Expansion Formula Research Articles

Asymptotics of orbifold Bergman kernels: mixed curvature case

We derive some asymptotic expansion formulas of the Bergman kernel of high tensor powers of an Hermitian orbifold line bundle with mixed curvature tensored with an orbifold vector bundle on a compact symplectic orbifold. In particular, when the orbifold has isolated singularities, we get an explicit formula for the asymptotic expansion of the Bergman kernel in the distribution sense. Finally, by applying our results to the complex case, we get a Riemann–Roch–Kawasaki type formula.

Screening-Constant-by-Unit-Nuclear-Charge method investigations of high lying ([formula omitted]) [formula omitted], [formula omitted] Rydberg series in the photoionization spectra of the halogen-like ion Kr+

Energy positions and quantum defects of the 4s24p4 (1D2,1S0) ns, nd Rydberg series originating from the 4s24p52P3/2∘ ground state and from the 4s24p52P1/2∘ metastable state of Kr+ are reported. Calculations are performed using the Screening Constant by Unit Nuclear Charge (SCUNC) method. The results obtained are in suitable agreement with recent experimental data from the combined ASTRID merged-beam set up and Fourier Transform Ion Cyclotron Resonance device (Bizau et al., 2011), ALS measurements (Hinojosa et al., 2012), and multi-channel R-matrix eigenphase derivative calculations (McLaughlin and Balance, 2012). In addition, analysis of the 4s24p4(1D2)nd and the 4s24p4(1S0)nd resonances is given via the SCUNC procedure. The excellent results obtained from our work point out that the SCUNC formalism may be used to confirm the results of the analysis from the standard quantum-defect expansion formulas. Eventual errors occurring in the analysis can then be automatically detected and corrected via the SCUNC procedure.

Multi-variable Hermite matrix polynomials: Properties and applications

In this paper, the multi-variable Hermite matrix polynomials are introduced by algebraic decomposition of exponential operators. Their properties are established using operational methods. The matrix forms of the Chebyshev and truncated polynomials of two variable are also introduced, which are further used to derive certain operational representations and expansion formulae.

Explicit energy expansion for general odd-degree polynomial potentials

In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd-degree polynomial potentials of the form V (x) = (ix)2N+1 + β1x2N + β2x2N−1 + ··· + β2Nx, where β′k are real or complex for 1 ⩽ k ⩽ 2N. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order, very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters β1,β2… and β2N of the potential. Unlike in the even-degree polynomial case, the highest-order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex branch points, which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobiʼs triple product identity and the t-coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewellʼs and Chen–Chen–Huangʼs octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system A2 as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.

Dirac multimode ket-bra operators’ $\mathfrak{Q}$ -ordered and $\mathfrak{P}$ -ordered integration theory and general squeezing operator

We develop quantum mechanical Dirac ket-bra operator’s integration theory in \(\mathfrak{Q}\)-ordering or \(\mathfrak{P}\)-ordering to multimode case, where \(\mathfrak{Q}\)-ordering means all Qs are to the left of all Ps and \(\mathfrak{P}\)-ordering means all Ps are to the left of all Qs. As their applications, we derive \(\mathfrak{Q}\)-ordered and \(\mathfrak{P}\)-ordered expansion formulas of multimode exponential operator \(e^{ - iP_l \Lambda _{lk} Q_k } \). Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation \(\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \) is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.

Wave structure interaction problems in three-layer fluid

Wave structure interaction problems in a three-layer fluid having an elastic plate covered free surface are studied in a three-dimensional fluid domain in both the cases of finite and infinite water depths. Wave characteristics are analyzed from the dispersion relation of the associated wave motion, and approximate results are derived in both the cases of deep water and shallow water waves. Further, the expansion formulae and the associated orthogonal mode-coupling relations are derived for the velocity potentials for the wave structure interaction problems in channels of finite and infinite depths. The utility of the expansion formulae is demonstrated by (1) deriving the source potentials associated with the wave structure interaction problems in a three-layer fluid medium of finite and infinite water depths and (2) analyzing the wave scattering by a partially frozen crack in a floating ice sheet in the three-layer fluid medium in a three-dimensional channel of finite water depth. Various results derived can be used to deal with acoustic wave interaction with flexible structures and other wave structure interaction problems of similar nature arising in different branches of physics and engineering.

