methods were shown to be useful for academic study and can be applied in educational scenarios, or even in the musical instruments industry.

Stick–slip vibration is a common phenomenon in ultra-deep drilling that significantly impacts the failure of both drill bits and drill tools. The most direct and efficacious approach to alleviating the stick–slip vibration of the drill string in the downhole is to modify its external excitation. In recent years, the composite impact tools that can simultaneously offer axial and torsional excitation in the downhole have been applied, effectively reducing the stick–slip vibration of the drill string. However, the mechanical mechanism thereof remains undefined. In order to understand the nature of this phenomenon, A dynamic model of the drill string taking into account multi-directional excitations is presented. The governing nonlinear equations are obtained by using the Lagrangian approach, which take the work done by the multidirectional excitation into consider. The Hertz contact model is introduced considering the constraints of the wellbore, and the finite element node iteration method is employed to solve the dynamics equation of drill string. The axial vibration, torsion vibration and phase trajectory characteristics of the drill string under multidirectional excitation are analyzed, and the inhibitory effect of the excitations on stick–slip vibration is clarified. The results show that the vibration characteristics of the bottom hole assembly can be significantly altered through periodic axial and torsional excitations at higher frequencies, resulting in the emergence of high-frequency vibration responses. These responses exhibit a pronounced inhibitory effect on stick–slip suppressed.

This study explores the flow-induced vibration characteristics of composite tubing under the action of gas-liquid multiphase flow. Based on the Hamilton variational principle, the longitudinal-transverse coupling vibration control equation of the pipe string is established, which is solved by the Galerkin method and verified by the benchmark case. The parameter analysis shows that the stability of the system is affected by the internal flow velocity, gas content, fiber ply angle, bottom hole pressure and packer position. It is found that the critical velocity of multiphase flow is significantly higher than that of single-phase flow, and there is a linear relationship between the critical velocity and the gas fraction in a specific void fraction range. The critical velocity curve and natural frequency curve with the change of fiber ply angle show symmetrical distribution. Optimizing the position of the packer can generate extreme points in the critical flow rate and natural frequency, and achieve a synergistic peak of a specific configuration. The comparative analysis shows that the bottom hole pressure is the dominant stability factor, which significantly affects the vibration energy distribution. The numerical analysis data in this paper provide a theoretical basis for offshore oil exploitation and geothermal well application.

The aim of our work was to investigate how electric guitar strings wear out. There are many myths about string wear. We decided to investigate what the wear process looks like in real life. In our work, sound processing methods such as DTFT and spectrogram were used. However, the most important research method is the use of time-frequency analysis to study the sound of the string and its wear process. Another key method used in our work is the application of a phenomenon known from psychoacoustics, pitch. In our work, we have been able to show that the use of pitch in combination with time-frequency analysis makes it possible to demonstrate string wear. This was not achievable for previously known methods. We have also shown that the string yield limit is exceeded immediately when the strings are placed on the guitar neck. This affects the sound equation of the string. In this work, we have proposed a transformation of the classical string equation so that it correctly describes the sound of the string as it is worn. The research method we have developed, combining pitch and time-frequency analysis, could presumably be used in the future to study the wear and tear of other vibrating systems, such as bridges and viaducts.

We study the problem of minimizing the ratio of the first eigenvalues of vibrating string equations subject to the Robin boundary conditions for the class of concave weights. We show that, unlike the Dirichlet, Neumann and mixed boundary conditions, the constant weight is not minimizing for the class of concave weights. In addition, we prove a relation between the eigenvalues and real roots of the first Airy functions and their derivatives.

Generalised Canonical Systems Related to Matrix String Equations: Corresponding Structured Operators and High-Energy Asymptotics of Weyl Functions

Abstract In this paper, we investigate a series of W‐type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W‐constraints for tau‐functions of higher KdV hierarchies that satisfy the string equation. We will give simple uniform formulas for actions of these operators on all ordinary Schur functions and Schur Q‐functions. As applications of such formulas, we will give new simple proofs for Alexandrov's conjecture and Mironov–Morozov's formula, which express the Brézin–Gross–Witten and Kontsevich–Witten tau‐functions as linear combinations of Q‐functions with simple coefficients, respectively.

