Search results for "Computational Mathematic"
showing 10 items of 987 documents
The planar two-body problem for spheroids and disks
2021
We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not invol…
A computational study of the lowest singlet and triplet states of neutral and dianionic 1,2-substituted icosahedral and octahedralo-carboranes
2006
This work introduces a calibrated B3LYP/6-31G(d) study on the electronic structure of singlet and triplet neutral species of 1,2-substituted icosahedral 1,2-R(2)-1,2-C(2)B(10)H(10) and octahedral 1,2-R(2)-1,2-C(2)B(4)H(4) molecules with R = {H, OH, SH, NH(2), PH(2), CH(3), SiH(3)} and their respective dianions formed by proton removal on each R group. A variety of small adiabatic singlet-triplet gaps DeltaE(ST) are obtained from these systems ranging from 2.93 eV (R = NH(2)) <or= DeltaE(ST) <or= 3.98 eV (R = SiH(3)) for the icosahedral neutrals and 1.56 eV (R = NH(2)) <or= DeltaE(ST) <or= 4.13 eV (R = SiH(3)) for the octahedral neutrals, these gaps being globally smaller for the dianionic s…
Understanding the ring current effects on magnetic shielding of hydrogen and carbon nuclei in naphthalene and anthracene
2008
The local response to an external magnetic field normal to the molecular plane of naphthalene and anthracene was investigated via current density and magnetic shielding density maps. The Biot-Savart law shows that the deshielding caused by pi-ring currents in naphthalene is stronger for alpha- than for beta-protons due to geometrical factors. The shielding tensor of the carbon nuclei in both molecules is strongly anisotropic and its out-of-plane component determines the up-field chemical shift of (13)C in nuclear magnetic resonance spectra. The pi-ring currents flowing beyond the C-skeleton in front of a probe carbon nucleus, and on remote parts of the molecular perimeter, yield positive co…
Simple connections between generalized hypergeometric series and dilogarithms
1997
AbstractConnections between generalized hypergeometric series and dilogarithms are investigated. Some simple relations of an Appell's function and dilogarithms are found.
Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams
2000
Present and future high-precision tests of the Standard Model and beyond for the fundamental constituents and interactions in Nature are demanding complex perturbative calculations involving multi-leg and multi-loop Feynman diagrams. Currently, large effort is devoted to the search for closed expressions of loop integrals, written whenever possible in terms of known - often hypergeometric-type - functions. In this work, the scalar three-point function is re-evaluated by means of generalized hypergeometric functions of two variables. Finally, use is made of the connection between such Appell functions and dilogarithms coming from a previous investigation, to recover well-known results.
New indefinite integrals from a method using Riccati equations
2018
ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which ...
Energy dissipative solutions to the Kobayashi-Warren-Carter system
2017
In this paper we study a variational system of two parabolic PDEs, called the Kobayashi-Warren-Carter system, which models the grain boundary motion in a polycrystal. The focus of the study is the existence of solutions to this system which dissipate the associated energy functional. We obtain existence of this type of solutions via a suitable approximation of the energy functional with Laplacians and an extra regularization of the weighted total variation term of the energy. As a byproduct of this result, we also prove some $\Gamma$-convergence results concerning weighted total variations and the corresponding time-dependent cases. Finally, the regularity obtained for the solutions togethe…
The Euler–Lagrange equation for the Anisotropic least gradient problem
2016
Abstract In this paper we find the Euler–Lagrange equation for the anisotropic least gradient problem inf { ∫ Ω ϕ ( x , D u ) : u ∈ B V ( Ω ) , u | ∂ Ω = f } being ϕ a metric integrand and f ∈ L 1 ( ∂ Ω ) . We also characterize the functions of ϕ -least gradient as those whose boundary of the level set is ϕ -area minimizing in Ω .
Indefinite integrals of products of special functions
2016
ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals o…
More indefinite integrals from Riccati equations
2019
ABSTRACTTwo new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of...