Approximate Analytical Formula Research Articles

Alpha -attractor from superconformal E-models in brane inflation

In the large extra dimensional braneworld inflation, Friedmann equation is modified to include a quadratic term in energy density with an additional parameter lambda called brane tension in addition to the usual linear term. The high energy brane corrections modify the slow-roll parameters and affect the behaviour of inflation. We analyse the superconformal inflation for E-models and find that there exist alpha -attractors in brane inflation. The predictions for the scalar spectral index n_s and the tensor-to-scalar ratio r are computed numerically, and approximate analytic formulas in the high energy limit have been given for the observable n_s and r. The constraints on the model parameters are obtained by using Planck 2018 and BICEP2 observational data.

Спектроскопия слоев на плоскопараллельных подложках

Integral expressions for the reflectance and transmittance spectra of the structure consisted of two thin layers deposited on opposite faces of a plane-parallel substrate at oblique illumination of the structure with partially coherent light are obtained. As a result of the asymptotic analysis of the integrals, approximate analytical formulae are established for calculating the indicated spectra, convenient for use in solving inverse spectrophotometry problems. An aluminum doped zinc oxide layer deposited on a glass substrate is studied. The spectra of the refractive index and absorption coefficient of the layer and the substrate, as well as the thickness of the layer, are determined by processing the reflectance and transmission spectra of the structure, measured for s- and p- polarized waves at two angles of light incidence. The found parameters are used in computational experiments to estimate the applicability limits of the formulated approximations.

Accuracy of the analytical escape rate for a cusp barrier in the overdamping regime

For the first time, the accuracy of the approximate analytical Kramers formula for the thermal decay rate over a cusp barrier, RK, is checked numerically for the overdamping regime. The numerical quasistationary rate, RD, which is believed to be exact within the statistical errors is evaluated by means of computer modeling of the stochastic Langevin-type dynamical equations. The agreement between RK and RD significantly depends upon the friction strength and the height of the barrier in comparison to the thermal energy. The difference between RK and RD decreases with the dimensionless damping parameter φ, however, does not become smaller than 10-20%. The unexpected growth of the difference between RK and RD with the governing parameter is observed.

A closed-form approximation for pricing geometric Istanbul options

The Istanbul options were first introduced by Michel Jacques in 1997. These derivatives are considered as an extension of the Asian options. In this paper, we propose an analytical approximation formula for a geometric Istanbul call option (GIC) under the Black-Scholes model. Our approximate pricing formula is obtained in closed-form using a second-order Taylor expansion. We compare our theoretical results with those of Monte-Carlo simulations using the control variates method. Finally, we study the effects of changes in the price of the underlying asset on the value of GIC.

A closed-form approximation for pricing geometric Istanbul options

The concept of Istanbul options were first introduced by Michel Jacques in 1997. These derivatives products are considered as an extension of the Asian options. In this paper, we propose an analytical approximation formula for a geometric Istanbul call option (GIC) under the standard Black-Scholes model. Our approximate pricing formula is obtained in closed-form using a second-order Taylor expansion. We compare our theoretical results with those of Monte Carlo simulations using the control variates method. We also carry out a comparative price study with an arithmetic Istanbul call option. Finally, we study the effects of changes in the price of the underlying asset on the value of GIC.

Оценка гарантированного срока службы для модели системы с резервированием разнотипными элементами

The paper focuses on the problem of confidence estimation of reliability indicators for a system model with loaded redundancy of elements of various sub-systems. The lower confidence limits are constructed for the system reliability function, as well as for the indicator associated with it, the indicator having a given guaranteed level of the system uptime, i.e., its gamma-percentile life. Within the research, we obtained approximate asymptotic --- for the case of high reliability --- expressions for confidence estimates of these basic indicators of system reliability. Rather simple approximate analytical calculation formulas based on these asymptotic expressions are given for the lower confidence boundary of the system reliability function and a similar confidence boundary for the guaranteed system life.

Current Limitations for the Axially Symmetric Hollow Electron Beam in the Drift Tubes of Millimeter-Wave Vacuum Electronic Devices

The conditions for ensuring the equilibrium steady-state of a high-density hollow electron beam transported in cylindrical drift tubes and focused by a uniform magnetic field are investigated. The results of numerical calculations of the currents limited due to the longitudinal deceleration of electrons by the space charge and the violation of equilibrium transverse magnetic focusing conditions when transporting beams of different configurations in klystron-type devices are presented. The limitations of the working length of the device associated with the excitation of diocotron instability are estimated. The calculation results were compared with the data obtained by the quasi-3D Arsenal—MSU computer code and by the approximate analytical formulas of other authors.