<p>This paper is concerned with a coupled system modeling the oscillations of suspension bridges, which consists of a beam equation and a viscoelastic string equation. We first transformed the original initial-boundary value problem into an equivalent one in the history space framework. Then we obtained the global well-posedness and regularity of mild solutions by using the semigroup theory. In addition, we employed the perturbed energy method to establish a stabilizability estimate. By verifying the gradient property and quasi-stability of the corresponding dynamical system, we derived the existence of a global attractor with finite fractal dimension.</p>

Due to its significance in numerous scientific and engineering domains, discrete fractional calculus (DFC) has received much attention recently. In particular, it seems that the exploration of the stability of DFC is crucial. A mathematical model of the discrete fractional equation describing the deflection of a vertical column along with two-point boundary conditions featuring the Riemann–Liouville operator is constructed to study several kinds of Ulam stability results in this research work. In addition, we developed Lyapunov-type inequality and its application to an eigenvalue problem for discrete fractional rotating string equations. Finally, the effectiveness of the theoretical findings is demonstrated with numerical examples.

On the completeness property of root vector systems for 2 × 2 Dirac type operators with non-regular boundary conditions

The model of two dimensional quantum gravity defining the Virasoro minimal string, presented recently by Collier, Eberhardt, Mühlmann, and Rodriguez, was also shown to be perturbatively (in topology) equivalent to a random matrix model. An alternative definition is presented here, in terms of double-scaled orthogonal polynomials, thereby allowing direct access to nonperturbative physics. Already at leading order, the defining string equation’s properties yield valuable information about the nonperturbative fate of the model, confirming that the case (c=25,c^=1) (central charges of spacelike and timelike Liouville sectors) is special, by virtue of sharing certain key features of the N=1 supersymmetric JT gravity string equation. Solutions of the full string equation are constructed using a special limit, and the (Cardy) spectral density is completed to all genus and beyond. The distributions of the underlying discrete spectra are readily accessible too, as is the spectral form factor. Some examples of these are exhibited. Published by the American Physical Society 2024

A random matrix model definition of a family of N=1 supersymmetric extensions of the Virasoro minimal string of Collier, Eberhardt, Mühlmann, and Rodriguez is presented. An analysis of the defining string equations shows that the models all naturally have unambiguous nonperturbative completions, which are explicitly supplied by the double-scaled orthogonal polynomial techniques employed. Perturbatively, the multiloop correlation functions of the model define a special supersymmetric class of “quantum volumes,” generalizing the prototype case, some of which are computed. Published by the American Physical Society 2024

Global stability of the plane wave solutions to the relativistic string equation with non-small perturbations

Recent measurements at the LHC have revealed heavy-flavour baryon fractions much larger than those observed at LEP, with e.g., Λc+/D0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Lambda}_c^{+}/{\ extrm{D}}^0 $$\\end{document} and Λb0/B0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Lambda}_b^0/{\ extrm{B}}^0 $$\\end{document} reaching ∼ 0.5 at low p⊥. One scenario that has been at least partly successful in predicting observed trends is QCD colour reconnections with string junctions. In previous work, however, the limit of a low-p⊥ heavy quark was not well defined. We reconsider the string equations of motion for junction systems in this limit, and find that the junction effectively becomes bound to the heavy quark, a scenario we refer to as a “pearl on a string”. We extend string-junction fragmentation in Pythia with a dedicated modelling of this limit for both light- and heavy-quark “pearls”.

