Form Of Analytical Functions Research Articles

Application of probabilistic thermomechanical models to large and/or complex structures

Abstract An affine probabilistic model is used to show how the cumulative and local probabilities of fracture of yielding can be determined in complex and/or large structures. The theory of probabilistic strength of materials supplies an expression allowing the cumulative probability of fracture, or yielding of a large and/or complex structure to be computed, taking into account the knowledge of the said probabilities in a material similarly manufactured but smaller, using an affine transformation. The result is independent of the stress field and of the analytical form of Weibull's specific risk function. The system must be loaded gradually, and if there is some thermal shock then the time in which the thermal perturbation is dissipated into the structure must be greater than the time occupied by the elastic wave in going through the structure, such that the assumption of gradual loading still holds. Costs in terms of time and money are reduced by using smaller models and by considering a time-transformation that allows the testing of structures subjected to thermal stresses to be speeded up.

The Green's function of confined electrons in an external magnetic field: Analytical formulation

We derive the analytical form of the Green's function of 2-dimensional electrons with lateral confinement in a perpendicular magnetic field. The confinement potentials considered are infinite barriers at radius R (quantum dot) and at r and R (quantum ring).

Matching network design of microstrip antennas with simplified real frequency technique

A new approach for increasing the bandwidth of microstrip antennas by using the simplified real frequency technique is presented. The method is numerical, its most significant feature is that it does not require any form of equivalent circuit or analytical impedance functions for the antennas and generators, and the measured or calculated impedance data are processed directly. Furthermore, neither an a priori choice of a matching network topology nor an analytic form of the system transfer function is needed. A representative example shows the validity of the design procedure.

An improved H3 potential energy surface

We report ab initio calculations of the ground state energy for 404 new conformations of H3, supplementing the set of 368 conformations reported previously by others. The entire dataset has been used to constrain an analytical functional form for the potential energy surface, building on that of Truhlar and Horowitz. The new surface extends the Truhlar and Horowitz surface to higher energies and offers some modest improvement at lower energies. In addition, we have eliminated a problem with derivatives of the London equation that was pointed out by Johnson. The new surface matches the 772 ab initio energies with an overall root-mean-square (rms) error of 0.25 mhartree (i.e., 0.16 kcal/mol) and a maximum absolute deviation of 1.93 mhartree (1.21 kcal/mol); for ‘‘noncompact’’ conformations (no interatomic distance smaller than 1.15 bohr), the rms error is 0.17 mhartree (0.11 kcal/mol) and the maximum absolute deviation is 1.10 mhartree (0.69 kcal/mol). The classical barrier height for H+H2→H2+H is estimated to be 15.20±0.15 mhartree (i.e., 9.54±0.09 kcal/mol).

The analytical form of the length distribution function for computer interconnection

An analysis of a hierarchical computer interconnection model that yields the analytical form of the interconnection distribution function is presented. It is shown that this function is consistent with the previously derived equation for the average interconnection length and that the distribution function accurately describes the distribution of interconnections within previously constructed computer systems. The distribution function is then used to investigate the proposed relationship between the exponent of the Rent equation and the gradient of the length distribution function. It is confirmed that the pin-limited partitioning of computer systems results in an approximate power-law length distribution function, and for large numbers of gates the characteristic exponent of the length distribution function gamma is related to the Rent exponent by gamma approximately=3-2 p. In addition, it is shown that the theoretical equations are a good approximation to experimentally observed interconnection distributions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Kinetic invariant model of dissolution with chemical reaction of large particles

AbstractThe kinetic invariant model has been reformulated and an approximate analytical form of a kinetic function is obtained for the dissolution accompanied by a chemical reaction of large solid particles. A restriction on the analytical solution is provided in the form of a sufficient condition for the convergence of the series involved. Expressions for the concentration of unreacted, dissolved solid reactant, the time of complete conversion, and the ratio of driving forces available for the reaction and mass transfer steps are presented. Solutions for the rate‐limiting regimes are deduced as special cases of the general solution. The small‐particle solution is recovered as a special case in the limit of a vanishingly small Reynolds number. Rigorous numerical solution has been obtained to assess the accuracy of the pseudostationary‐state solution and to discern the validity of the approximations.

On a generalization of the Kubo-Toyabe formula

We have derived an analytical form of the muon depolarization function G z ( t) when the magnetic field on the muon can be treated as a sum of two fields with Gaussian and δ-function distributions. We discuss the application of our results to processing μSR data on LaSrCuO polycrystal samples.

