Narrow Results

The existence and stability of the Shapiro steps in the ac driven dissipative Frenkel-Kontorova model are studied. The particular attention has been focused on, the variations of the step width and critical depinning force with the ac amplitude. The amplitude dependence is strongly influenced by the frequency of ac force where at the higher frequencies, the oscillations have the Bessel like form.

Aspects of both hybrid and fractional calculus are combined in the (Proportional Caputo-Hybrid) $P_{\mathrm{\text{cap}}}$ operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the $P_{\mathrm{\text{cap}}}$ operators via multiplicative calculus which are termed as multiplicative $P_{\mathrm{\text{cap}}}$ operators. we initially formulate two H.H (Hermite–Hadamard)-type inequalities applicable to multiplicative (geometric) convex function via multiplicative $P_{\mathrm{\text{cap}}}$ operators. Subsequently, by leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex function via multiplicative $P_{\mathrm{\text{cap}}}$ operators also demonstrating two novel identities applicable to multiplicatively differentiable functions. By leveraging these identities, we then establish inequalities of trapezoid and midpoint types specifically designed for multiplicatively convex functions. Additionally, we explore applications of these findings to special functions and special means.

We prove in this paper that the minimizer of Lennard–Jones energy per particle among Bravais lattices is a triangular lattice, i.e. composed of equilateral triangles, in ℝ² for large density of points, while it is false for sufficiently small density. We show some characterization results for the global minimizer of this energy and finally we also prove that the minimizer of the Thomas–Fermi energy per particle in ℝ² among Bravais lattices with fixed density is triangular.

In this paper, buckling of a nanowire column subjected to self-weight and tip load is investigated. One end of the nanowire is free, while the other end is attached to a rotational spring support. Considering the equilibrium equations together with the Euler–Bernoulli beam theory, the governing differential equation describing the behavior of the column can be obtained. Effect of surface stress is also incorporated into the formulations in terms of transverse distributed loading. The differential equation has been solved analytically and the general solution can be presented in the terms of Bessel function of the first kind. Applying the boundary conditions, the characteristic equations influenced by surface stress and stiffness of the rotational spring at the support can be expressed and then the critical load can be determined using the Newton–Raphson iterative scheme. From the results, they reveal that the positive surface stress could strengthen the nanowire against the buckling. Fixity at the base is also influenced to the critical load where the increase of the stiffness of the spring results in the increase of critical load as well.

Myringoplasty is one of the routine surgeries in the treatment of tympanic membrane (TM) perforation. Since the anatomic structure of the middle ear cannot be simulated in clinical treatment, the surgery is mainly directed by experiences. Based on the mechanical properties of TM in the anatomy, four hypotheses are presented and TM is simplified as a sectorial annulus plate with fixed boundary condition. This paper proposes a free vibration model of TM. Its natural frequencies of free vibration are obtained by variables separation method and Bessel function. The system of fundamental solutions of fourth-order homogeneous equations can be solved for the analytical expressions of corresponding natural vibration mode. The theoretical model is proved to be valid since the natural frequency of the model is consistent with the experimental data. The effect of geometric parameters and material parameters on TM natural frequency is subsequently discussed in the numerical examples. Especially, the diameter and thickness of TM will cause different natural frequency errors above 40%, while the Young’s modulus and density of TM cause errors below 15% as well.

According to the vibration characteristics of the round window membrane, a mechanical model that contains round window membrane and the soft tissue is established. The Euler equation of the whole of round window membrane and the soft tissue and the complementary boundary conditions are derived by the variational principle. Combined with the Bessel function, the analytical solution of the total displacement of round window membrane and the soft tissue is obtained by using Mathematica. The results are in good agreement with experimental data, which confirms the validity of the analytical solution of the model. At the same time, the effect of different thicknesses and different elastic modulus of soft tissue on the total displacement of round window membrane and soft tissue is studied by analytical method. The results show that with the thickening of the soft tissue, the total displacement of round window membrane and the soft tissue decreased gradually. However, with the decrease of elastic modulus of the soft tissue, the total displacement of round window membrane and the soft tissue increased gradually. Furthermore, the relationship between thickness and elastic modulus of the soft tissue and the corresponding range selection is achieved, which can evaluate the transmission performance of round window membrane efficiently and provide theoretical basis for the reverse excitation of artificial prosthesis.

