Semi-analytical Formula Research Articles

A method of treating the nearly singular integral in calculation of sound radiation with BEM

For the nearly singular integral of three-dimensional acoustic boundary element method (BEM), based on the 6-noded triangular isoparametric element, a new semi-analytical algorithm of three-dimensional high order element is proposed in this paper. Using Taylor expansion of trigonometric functions in the three dimensional acoustic fundamental solutions, the singular part of the fundamental solutions is separated. Based on the geometric characteristics of the 6-noded triangular element, an approximate singular kernel function is constructed which has the same singularity as singular integral kernel function. Subtracting the approximate kernel function from the kernel function of the singular integral, the latter is decomposed into a regular kernel function and an approximate singular kernel function. The integral of the regular kernel function can be calculated accurately by using the conventional Gauss numerical quadrature. The integral of the new singular part is calculated by the semi-analytic formula derived in this paper. In the surface of the integral element, the local coordinate system ρθ is established and the approximate singular integral is transformed into the integrals of variables ρ and θ which are already separated in ρθ system. The integral with respect to polar variable ρ is expressed by the analytic formulations first. Then the new singular integral which is a surface integral is transformed into the line integral with respect to variable θ , which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly singular surface integrals in 3D acoustic BEM. Some examples are given in the last part of this paper to show the accuracy and the effectiveness of the present algorithm. The computed results demonstrate that the semi-analytic algorithm with high order element presented in this paper is more effective than linear regularization BEM to solve nearly singular integrals for 3D acoustic BEM.

Going beyond PFA: A precise formula for the sphere-plate Casimir force

Quantum fluctuations of the electromagnetic field in the medium surrounding two discharged macroscopic polarizable bodies induce a force between the two bodies, the so-called Casimir force. In the last two decades many experiments have accurately measured this force, and significant efforts are made to harness it in the actuation of micro- and nano-machines. The inherent many-body character of the Casimir force makes its computation very difficult in non-planar geometries, like the standard experimental sphere-plate configuration. Here we derive an approximate semi-analytic formula for the sphere-plate Casimir force, which is both easy to compute numerically and very accurate at all distances. By a comparison with the fully converged exact scattering formula, we show that the error made by the approximate formula is indeed much smaller than the uncertainty of present and foreseeable Casimir experiments.

Pricing currency options in the Heston/CIR double exponential jump-diffusion model

We examine currency options in the double exponential jump-diffusion version of the Heston stochastic volatility model for the exchange rate. We assume, in addition, that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semi-analytical formula for the price of the European currency call option in the hybrid foreign exchange/interest rates model.

Electrostatic roles in electron transfer from [NiFe] hydrogenase to cytochrome c3 from Desulfovibrio vulgaris Miyazaki F

Electrostatic interactions between proteins are key factors that govern the association and reaction rate. We spectroscopically determine the second-order reaction rate constant (k) of electron transfer from [NiFe] hydrogenase (H2ase) to cytochrome (cyt) c3 at various ionic strengths (I). The k value decreases with I. To analyze the results, we develop a semi-analytical formula for I dependence of k based on the assumptions that molecules are spherical and the reaction proceeds via a transition state. Fitting of the formula to the experimental data reveals that the interaction occurs in limited regions with opposite charges and with radii much smaller than those estimated from crystal structures. This suggests that local charges in H2ase and cyt c3 play important roles in the reaction. Although the crystallographic data indicate a positive electrostatic potential over almost the entire surface of the proteins, there exists a small region with negative potential on H2ase at which the electron transfer from H2ase to cyt c3 may occur. This local negative potential region is identical to the hypothetical interaction sphere predicted by the analysis. Furthermore, I dependence of k is predicted by the Adaptive Poisson–Boltzmann Solver considering all charges of the amino acids in the proteins and the configuration of H2ase/cyt c3 complex. The calculation reproduces the experimental results except at extremely low I. These results indicate that the stabilization derived from the local electrostatic interaction in the H2ase/cyt c3 complex overcomes the destabilization derived from the electrostatic repulsion of the overall positive charge of both proteins.

Dynamic encapsulation of corannulene molecules into a single-walled carbon nanotube.

