Exact Analytical Result Research Articles

On the role of shear in cosmological averaging II: large voids, non-empty voids and a network of different voids

We study the effect of shear on the cosmological backreaction in the context of matching voids and walls together using the exact inhomogeneous Lemaȋtre-Tolman-Bondi solution. Generalizing JCAP 1010 (2010) 021, we allow the size of the voids to be arbitrary and the densities of the voids and walls to vary in the range 0 ⩽ Ωv ⩽ Ωw ⩽ 1. We derive the exact analytic result for the backreaction and consider its series expansion in powers of the ratio of the void size to the horizon size, r0/t0. In addition, we deduce a very simple fitting formula for the backreaction with error less than 1% for voids up to sizes r0≳t0. We also construct an exact solution for a network of voids with different sizes and densities, leading to a non-zero relative variance of the expansion rate between the voids. While the leading order term of the backreaction for a single void-wall pair is of order (r0/t0)2, the relative variance between the different voids in the network is found to be of order (r0/t0)4 and thus very small for voids of the observed size. Furthermore, we show that even for very large voids, the backreaction is suppressed by an order of magnitude relative to the estimate obtained by treating the walls and voids as disjoint Friedmann solutions. Whether the suppression of the backreaction due to the shear is just a consequence of the restrictions of the used exact models, or a generic feature, has to be addressed with more sophisticated solutions.

Dynamical polarization of graphene in a magnetic field

The one-loop dynamical polarization function of graphene in an external magnetic field is calculated as a function of wavevector and frequency at finite chemical potential, temperature, band gap, and width of Landau levels. The exact analytic result is given in terms of digamma functions and generalized Laguerre polynomials, and has the form of double sum over Landau levels. Various limits (static, clean, etc) are discussed. The Thomas-Fermi inverse length $q_F$ of screening of the Coulomb potential is found to be an oscillating function of a magnetic field and a chemical potential. At zero temperature and scattering rate, it vanishes when the Fermi level lies between the Landau levels.

Lyapunov exponents and the natural invariant density determination of chaotic maps: an iterative maximum entropy ansatz

We apply the maximum entropy principle to construct the natural invariant density and the Lyapunov exponent of one-dimensional chaotic maps. Using a novel function reconstruction technique, that is based on the solution of the Hausdorff moment problem via maximizing Shannon entropy, we estimate the invariant density and the Lyapunov exponent of nonlinear maps in one dimension from a knowledge of finite number of moments. The accuracy and the stability of the algorithm are illustrated by comparing our results to a number of nonlinear maps for which the exact analytical results are available. Furthermore, we also consider a very complex example for which no exact analytical result for the invariant density is available. A comparison of our results to those available in the literature is also discussed.

Inclusive heavy flavor hadroproduction in NLO QCD: The exact analytic result

We present the first exact analytic result for all partonic channels contributing to the total cross section for the production of a pair of heavy flavors in hadronic collisions in NLO QCD. Our calculation is a step in the derivation of the top quark pair production cross section at NNLO in QCD, which is a cornerstone of the precision LHC program. Our results uncover the analytical structures behind observables with heavy flavors at higher orders. They also reveal surprising and non-trivial implications for kinematics close to partonic threshold.

NanoSQUID sensitivity for isolated dipoles and small spin populations

The sensitivity of nanoscale SQUIDs (nanoSQUIDs; SQUID: superconducting quantuminterference device) to single and small populations of magnetic dipoles is considered. Thesimple estimate given previously for the atomic spin sensitivity of a nanoSQUIDcoupled to an isolated magnetic dipole at its centre is confirmed. It is demonstratedthat the sensitivity is constrained in most practical situations by the finite sizeof the SQUID loop and nanobridges. An exact analytic result is obtained for ananoSQUID composed of an idealized filamentary circular loop. The issue of optimumplacement and orientation of the dipole with respect to the nanoSQUID hole is alsoconsidered, and it is shown that the optimum position for the dipole depends upon theheight of the dipole above the plane of the nanoSQUID. It is pointed out that theconclusion quoted by previous authors, that the optimum position is above one of theJosephson junctions or Dayem bridges, although true in the limit of very narrowbridges with tight magnetic coupling, is not true in general, and estimates of thepotential sensitivity when the dipole is placed in this region based on simplefilamentary models are likely to overestimate the sensitivity achievable in practice.

Analytical solutions to rate equations including losses describing threshold pumped fiber lasers

An exact analytical expression to the threshold pump power has been deduced for rate equations including loss coefficients applicable to fiber lasers doped with quasi two-energy-level ions. Following the derivation of a closed-form expression for the variation of the pump power along the fiber, together with analytical solutions to the position dependent atom population density and laser power in terms of the pump power, we have analytically and fully solved rate equations describing threshold pumped fiber lasers. A comparison has also been made between the exact analytical result with the previous reported results, which indicates that various approximation have underestimated the threshold pump powers.

Effect of discontinuity in the threshold distribution on the critical behavior of a random fiber bundle

The critical behavior of a random fiber bundle model with mixed uniform distribution of threshold strengths and global load sharing rule is studied with a special emphasis on the nature of distribution of avalanches for different parameters of the distribution. The discontinuity in the threshold strength distribution of fibers nontrivially modifies the critical stress as well as puts a restriction on the allowed values of parameters for which the recursive dynamics approach holds good. The discontinuity leads to a nonuniversal behavior in the avalanche size distribution for smaller values of avalanche size. We observe that apart from the mean field behavior for larger avalanches, a new behavior for smaller avalanche size is observed as a critical threshold distribution is approached. The phenomenological understanding of the above result is provided using the exact analytical result for the avalanche size distribution. Most interestingly, the prominence of nonuniversal behavior in avalanche size distribution depends on the system parameters.

Critical behavior of random fibers with mixed Weibull distribution

A random fiber bundle model with a mixed Weibull distribution is studied under the global load sharing scheme. The mixed model consists of two sets of fibers. The threshold strength of one set of fibers is randomly chosen from a Weibull distribution with a particular Weibull index, and another set of fibers with a different index. The mixing tunes the critical stress of the bundle and the variation of critical stress with the amount of mixing is determined using a probabilistic method where the external load is increased quasistatically. In a special case which we illustrate, the critical stress is found to vary linearly with the mixing parameter. The critical exponents and power-law behavior of burst avalanche size distribution is found to remain unaltered due to mixing.

An application of comonotonicity and convex ordering to present values with truncated stochastic interest rates

A subject often recurring in recent financial and actuarial research is the investigation of present value functions with stochastic interest rates. Only in the case of uncomplicated payment streams and rather basic interest rate models is an exact analytical result for the distribution function available. In the present contribution, we introduce the concept of truncated stochastic interest rates, useful to adapt general stochastic models to specific financial requirements, and we show how to obtain analytical results for bounds for the present value. We elaborate our method in extension for the Hull and White model and related models. We illustrate the accuracy of the approximations graphically, and we use the bounds to estimate the Value-at-Risk.

Numerical solution of KdV equation using modified Bernstein polynomials

Here we present an algorithm for approximating numerical solution of Korteweg–de Vries (KdV) equation in a modified B-polynomial basis. A set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable. Galerkin method is used to determine the expansion coefficients to construct initial trial functions. For the time variable, the system of equations is solved using fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. We have presented our numerical result with an exact analytical result. Excellent agreement is found between exact and approximate solutions. This procedure has a potential to be used in more complex system of differential equations where no exact solution is available.