An expansion formula for fractional derivatives of variable order

Abstract In this work we extend our previous results and derive an expansion formula for fractional derivatives of variable order. The formula is used to determine fractional derivatives of variable order of two elementary functions. Also we propose a constitutive equation describing a solidifying material and determine the corresponding stress relaxation function.

Elasticity solutions for functionally graded annular plates subject to biharmonic loads

Based on England’s expansion formula for displacements, the elastic field in a transversely isotropic functionally graded annular plate subjected to biharmonic transverse forces on its top surface is investigated using the complex variables method. The material parameters are assumed to vary along the thickness direction in an arbitrary fashion. The problem is converted to determine the expressions of four analytic functions α (ζ), β (ζ), ϕ (ζ) and ψ (ζ) under certain boundary conditions. A series of simple and practical biharmonic loads are presented. The four analytic functions are constructed carefully in a biconnected annular region corresponding to the presented loads, which guarantee the single-valuedness of the mid-plane displacements of the plate. The unknown constants contained in the analytic functions can be determined from the boundary conditions that are similar to those in the plane elasticity as well as those in the classical plate theory. Numerical examples show that the material gradient index and boundary conditions have a significant influence on the elastic field.

Wave interaction with a floating and submerged elastic plate system

Surface gravity wave interaction with a floating and submerged elastic plate system is analyzed under the assumption of small-amplitude surface water wave theory and structural response. The plane progressive wave solution associated with the plate system is analyzed to understand the characteristics of the flexural gravity waves in different modes. Further, linearized long-wave equations associated with the wave interaction with the elastic plate system are derived. The dispersion relations are derived based on small-amplitude wave theory and shallow-water approximation and are compared to ensure the correctness of the mathematical formulation. To deal with various types of problems associated with gravity wave interaction with a floating and submerged flexible plate system, Fourier-type expansion formulae are derived in the cases of water of both finite and infinite depths in two dimensions. Certain characteristics of the eigensystems of the developed expansion formulae are derived. Source potentials for surface wave interaction with a floating flexible structure in the presence of a submerged flexible structure are derived and used in Green’s identity to obtain the expansion formulae for flexural gravity wavemaker problems in the presence of submerged flexible plates. The utility of the expansion formulae and associated orthogonal mode-coupling relations is demonstrated by investigating the diffraction of surface waves by floating and submerged flexible structures of two different configurations. The accuracy of the computational results is checked using appropriate energy relations. The present study is likely to provide fruitful solutions to problems associated with floating and submerged flexible plate systems of various configurations and geometries arising in ocean engineering and other branches of mathematical physics and engineering including acoustic structure interaction problems.

On an Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver

This paper presents a mathematical validity for an asymptotic expansion scheme of the solutions to the forward-backward stochastic differential equations (FBSDEs) in terms of a perturbed driver in the BSDE and a small diffusion in the FSDE. This computational scheme was proposed by Fujii-Takahashi (2012a), which has been successfully employed to solve the derivatives and optimal portfolio problems in Fujii-Takahashi (2012b, c) and Fujii et al. (2012). In particular, we represent the coefficients up to an arbitrary order expansion of the BSDE by the solution to a system of the associated BSDEs with the FSDE, and obtain the error estimate of the expansion with respect to the driver perturbation. Accordingly, we show a concrete representation for each expansion coefficient of the volatility component, that is the martingale integrand in the BSDE. Then, we apply our proposed FSDE expansion formula with its precise error estimate to the BSDE expansion coefficients to finally obtain the total residual estimate.

Asymptotic expansions of the gamma function and Wallis function through polygamma functions

A new view on classical asymptotic expansions of the logarithm of gamma function is given. Then general formulae for the asymptotic expansions of the logarithm of gamma function and the Wallis power function through polygamma functions are derived and analysed.

