Two-dimensional Riemann problem of the Euler equations to the Van der Waals gas around a sharp corner

In this paper, we study the Riemann problem of the two-dimensional (2D) pseudo-steady supersonic flow with Van der Waals gas around a sharp corner expanding into vacuum. The essence of this problem is the interaction of the centered simple wave with the planar rarefaction wave, which can be solved by a Goursat problem or a mixed characteristic boundary value and slip boundary value problem for the 2D self-similar Euler equations. We establish the hyperbolicity and a priori C¹ estimates of the solution through the methods of characteristic decompositions and invariant regions. Moreover, we construct the pentagon invariant region in order to obtain the global solution. In addition, based on the generality of the Van der Waals gas, we construct the subinvariant regions and get the hyperbolicity of the solution according to the continuity of the subinvariant region. At last, the global existence of solution to the gas expansion problem is obtained constructively.     

GM(1,1)改进模型及其应用

根据 GM( 1 ,1 )灰色模型的指数特性 ,通过在区间上求积分给出了关于背景值的一个比较确切的计算公式 ,讨论了由此建立的 GM( 1 ,1 )改进模型的适用范围和预测精度 .结果表明改进模型比原 GM( 1 ,1 )模型适用性要强、模拟和预测精度要高 ,不仅适用于低增长序列、也适用于高增长序列 ,不仅适用于短期预测 ,同样也适用于中、长期预测     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

Local minima of the trust region problem

We consider the minimization of a quadratic formzVz+2zq subject to the two-norm constraint z=. The problem received considerable attention in the literature, notably due to its applications to a class of trust region methods in nonlinear optimization. While the previous studies were concerned with just the global minimum of the problem, we investigate the existence of all local minima. The problem is approached via the dual Lagrangian, and the necessary and sufficient conditions for the existence of all local minima are derived. We also examine the suitability of the conventional numerical techniques used to solve the problem to a class of single-instruction multiple-data computers known as processor arrays (in our case, AMT DAP 610). Simultaneously, we introduce certain hardware-oriented multisection algorithms, showing their efficiency in the case of small to medium size problems.This research was partially supported by the National Physical Laboratories of England under Contract RTP2/155/127.     

SUPPLEMENT TO “PASSAGE, BLOCKADE, SINK AND SOURCE OF A PLANAR DYNAMICAL SYSTEM”

A Theorem is given on the number of passages passing throgh a multiply-connected region,which corrects a wrong conjecture in a former paper of the author.     

A NONMONOTONE TRUST REGION METHOD FOR NONLINEAR LEAST SQUARES PROBLEMS

In this paper we present a nonmonotone trust region method for nonlinear least squares problems with zero-residual and prove its convergence properties. The extensive numerical results are reported which show that the nonmonotone trust region method is generally superior to the usual trust region method.     

The crossover from classical to quantum regime in the problem of the decay of the metastable state

The probability of the decay of the metastable state has been found as a function of viscosity and temperature. At low temperatures, a classical overbarrier transition changes to the quantum tunneling. At low viscosity in classical and transitive semiquantum region of temperatures a depopulation of the distribution function is significant. The distribution function is shown to satisfy the integral kinetic equation, the kernel of which equals the transition probability. The probability of transitions, induced by the red noise, with the frequency comparable to the transition frequency, is found.     

In search of the Griffiths shield region

The van der Waals equation of state for binary mixtures has been used to determine the location and shape of the Griffiths shield region (where three tricritical lines intersect). If one takes the geometric mean fora ₁₂, the arithmetic mean forb ₁₂, and the configurational free energy as ideal, the center of the Griffiths shield region is found only when the ratio of molecular sizes is infinite. When the Flory equation for the configurational free energy for mixtures of chain molecules is substituted for the ideal form, the results appear to be somewhat different. However, for all the cases studied, with systems which approach geometric mean behavior one finds the shield region only when the ratio of molecular size is very large and when the internal pressure of the small molecule is very much greater than that of the long-chain molecule.This paper is dedicated to our colleague Howard Reiss on the occasion of his 66th birthday.     

Atomistic–continuum interphase model for effective properties of composite materials containing nano-inhomogeneities

Classical micromechanics were revised to study the elastic properties of heterogeneous materials containing nano-inhomogeneities. Contrary to previous studies, this work introduces the concept of an interphase, in contrast to a sharp interface, to account for the interface excess stress effect at the nano-scale. The interphase's constitutive properties are derived from atomistic simulations within the continuum framework. These properties are then incorporated in a micromechanics-based interphase model to compute the effective properties of nano-composites. This scale transition approach bridges the gap between discrete systems (atomic level interactions) and continuum mechanics. An advantage of this approach is that it combines atomistic with continuum models that consider inhomogeneity and interphase morphology. It thereby enables us to account simultaneously for both the shape and the anisotropy of a nano-inhomogeneity and interphase at the continuum level when we compute a material's overall properties. In so doing, it frees us from making any assumptions about the interface characteristics between matrix and the nano-inhomogeneity.     

A linear algorithm of the reference region model for DCE-MRI is robust and relaxes requirements for temporal resolution

Dynamic contrast enhanced MRI (DCE-MRI) has utility for improving clinical diagnoses of solid tumors, and for evaluating the early responses of anti-angiogenic chemotherapies. The Reference Region Model (RRM) can improve the clinical implementation of DCE-MRI by substituting the contrast enhancement of muscle for the Arterial Input Function that is used in traditional DCE-MRI methodologies. The RRM is typically fitted to experimental results with a non-linear least squares algorithm. This report demonstrates that this algorithm produces inaccurate and imprecise results when DCE-MRI results have low SNR or slow temporal resolution. To overcome this limitation, a linear least-squares algorithm has been derived for the Reference Region Model. This new algorithm improves accuracy and precision of fitting the Reference Region Model to DCE-MRI results, especially for voxel-wise analyses. This linear algorithm is insensitive to injection speeds, and has 300- to 8000-fold faster calculation speed relative to the non-linear algorithm. The linear algorithm produces more accurate results for over a wider range of permeabilities and blood volumes of tumor vasculature. This new algorithm, termed the Linear Reference Region Model, has strong potential to improve clinical DCE-MRI evaluations.