Efficiency of encounter-controlled reaction between diffusing reactants in a finite lattice: Non-nearest-neighbor effects

The influence of non-nearest-neighbor displacements on the efficiency of diffusion–reaction processes involving one and two mobile diffusing reactants is studied. An exact analytic result is given for dimension d = 1 from which, for large lattices, one can recover the asymptotic estimate reported 30 years ago by Lakatos-Lindenberg and Shuler. For dimensions d = 2 , 3 we present numerically exact values for the mean time to reaction, as gauged by the mean walklength before reactive encounter, obtained via the theory of finite Markov processes and supported by Monte Carlo simulations. Qualitatively different results are found between processes occurring on d = 1 versus d > 1 lattices, and between results obtained assuming nearest-neighbor (only) versus non-nearest-neighbor displacements.

Conditional Lagrangian acceleration statistics in turbulent flows with Gaussian-distributed velocities

The random intensity of noise approach to the one-dimensional Laval-Dubrulle-Nazarenko-type model having deductive support from the three-dimensional Navier-Stokes equation is used to describe Lagrangian acceleration statistics of a fluid particle in developed turbulent flows. Intensity of additive noise and cross correlation between multiplicative and additive noises entering a nonlinear Langevin equation are assumed to depend on random velocity fluctuations in an exponential way. We use an exact analytic result for the acceleration probability density function obtained as a stationary solution of the associated Fokker-Planck equation. We give a complete quantitative description of the available experimental data on conditional and unconditional acceleration statistics within the framework of a single model with a single set of fit parameters. The acceleration distribution and variance conditioned on Lagrangian velocity fluctuations and the marginal distribution calculated by using independent Gaussian velocity statistics are found to be in a good agreement with the recent high-Reynolds-number Lagrangian experimental data. The fitted conditional mean acceleration is very small, that is, in agreement with direct numerical simulations, and increases for higher velocities but it departs from the experimental data, which exhibit anisotropy of the studied flow.

Two-flux method for radiation heat transfer in anisotropic gas-particles media

Two-flux method can be used, as a simplification for the radiative heat transfer, to predict heat flux in a slab consisting of gas and particles. In the original two-flux method (Schuster, 1905 and Schwarzschild, 1906), the radiation field was assumed to be isotropic. But for gas-particles mixture in combustion environments, the scatterings of particles are usually anisotropic, and the original two-flux method gives critical errors when ignoring this anisotropy. In the present paper, a multilayer four-flux model developed by Roze et al. (2001) is extended to calculate the radiation heat flux in a slab containing participating particles and gas mixture. The analytic resolution of the radiative transfer equation in the framework of a two-flux approach is presented. The average crossing parameter e and the forward scattering ratio ζ are defined to describe the anisotropy of the radiative field. To validate the model, the radiation transfer in a slab has been computed. Comparisons with the exact analytical result of Modest (1993) and the original two-flux model show the exactness and the improvement. The emissivity of a slab containing flyash/CO2/H2O mixture is obtained using the new model. The result is identical with that of Goodwin (1989).

A Simplified Fourier Method for Nonhydrostatic Mountain Waves

Abstract A previously derived approximation to the standard Fourier integral technique for linear mountain waves is extended to include nonhydrostatic effects in a background flow with height-dependent wind and stratification. The approximation involves using ray theory to simplify the vertical eigenfunctions. The generalization to nonhydrostatic waves requires special treatment for resonant modes and caustics. Resonant modes are handled with a small amount of damping, and caustics are handled with a uniformly valid approximation involving the Airy function. This method is developed for both two- and three-dimensional flows, and its results are shown to compare well with an exact analytical result for two-dimensional mountain waves and with a numerical simulation for two- and three-dimensional mountain waves.

On a class of integrals of Legendre polynomials with complicated arguments—with applications in electrostatics and biomolecular modeling

The exact analytical result for a class of integrals involving (associated) Legendre polynomials of complicated argument is presented. The method employed can in principle be generalized to integrals involving other special functions. This class of integrals also proves useful in the electrostatic problems in which dielectric spheres are involved, which is of importance in modeling the dynamics of biological macromolecules. In fact, with this solution, a more robust foundation is laid for the Generalized Born method in modeling the dynamics of biomolecules.