The morphology of corannulene molecules encapsulated in a single-walled carbon nanotube (SWCNT) is addressed using atomistic simulations. Firstly, dynamic simulation (DS) of encapsulation of corannulene molecules into a SWCNT is performed using a molecular dynamics (MD) method. It is revealed that corannulene molecules encapsulated in a SWCNT tend to form concave-concave (CC) dimers, and these dimers make stacks tilting against the SWCNT axis or take an arrangement such that their convex surfaces face the inner wall of the SWCNT. This tendency arises from the fact that the van der Waals interactions between the convex surfaces of the corannulene molecules and the inner wall of the SWCNT dominate in their dynamic encapsulation into the SWCNT, and CC dimers are favored based on the energetics. Next, conjugate gradient (CG) energy minimizations starting from two kinds of initial arrangement of corannulene molecules in a SWCNT, concave-convex (CV) and CC/convex-convex (VV) arrangements, are performed. The CG energy minimizations confirm the result of DS that CC dimers are the structural motif of corannulene molecules in a SWCNT. From the final configurations of both the simulations, the tilt angles and intermolecular distances of the stacked molecules are calculated. With increasing the SWCNT diameter, the tilt angles decrease while the intermolecular distances remain almost constant. The tilt angles of the stacked corannulene molecules are approximately expressed by a semi-analytical formula which is derived on the basis of a geometrical constraint condition.

PRIMORDIAL BLACK HOLE FORMATION IN THE MATTER-DOMINATED PHASE OF THE UNIVERSE

ABSTRACT We investigate primordial black hole formation in the matter-dominated phase of the universe, where nonspherical effects in gravitational collapse play a crucial role. This is in contrast to the black hole formation in a radiation-dominated era. We apply the Zel’dovich approximation, Thorne’s hoop conjecture, and Doroshkevich’s probability distribution and subsequently derive the production probability of primordial black holes. The numerical result obtained is applicable even if the density fluctuation σ at horizon entry is of the order of unity. For , we find a semi-analytic formula , which is comparable to the Khlopov–Polnarev formula. We find that the production probability in the matter-dominated era is much larger than that in the radiation-dominated era for , while they are comparable with each other for . We also discuss how σ can be written in terms of primordial curvature perturbations.

An analytic expansion method for the valuation of double-barrier options under a stochastic volatility model

In this paper, we study a double-barrier option with a stochastic volatility model whose volatility is driven by a fast mean-reverting process, where the option's payoff is extinguished as the underlying asset crosses one of two barriers. By using an asymptotic analysis and Mellin transform techniques, we derive semi-analytic option pricing formulas with the sum of a leading-order term and a correction-order term, and then the accuracy of the first approximation price of the double-barrier option is verified by using Monte Carlo simulation. Moreover, we analyze the impact of stochastic volatility on the double-barrier option prices. Finally, we demonstrate that our results enhance the existing double-barrier option price structures in view of flexibility and applicability through the market price of volatility risk.

Modeling liquidation risk with occupation times

In this paper, we develop a new structural model that allows for a distinction between default and liquidation to be made. Default occurs when firm’s asset value process crosses a bankruptcy barrier. Here, we do not assume that default immediately triggers liquidation. Instead, the firm is allowed to continue operating even if it is in default. Liquidation is triggered as soon as the firm’s asset value has cumulatively spent a prespecified amount of time below the default barrier or has dropped below the liquidation barrier. The proposed model includes the Black–Cox model as a limiting case. A semi-analytical formula of the liquidation probability is derived for the case where firm’s asset value follows a geometric Brownian motion. Nonlinear volatility diffusion models are discussed as well.

Hybrid Monte Carlo and continuum modeling of electrolytes with concentration-induced dielectric variations.

The distribution of ions near a charged surface is an important quantity in many biological and material processes, and has been therefore investigated intensively. However, few theoretical and simulation approaches have included the influence of concentration-induced variations in the local dielectric permittivity of an underlying electrolyte solution. Such local variations have long been observed and known to affect the properties of ionic solution in the bulk and around the charged surface. We propose a hybrid computational model that combines Monte Carlo simulations with continuum electrostatic modeling to investigate such properties. A key component in our hybrid model is a semianalytical formula for the ion-ion interaction energy in a dielectrically inhomogeneous environment. This formula is obtained by solving for the Green's function Poisson's equation with ionic-concentration-dependent dielectric permittivity using a harmonic interpolation method and spherical harmonic series. We also construct a self-consistent continuum model of electrostatics to describe the effect of ionic-concentration-dependent dielectric permittivity and the resulting self-energy contribution. With extensive numerical simulations, we verify the convergence of our hybrid simulation scheme, show the qualitatively different structures of ionic distribution due to the concentration-induced dielectric variations, and compare our simulation results with the self-consistent continuum model. In particular, we study the differences between weakly and strongly charged surfaces and multivalencies of counterions. Our hybrid simulations conform particularly the depletion of ionic concentrations near a charged surface and also capture the charge inversion. We discuss several issues and possible further improvement of our approach for simulations of large charged systems.

