Two efficient algorithms for the computation of ideal sums in quadratic orders

This paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders. If the class number of the quadratic number field is equal to 1, these algorithms can be used to calculate the GCD in the quadratic order. We show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in up to a constant factor, i.e., in where bounds the size of the operands and denotes an upper bound for the multiplication time of -bit integers. Using Schönhage-Strassen's asymptotically fast multiplication for -bit integers, we achieve

     

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Let M ⁿ be an n-dimensional compact C ^∞-differentiable manifold, n ≥ 2, and let S be a C ¹-differential system on M ⁿ . The system induces a one-parameter C ¹ transformation group φ _t (−∞ < t < ∞) over M ⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M ⁿ . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M ⁿ of the differential system S. With respect to a given C ^∞ Riemannian metric of M ⁿ , we consider the bundle of all (n−2) spheres Q _x ⁿ⁻², x∈M, where Q _x ⁿ⁻² for each x consists of all unit tangent vectors of M ⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ _t ^# (−∞<t<∞) of . For an l-frame α = (u ₁, u ₂,⋯, u _l ) of M ⁿ at a point x in M, 1 ≥ l ≥ n−1, each u _i being in , we shall denote the volume of the parallelotope in the tangent space of M ⁿ at x with edges u ₁, u ₂,⋯, u _l by υ(α), and let . This is a continuous real function of t. Let

The Eulerian distribution on involutions is indeed unimodal

Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,…,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0,In,1,…,In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that

     

Operator Formulas Involving Generalized Stirling Number Derived by Virtue of Normal Ordering of Vacuum Projector

By virtue of the normal ordering of vacuum projector we directly derive some new complicated operator identities, regarding to the generalized Stirling number.     

Lepton-number-violating decays of singly-charged Higgs bosons in the type-（Ⅰ＋Ⅱ ） seesaw model

The lepton-number-violating decays of singly-charged Higgs bosons H^± are investigated in the minimal type-（ Ⅰ＋Ⅱ ） seesaw model with one SU（2）L Higgs triplet △ and one heavy Majorana neutrino N1 at the TeV scale. We find that the branching ratios B（H^＋ → 1α^＋υ^-） （for α = e,μ,τ） depend not only on the mass and mixing parameters of three light neutrinos υi （for i = 1, 2, 3） but also on those of N1. Assuming that the mass of N1 lies in the range of 200 GeV to 1 TeV, we figure out the generous interference bands for the contributions of υi and N1 to B（H^＋→ 1α^＋υ^-）. We illustrate some salient features of such interference effects by considering three typical mass patterns of υi, and show that the relevant Majorana CP-violating phases can affect the magnitudes of B（H^＋→ 1α^＋υ^-）） in this parameter region.     

A first-order system approach for diffusion equation. II: Unification of advection and diffusion

In this paper, we unify advection and diffusion into a single hyperbolic system by extending the first-order system approach introduced for the diffusion equation [J. Comput. Phys., 227 (2007) 315–352] to the advection–diffusion equation. Specifically, we construct a unified hyperbolic advection–diffusion system by expressing the diffusion term as a first-order hyperbolic system and simply adding the advection term to it. Naturally then, we develop upwind schemes for this entire   system; there is thus no need to develop two different schemes, i.e., advection and diffusion schemes. We show that numerical schemes constructed in this way can be automatically uniformly accurate, allow O(h)$O(h)$ time step, and compute the solution gradients (viscous stresses/heat fluxes for the Navier–Stokes equations) simultaneously to the same order of accuracy as the main variable, for all Reynolds numbers. We present numerical results for boundary-layer type problems on non-uniform grids in one dimension and irregular triangular grids in two dimensions to demonstrate various remarkable advantages of the proposed approach. In particular, we show that the schemes solving the first-order advection–diffusion system give a tremendous speed-up in CPU time over traditional scalar schemes despite the additional cost of carrying extra variables and solving equations for them. We conclude the paper with discussions on further developments to come.     

A robust numerical model for premixed flames with high density ratios based on new pressure correction and IMEX schemes

In this study we present a model for the interaction of premixed flames with obstacles in a channel flow. Although the flow equations are solved with Direct Numerical Simulation using a low Mach number approximation, the resolution used in the computation is limited (∼1 mm) hence the inner structure of the flame and the chemical scales are not solved. The species equations are substituted with a source term in the energy equation that simulates a one-step global reaction. A level set method is applied to track the position of the flame and its zero level is used to activate the source term in the energy equation only at the flame front. An immersed boundary method reproduces the geometry of the obstacles. The main contribution of the paper is represented by the proposed numerical approach: an IMEX (implicit–explicit) Runge–Kutta scheme is used for the time integration of the energy equation and a new pressure correction algorithm is introduced for the time integration of the momentum equations. The approach presented here allows to calculate flames which produce high density ratios between burnt and unburnt regions. The model is verified by simulating first simple solutions for one- and two-dimensional flames. At last, the experiments performed by Masri and Ibrahim with square and rectangular bodies are calculated.     

The influence of cell geometry on the Godunov scheme applied to the linear wave equation

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier–Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.     

Resolution effects and scaling in numerical simulations of passive scalar mixing in turbulence

The effects of finite grid resolution on the statistics of small scales in direct numerical simulations of turbulent mixing of passive scalars are addressed in this paper. Simulations at up to 2048³ grid points with grid spacing Δx varied from about 2 to 1/2 Batchelor scales (η_B) show that most conclusions on Schmidt number (Sc) dependence from prior work at less stringent resolution remain qualitatively correct, although simulations at resolution Δx≈η_B are preferred and will give adequate results for many important quantities including the scalar dissipation intermittency exponent and structure functions at moderately high orders. For Sc≥1, since η_B=ηSc^−1/2 (where η is the Kolmogorov scale), the requirement Δx≈η_B is more stringent than the corresponding criterion Δx≈η for the velocity field, which is thus well resolved in simulations aimed at high Schmidt number mixing. A simple argument is given to help interpret the effects of Schmidt and Reynolds numbers on trends towards local isotropy and saturation of intermittency at high Schmidt number. The present results also provide evidence for a trend to isotropy at high Reynolds number with fixed Sc=1.0. This is a new observation apparently not detected in less well resolved simulations in the past, and will require further investigation in the future.     

Triple diffusive mixed convection flow in a duct using convective boundary conditions

The motivation of the current article is to explore a numerical investigation on steady triply diffusive convection in a vertical channel. Heat is exchanged from the external fluid with the plates. The reference temperature is taken as equal and also as different for the external fluid. Solutions in the absence of viscous dissipation and buoyancy forces are also obtained as special cases. General solutions including the effects of viscous dissipation and buoyancy forces are obtained analytically using the method of perturbation. The analytical solutions can be used only if the Brinkman number is small. Hence to know the flow properties for all values of Brinkman number, we resort to numerical solutions. The effects of thermal Grashof number, solutal Grashof number, and the chemical reaction parameter on the flow field are evaluated numerically. The obtained results are validated against previously published results for special case of the problems.