Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.     

A least‐square weighted residual method for level set formulation

In this paper, a least‐square weighted residual method (LSWRM) for level set (LS) formulation is introduced to achieve interface capturing in two‐dimensional (2D) and three‐dimensional (3D) problems. An LSWRM was adopted for two semi‐discretized advection and reinitialization equations of the LS formulation. The present LSWRM provided good mathematical properties such as natural numerical diffusion and the symmetry of the resulting algebraic systems for the advection and reinitialization equations. The proposed method was validated by solving some 2D and 3D benchmark problems such as those involving a rotating slotted disk, the rotation of a slotted sphere, and a time‐reversed single‐vortex flow and a deformation problem of a spherical fluid. The numerical results were compared with those obtained from essentially non‐oscillatory type formulations and particle LS methods. Further, the proposed LSWRM for the LS formulation was coupled with a splitting finite element method code to solve the incompressible Navier–Stokes equations, and then, the collapse of a 3D broken dam flow was well simulated; in the simulation, the entrapping of air and the splashing of the surge front of water were reproduced. The mass conservation of the present method was found to be satisfactory during the entire simulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Edge-splittings preserving local edge-connectivity of graphs

Let G=(V+s,E) be a 2-edge-connected graph with a designated vertex s. A pair of edges rs,st is called admissible if splitting off these edges (replacing rs and st by rt) preserves the local edge-connectivity (the maximum number of pairwise edge disjoint paths) between each pair of vertices in V. The operation splitting off is very useful in graph theory, it is especially powerful in the solution of edge-connectivity augmentation problems as it was shown by Frank [Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5(1) (1992) 22-53]. Mader [A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978) 145-164] proved that if d(s)≠3 then there exists an admissible pair incident to s. We generalize this result by showing that if d(s)?4 then there exists an edge incident to s that belongs to at least ⌊d(s)/3⌋ admissible pairs. An infinite family of graphs shows that this bound is best possible. We also refine a result of Frank [On a theorem of Mader, Discrete Math. 101 (1992) 49-57] by describing the structure of the graph if an edge incident to s belongs to no admissible pairs. This provides a new proof for Mader's theorem.     

Hybrid Monte Carlo on Hilbert spaces

The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, high-dimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures π which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself.We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure Π in an enlarged phase space having the target π as a marginal, together with a Hamiltonian flow that preserves Π; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of Π when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space.     

Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient

A diffusive epidemic model for H1N1 influenza is formulated with a view to gain basic understanding of the virus behavior. All newborns are assumed to be susceptible. Mortality rate for infective individuals in the population is assumed to be greater than natural mortality rate. Latent, infectious and immune periods are assumed to be constants throughout this study. The numerical solutions of this model are carried out under three different initial populations distribution. In order to investigate the effect of the disease transmission coefficient on the spread of disease, β is taken to be constant as well as a function of seasonally varying time t and a function of spatial variable x  . The threshold quantity (_R0)$({\mathfrak{R}}_0)$ that governs the disease dynamics is derived. Numerical simulation shows that the system supports the existence of sustained and damped oscillations depending on initial populations distribution, the disease transmission rate and diffusion.     

Raman Spectrum of the Split ν4 Mode of CO[dbnd] 3 Ions in Aragonite

Infrared and Raman spectra of CO^[dbnd] ₃ ions are different in the two crystallographic forms of CaCO₃ Calcite and Aragonite owing to their different site symmetries. Due to this fact, i.r. has been used to assess the reversible Calcite ← → Aragonite transformation which occurs upon grinding of CaCO₃, and much work has been done in this field in the last few years^(1-4). In the present paper data are reported on Raman spectra of Calcite and Aragonite forms of CaCO₃ produced by the above mechanical procedure.     

The Splitting Representation of Matrix and Matrix Equation over a Central Division Algebra

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A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case

We present a numerical algorithm for the solution of the Vlasov–Poisson system of equations, in the magnetized case. The numerical integration is performed using the well-known “splitting” method in the electrostatic approximation, coupled with a finite difference upwind scheme; finally the algorithm provides second order accuracy in space and time. The cylindrical geometry is used in the velocity space, in order to describe the rotation of the particles around the direction of the external uniform magnetic field.Using polar coordinates, the integration of the Vlasov equation is very simplified in the velocity space with respect to the cartesian geometry, because the rotation in the velocity cartesian space corresponds to a translation along the azimuthal angle in the cylindrical reference frame. The scheme is intrinsically symplectic and significatively simpler to implement, with respect to a cartesian one. The numerical integration is shown in detail and several conservation tests are presented, in order to control the numerical accuracy of the code and the time evolution of the entropy, strictly related to the filamentation problem for a kinetic model, is discussed.     

大型稀疏矩阵的不完全因子分解法及预处理

本文对大型稀疏阵线性方程组的不完全因子分解及预处理法进行了研究。对对称正定阵和Ｌ阵分别提出了非对角元乘子不完全因子分解法的分解公式。对分解Ａ＝Ｍ－Ｎ，得到了当Ａ为对称正定时，Ｍ亦为对称正定，当Ａ为Ｌ阵时，分解为正规分裂等结果。并研究了预处理ＣＧ加速，最后的数值例子表明本文给出了的算法效果是良好的。