Coupled cooling method and application of latent heat thermal energy storage combined with pre-cooling of envelope: Method and model development

The traditional cooling methods cannot meet the requirements of safety, stability, reliability and no-power at the same time under some special circumstances. In this study, a new coupled cooling method of Latent Heat Thermal Energy Storage (LHTES) combined with Pre-cooling of Envelope (PE) is proposed and the numerical model of the coupled cooling method is developed. In the current study, a refuge chamber is selected as a case study. A semi-analytical method is used to analyze the cold storage performance of the Surrounding Rock (SR). Afterwards, a numerical model of the coupled cooling system, which takes the heat source, SR, Phase Change Material (PCM) and air heat transfer into consideration, is further established. The study identified that the simplified semi-analytical calculation formula with the diagram of the cold storage quantity of SR are very helpful for engineering calculation. The influence of the Fourier and Biot number on the cold storage capacity of SR can be easily analyzed. In addition, the whole-field model of the coupled cooling system is completely developed based on the PCM unit.

Semi-analytical valuation for discrete barrier options under time-dependent Lévy processes

Abstract Simple analytical solutions for the prices of discretely monitored barrier options do not yet exist in the literature. This paper presents a semi-analytical and fully explicit solution for pricing discretely monitored barrier options when the underlying asset is driven by a general Levy process. The explicit formula only involves elementary functions, and the Greeks are also explicitly available with little additional computation. By performing a Z -transform, we reduce the valuation problem to an integral equation. This equation is solved analytically with the solution expressed in terms of a Fourier cosine series. We then manage to analytically carry out the Z -transform inversion, and obtain a semi-analytical formula for pricing discrete barrier options. We establish the theoretical error bound and analyze the convergence order of our method. Numerical implementation demonstrates that our numerical results are accurate and efficient, and match up with the results from the benchmark methods in the literature.

Semi-Analytical Valuation for Discrete Barrier Options under Time-Dependent LLvy Processes

Simple analytical solutions for the prices of discretely monitored barrier options do not yet exist in the literature. This paper presents a semi-analytical and fully explicit solution for pricing discretely monitored barrier options when the underlying asset is driven by a general Levy process. The explicit formula only involves elementary functions, and the Greeks are also explicitly available with little additional computation. By performing a Z-transform, we reduce the valuation problem to an integral equation. This equation is solved analytically with the solution expressed in terms of a Fourier cosine series. We then manage to analytically carry out the Z-transform inversion, and obtain a semi-analytical formula for pricing discrete barrier options. We establish the theoretical error bound and analyze the convergence order of our method. Numerical implementation demonstrates that our numerical results are accurate and efficient, and match up with the results from the benchmark methods in the literature.

Pricing vulnerable path-dependent options using integral transforms

In the over-the-counter (OTC) markets, the holders of many contracts are vulnerable to counterparty credit risk. Because of this issue, vulnerable options must be considered. In addition, in a financial environment, the pricing of path-dependent options yields many interesting mathematical challenges. In this paper, we study the pricing of vulnerable path-dependent options using double Mellin transforms to investigate an explicit (closed) form pricing formula or semi-analytic formula in each path-dependent option.

Confinement-induced resonances in ultracold atom-ion systems

We investigate confinement-induced resonances in a system composed by a tightly trapped ion and a moving atom in a waveguide. We determine the conditions for the appearance of such resonances in a broad region -- from the "long-wavelength" limit to the opposite case when the typical length scale of the atom-ion polarisation potential essentially exceeds the transverse waveguide width. We find considerable dependence of the resonance position on the atomic mass which, however, disappears in the "long-wavelength and zero-energy" limit, where the known result for the confined atom-atom scattering is reproduced. We also derive an analytic and a semi-analytic formula for the resonance position in the "long-wavelength and zero-energy" limit and we investigate numerically how the position of the resonance is affected by a finite atomic colliding energy. Our results, which can be investigated experimentally in the near future, could be used to determine the atom-ion scattering length, the temperature of the atomic ensemble in the presence of an ion impurity, and to control the atom-phonon coupling in a linear ion crystal in interaction with a quasi one dimensional atomic quantum gas.

Smoothed dissipative particle dynamics model for mesoscopic multiphase flows in the presence of thermal fluctuations.