Analytical Approximation Formula for Barrier Option Prices under the Regime-Switching Model

In this article barrier options are analytically evaluated under the regime-switching model, with the volatility of the underlying price being allowed to jump between different states following a Markov chain. The target barrier option prices are expressed in a Fourier cosine series after a particular approximation formula is obtained. The accuracy and efficiency of the newly derived formula are demonstrated through numerical experiments, demonstrating the formula’s potential for practical applications. TOPICS:Analysis of individual factors/risk premia, factor-based models, options Key Findings • Barrier options are analytically evaluated under the regime-switching model. • This approximation formula is written in the form of a converged Fourier cosine series. • The formula is shown to be very accurate and efficient, and has a great potential to be applied in practice.

Inductance Formula for Square Spiral Inductors with Rectangular Conductor Cross Section

Planar spiral coils are used as inductors in radio frequency (RF) microelectronic integrated circuits (IC’s) and as antennas in both radio frequency identification (RFID) and telemetry systems. They must be designed to a specified inductance. From the literature, approximate analytical formulae for the inductance of such coils with rectangular conductor cross section are known. They yield the direct current (DC) inductance, which is considered as a good approximation for inductors in RF IC’s up to the GHz range. In principle, these formulae can simplify coil design considerably. But a recent comparative study of the most cited formulae revealed that their maximum relative error is often much larger than claimed by the author, and too large to be useful in circuit design.
 This paper presents a more accurate formula for the DC inductance of square planar spiral coils than was known so far. It is applicable to any design of such coils with up to windings. Owing to its scalability, this holds irrespectively of the coil size and the inductance range. It lowers the maximum error over the whole domain of definition from so far down to . This has been tested by the same method used in the comparative study mentioned above, where the precise reference inductances were computed with the help of the free standard software FastHenry2. A comparison to measurements is included. Moreover, the source code of a MATLAB® function to implement the formula is given in the appendix.

Interaction of a normally-incident plane wave with a nonlinear poroelastic fracture.

While it has been recognized that a large amplitude incident wave upon a dry fracture can exhibit nonlinear seismic wave scattering due to its stress-dependent mechanical compliance, the impact of pore fluid in the fracture and a fluid-filled poroelastic background medium-features common for fractures in the Earth-are not well understood. As a first step toward an understanding of the nonlinear poroelastic response of elastic waves in fractured media, analytical approximate formulas are used for the amplitude and phase of a normally incident plane wave using a perturbation method, assuming a fluid-filled, highly compliant nonlinear interface embedded in a linear poroelastic solid. The stress-closure behavior of the fracture is modeled by nonlinear, poroelastic displacement-discontinuity boundary conditions (a linear-slip interface). The theory predicts that the static ("Direct current," or DC) and higher-order-harmonic waves produced by the nonlinear scattering can be greatly reduced by the presence of fluid in the fracture. This, however, depends upon a number of parameters, including fracture compliance, fluid properties (compressibility and viscosity), and the permeability of the background medium, as well as environmental parameters such as the initial fluid pressure and stress acting on the fracture. The static effect produces low-frequency fluid pressure pulses when a finite-duration wave is incident upon the fracture-behavior unique to fluid-filled fractures within a poroelastic medium.

Pricing Weather Derivatives Index Based on Temperature: The Case of Bahir Dar, Ethiopia

In this paper we present a stochastic model for daily average temperature to calculate the temperature indices upon which temperature-based derivatives are written. We propose a seasonal mean and volatility model that describes the daily average temperature behavior using the mean-reverting Ornstein-Uhlenbeck process. We also use higher order continuous-time autoregressive process with lag 3 for modeling the time evolution of the temperatures after removing trend and seasonality. Our model is fitted to 11 years of data recorded, in the period 1 January 2005 to 31 December 2015, Bahir Dar, Ethiopia, obtained from Ethiopia National Meteorological Services Agency. The analytical approximation formulas are used to price heating degree days (HDD) and cooling degree days (CDD) futures. The suggested model is analytically tractable for derivation of explicit prices for CDD and HDD futures and option. The price of the CDD future is calculated, using analytical approximation formulas. Numerical examples are presented to indicate the accuracy of the method. The results show that our model performs better to predict CDD indices.

