Numerical Solution of Internet Pricing Scheme Based on Perfect Substitute Utility Function

In this paper we will analyze the internet pricing schemes based on Perfect Substitute utility function for homogeneous and heterogeneous consumers. The pricing schemes is useful to help internet service providers (ISP) in maximizing profits and provide better service quality for the users. The models on every type of consumer is applied to the data traffic in Palembang server in order to obtain the maximum profit to obtain optimal. The models are in the form of nonlinear optimization models and can be solved numerically using LINGO 11.0 to get the optimal solution. The results show that the case when we apply flat fee, USAge-based and two part tariff scheme for homogenous we reach the same profit and heterogeneous on willingness to pay we got higher profit if we apply USAge based and two part tariff schemes. Meanwhile, for the case when we apply USAge based and two part tariff schemes for heterogeneous on demand, we reach better solution than other scheme

SU(3) sphaleron: Numerical solution

We complete the construction of the sphaleron $\widehat{S}$ in $SU(3)$ Yang-Mills-Higgs theory with a single Higgs triplet by solving the reduced field equations numerically. The energy of the $SU(3)$ sphaleron $\widehat{S}$ is found to be of the same order as the energy of a previously known solution, the embedded $SU(2)\times U(1)$ sphaleron $S$. In addition, we discuss $\widehat{S}$ in an extended $SU(3)$ Yang-Mills-Higgs theory with three Higgs triplets, where all eight gauge bosons get an equal mass in the vacuum. This extended $SU(3)$ Yang-Mills-Higgs theory may be considered as a toy model of quantum chromodynamics without quark fields and we conjecture that the $\widehat{S}$ gauge fields play a significant role in the nonperturbative dynamics of quantum chromodynamics (which does not have fundamental scalar fields but gets a mass scale from quantum effects).Comment: 36 pages, 6 figures, v5: published versio

Modified energy for split-step methods applied to the linear Schr\"odinger equation

We consider the linear Schr\"odinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long tim

Numerical solution of large Lyapunov equations

A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems

Numerical solution of Riemann-Hilbert problems: Painleve II

We describe a new spectral method for solving matrix-valued Riemann-Hilbert problems numerically. We demonstrate the effectiveness of this approach by computing solutions to the homogeneous Painleve II equation. This can be used to relate initial conditions with asymptotic behaviour

Numerical Solution of Quantum-Mechanical Pair Equations

We discuss and illustrate the numerical solution of the differential equation satisfied by the first‐order pair functions of Sinanoğlu. An expansion of the pair function in spherical harmonics and the use of finite difference methods convert the differential equation into a set of simultaneous equations. Large systems of such equations can be solved economically. The method is simple and straightforward, and we have applied it to the first‐order pair function for helium with 1 / r_(12) as the perturbation. The results are accurate and encouraging, and since the method is numerical they are indicative of its potential for obtaining atomic‐pair functions in general