Let Φ be a trace-preserving, positivity-preserving linear map on the algebra of complex 2 × 2 matrices, and let Ω be any finite-dimensional completely positive map. For p = 2 and p ≥ 4, we prove that the maximal p-norm of the product map Φ⊗Ω is the product of the maximal p-norms of Φ and Ω. Restricting Φ to the class of completely positive maps, this settles the multiplicativity question for al...

We propose an asymptotic-preserving (AP) scheme for kinetic equations that is efficient also in the hydrodynamic regimes. This scheme is based on the BGK-penalty method introduced by Filbet-Jin [14], but uses the penalization successively to achieve the desired asymptotic property. This method possesses a stronger AP property than the original method of Filbet-Jin, with the additional feature o...

In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including ?-method, mixed method, Richardson Du Fort Frankel MADE (modified alternating directional explicit) method. These produce drawbacks such as spurious oscillations negative values in solution when volatility is much smaller than interest rate. The method sacrifices accuracy obtain sta...

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property crucial to ability of a scheme simulate perturbation waves over lake-at-rest steady state such as on lake or tsunami in deep ocean. combines method, hydrostatic reconstruction technique, and change unknown...

We develop a high-order positivity-preserving discontinuous Galerkin (DG) scheme for linear Vlasov-Boltzmann transport equations (Vlasov-BTE) under the action of quadratically confined electrostatic potentials. The solutions of such BTEs are positive probability distribution functions and it is very challenging to have a mass-conservative, high-order accurate scheme that preserves positivity of...

We revisit Bressoud's generalized Borwein conjecture. Making use of certain positivity-preserving transformations for q-binomial coefficients, we establish the truth infinitely many new cases Bressoud In addition, prove doubly-bounded refinement Foda-Quano identities. Finally, discuss companions to even moduli particular, all 10mod20 identities are derived.

In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for film height h <mml:annotation encoding="application/x-tex...

A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of Ripa model -- a generalization shallow water equations that accounts effects temperature variations. Thermodynamic processes are important particularly in upper layers ocean where variations sea surface play fundamental role climate change. The well-balance property which requires schemes to...