Positive numerical splitting method for the Hull and White 2D Black–Scholes equation

We consider the locally one‐dimensional backward Euler splitting method to solve numerically the Hull and White problem for pricing European options with stochastic volatility in the presence of a mixed derivative term. We prove the first‐order convergence of the time‐splitting. The parabolic equation degenerates on the boundary x = 0 and we apply a fitted finite volume scheme to the equation to resolve the degeneracy and derive the fully discrete problem as we also investigate the discrete maximum principle. Numerical experiments illustrate the efficiency of our difference scheme. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 822–846, 2015     

Density Functional Theory as a Data Science

The development of density functional theory (DFT) functionals and physical corrections are reviewed focusing on the physical meanings and the semiempirical parameters from the viewpoint of data science. This review shows that DFT exchange‐correlation functionals have been developed under many strict physical conditions with minimizing the number of the semiempirical parameters, except for some recent functionals. Major physical corrections for exchange‐correlation function‐ als are also shown to have clear physical meanings independent of the functionals, though they inevitably require minimum semiempirical parameters dependent on the functionals combined. We, therefore, interpret that DFT functionals with physical corrections are the most sophisticated target functions that are physically legitimated, even from the viewpoint of data science.     

Entropy and equilibrium property of QCD-instanton induced final state in deep-inelastic scattering

The scaling and additivity properties of Rényi entropy in rapidity space of the instanton final state (IFS) and current jet identified by the r-sorting method from the QCDINS Monte Carlo event sample are to saturation with decreasing phase space scale. Furthermore, it is found that the additivity of H2 holds well for the IFS in narrow rapidity windows at different positions. These results indicate that the IFS produced in the instanton-induced process of deep inelastic scattering has reached local equilibrium.     

QED Theory of Transition Probabilities and Line Profiles in Highly Charged Ions

A rigorous QED theory of the spectral line profiles is applied to transition probabilities in few-electron highly charged ions. Interelectron interaction corrections are included as well as radiative corrections. Parity nonconserving (PNC) amplitudes with effective weak interactions between the electrons and nucleus are also considered. QED and interelectron interaction corrections to the PNC amplitudes are derived.     

Pseudoinstantons in Metric-Affine Field Theory

In abstract Yang–Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible (pseudoinstanton). We suggest a metric-affine action which is a natural generalization of the Yang–Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino.     

QCD corr ec tions to d ecay-l ep ton polar and azimu thal angular dis tribu tions in e+ e− → t $$\ov erlin e t $$ in th e sof t-gluon approxima tionin th e sof t-gluon approxima tion

QCD corrections to order α _s in the soft-gluon approximation to angular distributions of decay charged leptons in the process e ⁺ e ⁻ → t , followed by semileptonic decay of t , are obtained in the esu− e ⁻ centre-of-mass frame. As compared to distributions in the top rest frame, these have the advantage that they would allow direct comparison with experiment without the need to reconstruct the top rest frame. The results also do not depend on the choice of a spin quantization axis for t or t. Analytic expression for the triple distribution in the polar angle of t and polar and azimuthal angles of the lepton is obtained. Analytic expression is also derived for the distribution in the charged-lepton polar angle. Numerical values are discussed for √s=400, 800 and 1500 GeV.     

Unusual Corrections to Scaling in the 3-State Potts Antiferromagnet on a Square Lattice

At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2 and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for example, the correlation length diverges as J/kT according to Ae ² (1+be ^– +···), where b is a nonuniversal constant that may nevertheless be determined independently. A similar result holds for the staggered susceptibility. These results are shown to be consistent with the anomalous behavior found in the Monte Carlo simulations of Ferreira and Sokal.     

Non-Violation of Energy Conditions in the Future Accelerated Universe Due to Quantum Effects

Here, an accelerated phantom model for the late universe is explored, which is free from future singularity. It is interesting to see that this model exhibits strong curvature for all time in future, unlike models with ‘big-rip singularity’ showing high curvature near singularity time only. So, quantum gravity effects grow dominant as time increases in late universe too. More importantly, it is demonstrated that quantum corrections to FRW equations lead to non-violation of ‘cosmic energy conditions’ of general relativity, which are violated for accelerating universe without these corrections.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Family of instantons of the Kraichnan model with a frozen velocity field

We find the family of instantons of the Kraichnan model with a frozen velocity field. Using these instantons, we investigate the asymptotic behavior of higher orders of the perturbation theory series constructed for the response function. We demonstrate that although the number of diagrams increases factorially with the degree of the perturbation series term, the perturbation series itself has a finite and sometimes even infinite convergence radius. We thus disprove the commonly accepted view that the type of series convergence can be determined by estimating the number of diagrams in higher orders of the perturbation theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 200–213, February, 2009.