Closed-form solution to the scattering by an infinite lossless or lossy elliptic cylinder coating a circular metallic core

An analytical, closed-form solution to the scattering problem from an infinite lossless or lossy elliptical cylinder coating a circular metal core is treated in this work. The problem is solved by expressing the electromagnetic field in both elliptical and circular wave functions, connected with one another by well-known expansion formulas. The procedure for solving the problem is cumbersome because of the nonexistence of orthogonality relations for Mathieu functions across the dielectric elliptical boundary. The solution obtained, which is free of Mathieu functions, is given in closed form, and it is valid for small values of the eccentricity h of the elliptical cylinder. Analytical expressions of the form S(h)=S(0)[1+g(2)h2+g(4)h4+O(h6] are obtained, permitting an immediate calculation for the scattering cross sections. The proposed method is an alternative one, for small h, to the standard exact numerical solution obtained after the truncation of the system matrices, composed after the satisfaction of the boundary conditions. Both polarizations are considered for normal incidence. The results are validated against the exact solution, and numerical results are given for various values of the parameters.

Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition

This work aims to examine a Sturm-Liouville operator with a piece-wise continuous coefficient and a spectral parameter in boundary condition. The orthogonality of the eigenfunctions, realness and simplicity of the eigenvalues are investigated. The asymptotic formula of the eigenvalues is found, and the resolvent operator is constructed. It is shown that the eigenfunctions form a complete system and the expansion formula with respect to eigenfunctions is obtained. Also, the evolution of the Weyl solution and Weyl function is discussed. Uniqueness theorems for the solution of the inverse problem with Weyl function and spectral data are proved.

Expansion formula for fractional derivatives in variational problems

We modify the expansion formula introduced in [T.M. Atanacković, B. Stanković, An expansion formula for fractional derivatives and its applications, Fract. Calc. Appl. Anal. 7 (3) (2004) 365–378] for the left Riemann–Liouville fractional derivative in order to apply it to various problems involving fractional derivatives. As a result we obtain a new form of the fractional integration by parts formula, with the benefit of a useful approximation for the right Riemann–Liouville fractional derivative, and derive a consequence of the fractional integral inequality ∫0Ty⋅0Dtαydt≥0. Further, we use this expansion formula to transform fractional optimization (minimization of a functional involving fractional derivatives) to the standard constrained optimization problem. It is shown that when the number of terms in the approximation tends to infinity, solutions to the Euler–Lagrange equations of the transformed problem converge, in a weak sense, to solutions of the original fractional Euler–Lagrange equations. An illustrative example is treated numerically.

PRICING STEP OPTIONS UNDER THE CEV AND OTHER SOLVABLE DIFFUSION MODELS

We consider a special family of occupation-time derivatives, namely proportional step options introduced by [18]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

A Numerical Scheme to Solve Fractional Optimal Control Problems

We review recent results obtained to solve fractional order optimal control problems with free terminal time and a dynamic constraint involving integer and fractional order derivatives. Some particular cases are studied in detail. A numerical scheme is given, based on expansion formulas for the fractional derivative. The efficiency of the method is illustrated through examples.

THE INTERSECTION BETWEEN EUROPEAN PUT PRICE AND ITS PAYOFF FUNCTION

In this paper, we study the intersection between the price of a European put and its payoff function. We derive asymptotic expansion formulas for the intersection near expiration date for three different cases of risk-free rate r and dividend yield q, i.e. r > q, r = q, and r < q. The comparison with those of the critical stock price of an American option enhances our understanding on the convergence of the asymptotic expansion near a singular point.

Edgeworth type expansion of ruin probability under Lévy risk processes in the small loading asymptotics

This paper presents an asymptotic expansion of the ultimate ruin probability under Lévy insurance risks as the loading factor tends to zero. The expansion formula is obtained via the Edgeworth type expansion for compound geometric distributions. We give higher-order expansion of the ruin probability, any order of which is available in explicit form, and discuss a certain type of validity of the expansion. We shall also give applications to evaluation of the VaR-type risk measure due to ruin, and the scale function of spectrally negative Lévy processes.

On the Bagley–Torvik Equation

We propose a new method for finding solution of Bagley–Torvik equation based on recently derived expansion formula for fractional derivatives. The case of nonlinear equations of Bagley–Torvik type is also discussed.