TOWARDS EXACT RESULTS IN NODAL ANTIFERROMAGNETIC PLANAR LIQUIDS

It has been argued in previous works by the authors that nodal excitations in (2 + 1)-dimensional doped antiferromagnets might exhibit, in the spin-charge separation framework and at specific regions of the parameter space, a supersymmetry between spinons and holons. This supersymmetry has been elevated to a N = 2 extended supersymmetry of composite operators of spinon and holons, corresponding to the effective "hadronic" degrees of freedom. In this work we elaborate further on this idea by describing in some detail the dynamics of a specific composite model corresponding to an Abelian Higgs model (SQED). The Abelian nature of the gauge group seems to be necessitated both by the composite structure used, and also by electric charge considerations for the various composites. We demonstrate the passage from a pseudogap to an unconventional superconducting phase, which notably is an exact non-perturbative analytic result, due to the underlying N = 2 supersymmetric Abelian gauge theory. We believe that these considerations may provide a first step towards a non-perturbative understanding of the phase diagrams of strongly-correlated electron systems.

Convergence of the linearδexpansion for the shift inTcfor Bose-Einstein condensation

The leading correction from interactions to the transition temperature T_c for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O(N) scalar field theory in 3 dimensions with N=2. We show that the linear delta expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the delta expansion. For N=2, we calculate the shift in T_c to fourth order in delta. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.

Some comments on models for field enhancement

To estimate the apex field-enhancement factor γ aassociated with a pointed protrusion on a flat planar surface, the simple physical models of a ‘floating sphere at emitter-plane potential’ and a ‘hemisphere on a post’ are often discussed. The corresponding mathematical expressions have the form: γ a= m+ h/ ρ, where ρ is the sphere or hemisphere radius, h is its ‘height above the emitter plane’, and m is a constant variously taken as 0, 2 or 3. Recent numerical simulations for the ‘hemisphere on a post’ model, reported elsewhere by two of us (CJE and GV) and by Kokkaris, Modinos and Xanthakis, have shown that all of these simple formulae significantly overpredict γ a if h/ ρ is large. This article first reexamines the basis of these simple formulae and confirms that they are less secure than is sometimes thought. The formulae reported elsewhere as fits to the numerical results are then quoted and compared with the simple formulae, and with the known exact analytical result for the ‘hemi-ellipsoid on a plane’ model. Discrepancies can be rationalised. Some general conclusions are drawn.

A method for correcting cosine-law errors in SEU test data

Single-event upset tests often change the angle of the ion beam relative to the device to mimic a change in ion linear energy transfer, and the data are then converted via an assumed cosine law. The converted data are intended to represent device susceptibility at normal incidence, but the cosine law sometimes contains considerable error. The standard method for correcting this error is based on the rectangular parallelepiped (RPP) model. However, exact analytical expressions derived from this model are not particularly simple, so specialized computer codes are needed unless approximations are used. This paper starts with an alternate physical model, utilizing a charge-collection efficiency function, and derives an exact analytical result (called the alpha law here) that replaces the cosine law but is almost as simple as the cosine law, even when device susceptibility has a strong azimuthal dependence. The same model can be used to calculate (via numerical integrations) rates in a known heavy-ion environment. An alternative is to use model parameters to construct the parameters for an integrated RPP calculation of rates.

Corley-Jacobson dispersion relation and trans-Planckian inflation

In this Letter we study the dependence of the spectrum of fluctuations in inflationary cosmology on possible effects of trans-Planckian physics, using the Corley/Jacobson dispersion relations as an example. We compare the methods used in previous work [1] with the WKB approximation, give a new exact analytical result, and study the dependence of the spectrum obtained using the approximate method of Ref. [1] on the choice of the matching time between different time intervals. We also comment on recent work subsequent to Ref. [1] on the trans-Planckian problem for inflationary cosmology.