The time-dependent diffusivity in the abdominal ganglion of Aplysia californica: experiments and simulations

The nerve cells of the Aplysia are much larger than mammalian neurons. Using the Aplysia ganglia to study the relationship between the cellular structure and the diffusion MRI signal can potentially shed light on this relationship for more complex organisms. We measured the dMRI signal of chemically-fixed abdominal ganglia of the Aplysia at several diffusion times. At the diffusion times measured and observed at low b-values, the dMRI signal is mono-exponential and can be accurately represented by the parameter ADC (Apparent Diffusion Coefficient). We performed numerical simulations of water diffusion for the large cell neurons in the abdominal ganglia after creating geometrical configurations by segmenting high resolution T2-weighted (T2w) images to obtain the cell outline and then incorporating a manually generated nucleus. The results of the numerical simulations validate the claim that water diffusion in the large cell neurons is in the short diffusion time regime at our experimental diffusion times. Then, using the analytical short time approximation (STA) formula for the ADC, we showed that in order to explain the experimentally observed behavior, it is necessary to consider the nucleus and the cytoplasm as two separate diffusion compartments. By using a two compartment STA model, we were able to illustrate the effect of the highly irregular shape of the cell nucleus on the ADC.

Three Calculation Methods of the WGM Cylindrical Cavity Diameter

WGM mode in cylindrical cavity can be rapidly calibrated with analytical approximate formulas based on cylindrical micro-cavity WGM resonance peaks position. On this basis, the diameter of circular cavity can be calculated by applying the three methods, which are the first-order approximation method, the second-order approximation method and mode field radius. When comparing the results which are got from the three methods with the actual geometric diameter, we find that the deviation is within 3%, 2%, 1% respectively. As a result, the diameter of circular cavity can be correctly worked out by making use of these three ways mentioned.

High-Voltage Diffused Step Recovery Diodes: II. Theory

Approximate analytical formulas are derived to evaluate the main characteristics of diffused step recovery diodes, which operate as current interrupters in high-power nanosecond pulse generators with intermediate inductive energy storage: the spurious prepulse voltage, characterizing the resistive loss in diffused step recovery diodes; amplitude; front duration; maximum voltage rise rate, and duration of the pulse formed on an active load. Similar formulas for epitaxial step recovery diodes, partially obtained for the first time and partially refining previous results, are given for comparison.

Focalizing slow neutron beams at and below micron scales and discussion on BNCT (II)

We studied the propagation of slow neutrons and their waveguiding properties in certain suitable thin films where Ti and Si play a key role, having in mind their possible use in Boron Neutron Capture Therapy (BNCT) of small tumors. In PBSi2017 we presented a general overview and some specific proposals for improved focalization of slow neutrons (at and a bit below one micron, with propagation up to about 1 m. with relatively small attenuation), by omitting a good number of quantitative aspects. The present work, based upon our presentation in PBSi2018, extends our previous proposals in PBSi2017 and provides more quantitative descriptions of part of them. Neutron wavelengths versus characteristic dimensions of the waveguides in our proposals typically require quantum-mechanical analysis: specifically, the description of the confined neutrons by means of propagation modes. The confined propagation of incoming waves along waveguides, the evolution of the resulting propagation modes and related issues are investigated. Those features give rise to computational challenges, depending on the smallness of (neutron wavelengths/characteristic dimensions). Approximate analytical formula is provided for the corresponding quantum-mechanical probability. Numerical estimates and the results of some simulations are presented. Consequences of the latter for BNCT will be discussed briefly.

Analytical shape functions and derivatives approximation formulas in local radial point interpolation methods with applications to financial option pricing problems

The radial point interpolation method is increasingly being applied for the numerical solution of partial differential equations in different fields. Most implementations in the literature for obtaining the matrix coefficients make use of numerical approximations of the shape functions and their derivatives. To avoid the solution of linear systems required for computation of derivative approximations, this work derives analytical shape functions for three and five-node support domains in a local radial point interpolation method (LRPIM) with multiquadrics as basis functions. A weak form algorithm for the Black–Scholes equation using three-node analytical shape functions is developed and its unconditional stability and convergence are theoretically established. LRPIM finite-difference (FD) formulas are derived and applied to the solution of one and two-asset financial options. A five-node LRPIM-FD method in one-dimension is shown to yield fourth-order accuracy and applications to two-asset problems also yield accurate prices for options on a minimum of two risky assets and exchange options.