Application of a simplified real-frequency synthesis method to distributed-element amplifier design

The simplified real-frequency method is probably the best microwave circuit synthesis method available for the design of broadband amplifiers for electronic countermeasure (ECM) or communications applications. When applied to the design of interstage equalizers for microwave amplifiers, the procedure, contrary to classical synthesis techniques, utilizes only the measured scattering parameters of the 2-port active device. The method thus directly includes device feedback effects, and neither an a priori choice of matching network topology nor an analytical form of the system transfer function is necessary. In the version described in this article, the optimization process is applied simultaneously to transducer power gain, input match (VSWR), and noise figure of a multistage amplifier. A procedure for directly extracting distributed transmission line elements for matching networks is briefly described. The program FREEL is then applied to the design of 3-stage 2-8-GHz and 1.15-1.5-GHz MESFET amplifiers in which three topologies are compared: lumped element, cascaded transmission line, and transmission line and open circuited stubs.

Dynamics of Antiferromagnetic Heisenberg Model at Low Temperatures

Dynamical correlation function Gr(t)=〈Sr(t)·S0(0)〉 and dynamical structure factor Sq(ω) for the Heisenberg antiferromagnet are calculated by the modified spin-wave theory. We use the Dyson-Maleev transformation, the ideal spin-wave states and the rotational averaging. The static correlation function Gr(t=0) coincides with that obtained in previous papers for the antiferromagnet. We rederive the Auerbach and Arovas dynamical structure factor. This satisfies the dynamic scaling hypothesis at low temperature and low momentum. Analytical form of dynamical scaling function is obtained. We find that the characteristic time τ is proportional to the correlation length ξ. In the classical limit our results are compared with the molecular dynamics calculation for 1D system. The agreement is good for the short time correlation, but it is not good for the long time correlation. This is in contrast to the agreement for a wide range of time in the case of ferromagnet. The analysis of magnetic susceptibility of La2CuO4 is given.

CATHODOLUMINESCENCE IN DOUBLE HETEROJUNCTION LASERS

The cathodoluminescence mode of the Scanning Electron Microscope is used to evaluate local properties of the active layer of double-heterojunction lasers. The analysis of the Cathodoluminescence line-scans and spectra from cleaved laser surfaces, as a function of the external bias and of the incident beam current, give information about the interface quality, the internal efficiency and the local gain of stimulated emission. In order to make a quantitative interpretation of our experimental measurements, an analytical form of the tridimensional generation function, derived from a Monte-Carlo trajectory simulation, is used.

Renormalized polarizability in the Maxwell Garnett theory

We develop a simple theory for the effective dielectric function of a system of identical spheres embedded in a homogeneous matrix within the dipolar long-wavelength approximation. We obtain a relationship analogous to the Clausius-Mossotti relation but with a renormalized polarizability for the spheres instead of the bare polarizability. This renormalized polarizability depends on the bare polarizability, the volume fraction and a functional of the two-particle correlation function of the spheres, and obeys a second order algebraic equation. We calculate the optical properties of metallic spheres within an insulating matrix and compare our results with previous theories and with experiment. We obtain a closed analytical form of the spectral function and check that it obeys Bergman's sum rules [1]. We construct a 2D version of the theory and calculate the optical properties of a disordered array of spheres located on top of a semi-infinite homogeneous substrate. Results are presented and the role of the image dipoles is analyzed.

Spectroscopy and photodissociation dynamics of H2O: Time-dependent view

We present a study of H2O photodissociation dynamics in the first excited state. The absorption and emission spectra are calculated quantum mechanically using a time-dependent wave packet propagation technique. The excited state potential energy surface is obtained from ab initio data generated by Palma et al., while the ground state is a model by Reimers and Watts cast in a simple analytical form and constructed by fitting to infrared data. The variation of the transition dipole moment is taken into account by using an analytical functional form which is adjusted to fit the low resolution experimental emission spectrum. The calculated absorption spectrum is very similar to both the experimental one and results obtained with other methods. The high resolution emission spectrum is calculated for the first time and shows excellent agreement with the recently measured experimental spectrum by Hudson and co-workers. Dynamics on both the ground and excited states are examined.

Stationary statistics and sequential properties of normal beam and global solar radiation on tilted surfaces

The stationary statistics and persistence characteristics of both hourly and daily total global solar radiation on tilted surfaces, as well as of hourly normal beam radiation, on a discrete monthly basis, are analyzed in a formalism akin to previous studies for global horizontal solar radiation. Our database includes three Australian locations of varying latitude and clearness index. We find that with a proper geometric/statistical transformation, the Probability Density Function (PDF) of global radiation on tilted surfaces can be accurately predicted from the corresponding statistics for global horizontal radiation, the latter typically being far more readily available. The PDF of hourly normal beam radiation seems to be well predicted via modification of a simple analytic functional form previously proposed for modeling global horizontal radiation. Finally, the persistence times and persistence strengths of global radiation on tilted surfaces and of normal beam radiation are found, not surprisingly, to be very similar to the corresponding values for global horizontal radiation.