Thermal fluctuations cause perturbations of fluid-fluid interfaces and highly nonlinear hydrodynamics in multiphase flows. In this work, we develop a multiphase smoothed dissipative particle dynamics (SDPD) model. This model accounts for both bulk hydrodynamics and interfacial fluctuations. Interfacial surface tension is modeled by imposing a pairwise force between SDPD particles. We show that the relationship between the model parameters and surface tension, previously derived under the assumption of zero thermal fluctuation, is accurate for fluid systems at low temperature but overestimates the surface tension for intermediate and large thermal fluctuations. To analyze the effect of thermal fluctuations on surface tension, we construct a coarse-grained Euler lattice model based on the mean field theory and derive a semianalytical formula to directly relate the surface tension to model parameters for a wide range of temperatures and model resolutions. We demonstrate that the present method correctly models dynamic processes, such as bubble coalescence and capillary spectra across the interface.

Entanglement entropy in a periodically driven Ising chain

In this work we study the entanglement entropy of a uniform quantum Ising chain in transverse field undergoing a periodic driving of period τ. By means of Floquet theory we show that, for any subchain, the entanglement entropy tends asymptotically to a value τ-periodic in time. We provide a semi-analytical formula for the leading term of this asymptotic regime: It is constant in time and obeys a volume law. The entropy in the asymptotic regime is always smaller than the thermal one: because of integrability the system locally relaxes to a generalized Gibbs ensemble (GGE) density matrix. The leading term of the asymptotic entanglement entropy is completely determined by this GGE density matrix. Remarkably, the asymptotic entropy shows marked features in correspondence to some non-equilibrium quantum phase transitions undergone by a Floquet state analog of the ground state.

APPROXIMATIONS OF BOND AND SWAPTION PRICES IN A BLACK–KARASIŃSKI MODEL

We derive semi-analytic approximation formulae for bond and swaption prices in a Black–Karasiński (BK) interest rate model. Approximations are obtained using a novel technique based on the Karhunen–Loève expansion. Formulas are easily computable and prove to be very accurate in numerical tests. This makes them useful for numerically efficient calibration of the model.

Stochastic finite element analysis of plates with the Certain Generalized Stresses Method

The Certain Generalized Stresses Method (CGSM) takes into account variability in static finite element analysis. The CGSM is dedicated to thin-walled structures: bars, beams, plates and shells. The objective of this paper is to present and evaluate a methodology based on the CGSM, for the static finite element analysis of plates with variability. The CGSM is a non-intrusive method that requires only one finite element analysis with some load cases to calculate the variability of mechanical quantities of interest. The statistical results: mean value, standard deviation and distribution are obtained by Monte Carlo simulations, using a semi-analytical formula. Two examples are treated: a square plate with a circular hole under tension and a simply supported square bending plate under uniformly distributed load. The method provides results of good quality and is very economical from a computational time point of view.

A new framework for solving partial differential equations using semi-analytical explicit RK(N)-type integrators

This paper develops a new explicit semi-analytical approach to solving PDEs based on the variation-of-constants formula, or the equivalent integration equation. These new schemes avoid the discretization of spatial derivatives. Therefore, the accuracy of the semi-analytical explicit scheme depends entirely on time integration. We first establish a semi-analytical formula for wave partial differential equations and show the consistency of the semi-analytical formula with different boundary conditions. We then present semi-analytical explicit RKN-type integrators for solving wave PDEs. This methodology is also extended to dealing with first-order hyperbolic PDEs and parabolic PDEs, and for both of them the corresponding semi-analytical explicit RK-type integrators are derived as well. Numerical simulations are implemented and the results show the effectiveness of the new semi-analytical explicit schemes. Meanwhile, an operator-variation-of-constants formula with applications is introduced for high-dimensional nonlinear wave equations before presenting the idea of semi-analytical explicit RKN integrators.

Interest Rate Swap Credit Valuation Adjustment

The credit valuation adjustment (CVA) of OTC derivatives is an important part of the Basel III credit risk capital requirements and current accounting rules. Its calculation is not an easy task—not only is it ­necessary to model the future value of the derivative, but also the probability of the default of a counterparty. Another complication in the calculation arises when the exposure to a counterparty is adversely, or favourably, correlated with the credit quality of that counterparty, i.e., when it is necessary to incorporate wrong-way, or right-way, risk. A semi-analytical CVA formula simplifying interest rate swap (IRS) valuation with the counterparty credit risk including the wrong-way risk is derived and analyzed in the article. The formula is based on the fact that the CVA of an IRS can be expressed using swaption prices. The link between the interest rates and the default time is represented by a Gaussian copula with a constant correlation coefficient. Finally, the results of the semi-analytical approach are compared with the results of a complex simulation study.