We consider matrix models exhibiting open-closed string duality in two-dimensional string theories with various amounts of supersymmetry. In particular, a relationship between matrix models in the β = 2 Wigner-Dyson class and models in the (1 + 2Γ, 2) Altland-Zirnbauer class relates the perturbative solutions of the two systems’ string equations. Point-like operator insertions in the closed string theory are mapped to the topological expansion of the free energy in the open string theory. We compute correlation functions of macroscopic loop operators and FZZT branes in a general topological gravity background. The relationship between the topological recursion of moduli space volumes and branes is discussed by analyzing the Virasoro conditions in the matrix models.

We propose a connection between the newly formulated Virasoro minimal string and the well-established (2,2m−1) minimal string via their dual matrix models by deriving the string equation of the Virasoro minimal string using the expansion of its density of states in powers of Em+1/2. The string equation is expressed as a power series involving double-scaled multicritical matrix models, which are dual to (2,2m−1) minimal strings. This reformulation of the Virasoro minimal string enables us to employ matrix theory tools to compute its n-boundary correlators. We analyze the scaling behavior of n-boundary correlators and quantum volumes V0,n(b)(ℓ1,…,ℓn) in the JT gravity limit.

Simulating surface conditions by solving the wave equation of a sucker-rod string is the theoretical basis of a sucker-rod pumping system. To overcome the shortcomings of the conventional finite difference method and analytical solution, this work describes a novel hybrid method that combines the analytical solution with the finite difference method. In this method, an analytical solution of the tapered rod wave equation with a recursive matrix form based on the Fourier series is proposed, a unified pumping condition model is established, a modified finite difference method is given, a hybrid strategy is established, and a convergence calculation method is proposed. Based on two different types of oil wells, the analytical solutions are verified by comparing different methods. The hybrid method is verified by using the finite difference method simulated data and measured oil data. The pumping speed sensitivity and convergence of the hybrid method are studied. The results show that the proposed analytical solution has high accuracy, with a maximum relative error relative to that of the classical finite difference method of 0.062%. The proposed hybrid method has a high simulation accuracy, with a maximum relative area error relative to that of the finite difference method of 0.09% and a maximum relative area error relative to measured data of 1.89%. Even at higher pumping speeds, the hybrid method still has accuracy. The hybrid method in this paper is convergent. The introduction of the finite difference method allows the hybrid method to more easily converge. The novelty of this work is that it combines the advantages of the finite difference method and the analytical solution, and it provides a convergence calculation method to provide guidance for its application. The hybrid method presented in this paper provides an alternative scheme for predicting the behavior of sucker-rod pumping systems and a new approach for solving wave equations with complex boundary conditions.

Abstract Two remarkable facts about Jackiw–Teitelboim (JT) two-dimensional dilaton-gravity have been recently uncovered: this theory is dual to an ensemble of quantum mechanical theories; and such ensembles are described by a random matrix model which itself may be regarded as a special (large matter-central-charge) limit of minimal string theory. This work addresses this limit, putting it in its broader matrix-model context; comparing results between multicritical models and minimal strings (i.e. changing in-between multicritical and conformal backgrounds); and in both cases making the limit of large matter-central-charge precise (as such limit can also be defined for the multicritical series). These analyses are first done via spectral geometry, at both perturbative and nonperturbative levels, addressing the resurgent large-order growth of perturbation theory, alongside a calculation of nonperturbative instanton-actions and corresponding Stokes data. This calculation requires an algorithm to reach large-order, which is valid for arbitrary two-dimensional topological gravity. String equations—as derived from the Gel’fand–Dikii construction of the resolvent—are analyzed in both multicritical and minimal string theoretic contexts, and studied both perturbatively and nonperturbatively (always matching against the earlier spectral-geometry computations). The resulting solutions, as described by resurgent transseries, are shown to be resonant. The large matter-central-charge limit is addressed—in the string-equation context—and, in particular, the string equation for JT gravity is obtained to next derivative-orders, beyond the known genus-zero case (its possible exact-form is also discussed). Finally, a discussion of gravitational perturbations to Schwarzschild-like black hole solutions in these minimal-string models, regarded as deformations of JT gravity, is included—alongside a brief discussion of quasinormal modes.