ON APPLICATION OF THE METHOD OF KINETIC INVARIANTS TO THE DESCRIPTION OF DISSOLUTION ACCOMPANIED BY A CHEMICAL REACTION

Abstract The dissolution, accompanied by chemical reaction, of monodisperse solid particles has been analysed. The resulting model, which accounts for the variation of mass transfer coefficient with the size of the dissolving particles, yields an approximate analytical form of a kinetic function. Rigorous numerical and approximate analytical solutions have been obtained for the governing system of nonlinear ordinary differential equations. The transient nature of the dissolution process as well as the accuracy of the analytical solution is brought out by the rigorous numerical solution. The analytical solution is fairly accurate for the major part of the range of operational times encountered in practice.

An analytical approximation to the HNC pair correlation functions: The case of 2:2 electrolytes

An analytical form of the HNC pair correlation function for the primitive ionic solution is proposed. This form interpolates between the EXP and MSA approximations. For 2:2 electrolytes it gives very good agreement to the thermodynamics and pair correlation function of the HNC.

Analysis and interpretation of a large body of 76Ge zero-neutrino double- beta -decay data.

Data from five recent $^{76}\mathrm{Ge}$ double-\ensuremath{\beta}-decay experiments were combined in a single spectrum equivalent to counting for 0.4 yr with 1770-${\mathrm{cm}}^{3}$ fiducial volume of Ge. The composite spectrum was analyzed using an approximate analytical form of the likelihood function. The maximum likelihood 68%-C.L. values for the half-life and Majorana \ensuremath{\nu} mass are ${\mathit{T}}_{1/2}^{\mathit{O}\mathit{v}}$${(}^{76}$Ge)\ensuremath{\gtrsim}4.1\ifmmode\times\else\texttimes\fi{}${10}^{23}$ yr and 〈${m}_{\ensuremath{\nu}}$〉\ensuremath{\lesssim}2.4 eV, respectively, when accounting for the slight depression in the spectrum, and 2.4\ifmmode\times\else\texttimes\fi{}${10}^{23}$ yr and 〈${m}_{\ensuremath{\nu}}$〉\ensuremath{\lesssim}3.2 eV when neglecting the depression. It is clearly demonstrated that experiments with current levels of background will soon reach their point of diminishing returns.

A water dimer potential based on a b i n i t i o calculations using Morokuma component analyses

We present a new water dimer potential. It is based on carrying out Morokuma component analyses at the ab initio level on 229 water dimer geometries and fitting the energy components to analytical functional forms. To these are then added a dispersion contribution, based on the analytical dispersion energy expression of Douketis et al. (Ref. 1). This potential gives a satisfactory description of the water dimer, with minimum R(O ⋅ ⋅ ⋅ O)=2.96 Å, ΔE=−5.2 kcal/mol, and minimum energy θ angle between H-bond donor and acceptor axis of 55°, in good agreement with experimental values of 2.98±0.04 Å, −5.4±0.2 kcal/mol, and 60±10°, respectively. The second virial coefficient B is also calculated in reasonable agreement with experiment, e.g., at 423 K, B (calculated) =−352, compared to B (experimental)=−332 cc/mol.

One-component plasma structure factor of liquid alkali metals

An analytical form of the direct correlation function is proposed to determine the structure factor of liquid alkali metals in the OCP reference system. Assuming that the interionic interaction can be modeled by the effective pair potential in the model potential approach, the structure factor is calculated using random phase approximation. It works well up to 2 k F, which we need to calculate the electronic transport properties, showing thus the significant role played by the electron gas response function.

Exact nonlinear dielectric function of the Resta model for semiconductors

Based on the Resta model we have obtained the exact analytical form of the nonlinear dielectric function for semiconductors. Calculations are performed for diamond and GaAs as examples.

Golden-rule approach to the soft-x-ray-absorption problem. III. The temperature dependence

The exact formulation of the soft-x-ray-absorption problem of previous work [Phys. Rev. B 28, 6833 (1983)] is generalized to the case of finite temperatures. By giving a closed-form expression of the response function in terms of the overlap integrals between the initial and final electron states, we demonstrate that the exactness can be pursued quite generally. The general formula, when applied to the case of a contact core-hole potential, leads to expressions of the dispersion integral and the frequency shift, both modified from their $T=0$ forms by the temperature effect. In the nearedge region, an analytical form of the response function can be derived, which reproduces precisely the time and temperature dependences of that of Yuval and Anderson [Phys. Rev. B 1, 1522 (1970)] and, in addition, provides an exact expression for the prefactor of the open-line contribution of the spectrum.