Oblique incidence of a plane wave on an inhomogeneously loaded grating of slots in a screen

In this paper, we demonstrate a new method to control the enhanced transmission due to Fabry–Perot resonances through an array of dielectric-loaded slots in a thick metallic screen. We obtain approximate analytical formulas for each polarization (TM and TE) using a mode-matching technique based on the assumption that the slots are small compared to a free-space wavelength and that the metallic screen is a perfect electric conductor as is appropriate for microwave frequency applications. We show that the bandwidth and locations of enhanced transmission can be controlled by the angle of incidence or by making the material filling the slots inhomogeneous. Computational results are given that demonstrate the influence of the angle of incidence on the enhanced transmission bandwidth. We show that the separation between enhanced transmission frequencies can be controlled by introducing a gap within the slot with a different dielectric constant. Also, we illustrate how the total transmission bandwidth can be increased by adjusting the angle of incidence and the gap size.

Perturbation calculations on interlayer transmission rates from symmetric to antisymmetric channels in parallel armchair nanotube junctions

Partially overlapping two parallel armchair nanotubes are investigated theoretically with the $\pi$ orbital tight bonding model. Considering the interlayer Hamiltonian as perturbation, we obtain approximate analytical formulas of the interlayer transmission rates $T_{\sigma',\sigma}$ from channel $\sigma$ to $\sigma'$ for all the four combinations $(\sigma',\sigma)=(\pm,\pm)$ and $(\pm,\mp)$, where suffixes $+$ and $-$ represent symmetric and anti-symmetric channels, respectively, with respect to the mirror plane of each tube. Landauer's formula conductance is equal to the sum of them in units of $2e^2/h$. According to the perturbation calculation, the interlayer Hamiltonian is transformed into the parameter $w_{\sigma',\sigma}$ that determines the analytical formula of $T_{\sigma',\sigma}$. By comparison with the exact numerical results, the effective range of the analytical formulas is discussed. In the telescoped coaxial contact, the off-diagonal part $T_{-,+}+T_{+,-}$ is very small compared to the diagonal part $T_{+,+}+T_{-,-}$. In the side contact, on the other hand, the off-diagonal part is more significant than the diagonal part in the zero energy peak of the conductance.

Very Fast Transients in a 500 kV Gas-Insulated Substation

This paper performs thorough investigations of very fast transients (VFTs) in a 500 kV gas-insulated substation (GIS) by adopting complete earth-return impedance and admittance formulas. VFT simulations are performed with wideband models in a simulation tool for electromagnetic transients. The effects of various GIS parameters on VFTs are investigated. The maximum core and pipe voltages along the GIS are presented. The frequency spectra of VFT voltages are evaluated, and the effects of GIS length and transformer capacitances on the VFT frequencies are analyzed. The frequency decreases as the GIS total length increases. The frequency is decreased significantly by stray capacitances of transformers because those are far greater than the GIB capacitance. The effect of pipe grounding on the pipe voltage is a very noticeable, but on the core voltage and the oscillating frequency is minor. The position of an operating DS influences the VFT over-voltages and frequency significantly. Approximate analytical formulas are derived for physical and qualitative interpretation of simulation results.

An analytical approach to bifurcations and stability in simplified mathematical models of nuclear reactors

Asymptotic analytical methods are applied to study some problems related with bifurcations (both local and global) and stability in three simple mathematical models of nuclear reactors.The first case is a reactor which is subcritical at rest and driven by a distributed external neutron source. Under suitable conditions in the temperature feedback reactivity, the mathematical model predicts a static global bifurcation with three critical states, two stable and one unstable, with the possibility of a runaway. The dynamics of this system is studied, in a framework of slow manifold theory, by methods of restricted nonlinear modal analysis, and the results of a digital simulation are summarized. The model could be modified and extended to study the space time dynamics of sub-critical multiplying systems driven by external neutron sources (neutron beams produced by accelerators).The second case is related with in phase and out of phase xenon oscillations in large thermal reactors. The known subcritical Hopf bifurcation that appears in the context of global mode oscillations is revisited. After applying a nonlinear modal analysis to the mathematical model, a normal form is derived by an averaging method. Approximate analytical formulae for the radii of the unstable limit cycles and for the trajectories of the state variables in a neighborhood of the bifurcation point are obtained.The third case is a non-trivial modification and development of a simple mathematical model intended to describe certain mechanical kinetic effects stemming from a possible coupling of nuclear, thermal and mechanical vibration processes. We show that, under suitable conditions, a dynamic supercritical Hopf bifurcation in the reactor power appears in the framework of our modified model. A normal form is derived by an averaging method. Analytical formulae for the radii of the stable limit cycles and the trajectories of the state variables in a neighborhood of the bifurcation are given.