The Szegő–Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann–Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369–376]. Temme's approach started from an explicit expression of the Szegő–Askey polynomials in terms of an ₂F₁-function, and followed by integral methods.

The Bessel discrete variable representation (DVR) method is tested to describe the interaction of atomic hydrogen with intense laser fields by numerically solving the time-dependent Schrödinger equation. Using the Bessel functions of the first kind, the singular terms, r^-2 or r^-1 at the origin, in the kinetic energy operators are analytically solved. As an illustration example, the high-order harmonic generation (HOHG) spectra in atomic hydrogen is calculated in length and acceleration forms. From the numerical results, it is concluded that this simple Bessel DVR may be a useful method for describing the interaction of atomic hydrogen with intense laser fields.

Denote by J_ν the Bessel function of the first kind of order ν and μ_ν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szegö states that the sequence is decreasing, another theorem of theirs states that the sequence has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x^-ν+1|J_ν-1(x)|, x ∈ (0, ∞), i.e. the sequence has higher monotonicity properties.

The research on information quantization is important in the field of information theory. As a result, based on the quantum theory, the information was quantified from the information receiving aspect in this report. First of all, several concepts were presented, such as the InfoBar, the Amount of Information and the Power of Information as well as the algorithm of the Power of Information. Then, according to the relationship between the InfoBar and the amount of Information, the wave equation was decided based on the receiving information, meanwhile, the equation of wave function was defined as well. Finally, via the numerical simulation, the received model results as well as the sample result were basically matched. Thus, the validity of the model can be proved.

The paper studies the zeros of 2mI_m/I _{m - 1} - (m+1)I₁/I₀, where I_m are the Bessel functions.

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues’ density, and write the universal limit in function of the Kontsevich matrix model (“matrix Airy function”, as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral (“matrix Bessel function”) known in the literature as generalized Kontsevich matrix model.

In this paper, we introduce a new approach to the study of finite integral formulas associated with special functions, bringing out the incomplete Aleph functions and other well-known special functions. Our method is based on the finite integral operator applied to incomplete Aleph functions. This approach makes it easier to derive integral formulas that connect the incomplete Aleph functions using well-known special functions and algebraic expressions. As a result, we can group integral formulas and special functions into different classes according to similar features. Furthermore, we can discover new integral formulas with this framework that are useful for computational applications.

The minimal representation π of the indefinite orthogonal group O(m+1, 2) is realized on the Hilbert space of square integrable functions on ℝ^m with respect to the measure |x|⁻¹dx₁ ⋯ dx_m. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m+3, ℂ) by means of the Bessel function. Taking its ‘boundary value’, we also find the integral kernel of the ‘inversion operator’ corresponding to the inversion element on the Minkowski space ℝ ^{m, 1}.

The E(2) dynamical symmetry is realized with both the differential operators and boson operators. It is shown that the discrete spectrum of the E(2) dynamical symmetry just correspond to the solutions of the infinite square well Hamiltonian in two dimensions, of which the eigenfunction can be related with the Bessel equation of integral order.

Two asymptotic expansions are obtained for the Laguerre polynomial for large n and fixed α > −1. These expansions are uniformly valid in two overlapping intervals covering the entire x-axis. The leading terms of both agree with the two asymptotic formulas given by Erdélyi who used the theory of differential equations. Our approach is based on two integral representations for the Laguerre polynomials. The phase function of one of these integrals has two coalescing saddle points, and to this one the cubic transformation introduced by Chester, Friedman, and Ursell is applied. The phase function of the other integral also has two coalescing saddle points, but in addition it has a simple pole. Moreover, the saddle points coalesce onto this pole. In this case a rational transformation is used, which mimics the singular behavior of the phase function. In both cases explicit expressions are given for the remainders associated with the asymptotic expansions.