A dual open coloring axiom

We discuss a dual of the Open Coloring Axiom introduced by Abraham et al. [U. Abraham, M. Rubin, S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of ₁-dense real order types, Ann. Pure Appl. Logic 29 (2) (1985) 123–206] and show that it follows from a statement about continuous colorings on Polish spaces that is known to be consistent. We mention some consequences of the new axiom and show that implies that all cardinal invariants in Cichoń’s diagram are at least ₂.     

On the Lipschitzian properties of polyhedral multifunctions

In this paper, we show that for a polyhedral multifunctionF:R ⁿ →R ^m with convex range, the inverse functionF ⁻¹ is locally lower Lipschitzian at every point of the range ofF (equivalently Lipschitzian on the range ofF) if and only if the functionF is open. As a consequence, we show that for a piecewise affine functionf:R ⁿ →R ⁿ ,f is surjective andf ⁻¹ is Lipschitzian if and only iff is coherently oriented. An application, via Robinson's normal map formulation, leads to the following result in the context of affine variational inequalities: the solution mapping (as a function of the data vector) is nonempty-valued and Lipschitzian on the entire space if and only if the solution mapping is single-valued. This extends a recent result of Murthy, Parthasarathy and Sabatini, proved in the setting of linear complementarity problems. Research supported by the National Science Foundation Grant CCR-9307685.     

A new outflow boundary condition

Boundary conditions come from Nature. Therefore these conditions exist at natural boundaries. Often, owing to limitations in computing power and means, large domains are truncated and confined between artificial synthetic boundaries. Then the required boundary conditions there cannot be provided naturally and there is a need to fabricate them by intuition, experience, asymptotic behaviour and numerical experimentation. In this work several kinds of outflow boundary conditions, including essential, natural and free boundar conditions, are evaluated for two flow and heat transfer model problems. A new outflow boundary condition, called hereafter the free boundary condition, is introduced and tested. This free boundary condition is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions. In the limit of zero Reynolds number the free boundary condition minimizes the energy functional among all possible choices of outflow boundary conditions. A review of results from applications of the same boundary conditions to several other flow situations is also presented and discussed.     

A second-order radiation boundary condition for the shallow water wave equations on two-dimensional unstructured finite element grids

A second-order radiation boundary condition (RBC) is derived for 2D shallow water problems posed in ‘wave equation’ form and is implemented within the Galerkin finite element framework. The RBC is derived by matching the dispersion relation for the interior wave equation with an approximate solution to the exterior problem for outgoing waves. The matching is correct to second order, accounting for curvature of the wave front and the geometry. Implementation is achieved by using the RBC as an evolution equation for the normal gradient on the boundary, coupled through the natural boundary integral of the Galerkin interior problem. The formulation is easily implemented on non-straight, unstructured meshes of simple elements. Test cases show fidelity to solutions obtained on extended meshes and improvement relative to simpler first-order RBCs.     

ATR-FTIR法比较米粉和若干种奶粉的成分含量

采用衰减全反射红外光谱(ATR-FTIR)法测定米粉和若干种奶粉的吸收光谱,通过计算峰面积的比值来比较所测样品中的脂肪、蛋白质和碳水化合物含量的不同。将此作为红外光谱开放式设计实验的教学内容,培养学生解决实际问题的综合能力和创新能力,收到了良好的教学效果。     

Path integral over reparametrizations: Lévy flights versus random walks

We investigate the properties of the path integral over reparametrizations (or the boundary value of the Liouville field in string theory). Discretizing the path integral, we apply the Metropolis–Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Lévy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby previous results on QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.     

Diffusive corrections to asymptotics of a strong-field quantum transport equation

The asymptotic analysis of a linear high-field Wigner-BGK equation is developed by a modified Chapman-Enskog procedure. By an expansion of the unknown Wigner function in powers of the Knudsen number ?, evolution equations are derived for the terms of zeroth and first order in ?. In particular, a quantum drift-diffusion equation for the position density of electrons, with an ?-order correction on the field terms, is obtained. Well-posedness and regularity of the approximate problems are established, and a rigorous proof that the difference between exact and asymptotic solutions is of order ?², uniformly in time and for arbitrary initial data is given.     

Elementary inductive dichotomy: Separation of open and clopen determinacies with infinite alternatives

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy.