A stable interface element scheme for the p-adaptive lifting collocation penalty formulation

This paper presents a procedure for adaptive polynomial refinement in the context of the lifting collocation penalty (LCP) formulation. The LCP scheme is a high-order unstructured discretization method unifying the discontinuous Galerkin, spectral volume, and spectral difference schemes in single differential formulation. Due to the differential nature of the scheme, the treatment of inter-cell fluxes for spatially varying polynomial approximations is not straightforward. Specially designed elements are proposed to tackle non-conforming polynomial approximations. These elements are constructed such that a conforming interface between polynomial approximations of different degrees is recovered. The stability and conservation properties of the scheme are analyzed and various inviscid compressible flow calculations are performed to demonstrate the potential of the proposed approach.     

Update of photoelectric absorption coefficients in the tables of McMaster

As a result of a systematic analysis of the compilation of McMaster's x‐ray cross‐sections, the polynomial coefficients of the N range (from edges N₇ to M₅) of elements Z = 61–69 have to be corrected. Furthermore, no detailed information on position and jumps of M edges of elements Z > 53 and N edges of elements Z > 85 are given. Elements Z = 84, 85, 87, 88, 89, 91 and 93 are missing. We have corrected the wrong values of polynomial coefficients and added missing M‐ and N‐edge data and missing elements. Copyright © 2005 John Wiley & Sons, Ltd.     

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

Factorizations of Polynomials over Noncommutative Algebras and Sufficient Sets of Edges in Directed Graphs

To directed graphs with unique sink and source we associate a noncommutative associative algebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when coefficients of the polynomial can be rationally expressed via elements of a given set of pseudo-roots (edges). Our results are based on a new theorem for directed graphs also proved in this paper. To the memory of Felix Alexandrovich Berezin. Vladimir Retakh was partially supported by NSA     

Application of group representation theory to derive Hermite interpolation polynomials on a triangle

Current methods used to devise sets of Hermite interpolation polynomials of minimal order that ensure C⁽ⁿ⁾ continuity across triangular element boundaries in two dimensions are not readily extensible to higher dimensions. The extension of such methods is especially difficult when the number of degrees of freedom afforded by data at points is different from the number of degrees of freedom determined by the coefficients of a complete polynomial basis to a particular order. This work introduces a formalism based on group representation theory that can accomplish this task in general. The method is introduced through the derivation of C⁽¹⁾ continuous Hermite polynomials that interpolate data at the three vertices of an equilateral triangular element. These interpolation polynomials are reported here for the first time. The polynomials derived here are compared to the standard polynomials defined in a right triangle by using the two sets of polynomials to solve the Laplace equation over finite elements. The methodology presented here is of use in higher dimensional elements when the complete polynomial degrees of freedom exceed the total C⁽ⁿ⁾ degrees of freedom at the nodes.     

Effect Algebras Which Can Be Covered by MV-Algebras

We exhibit effect algebras which can be covered by MV-subalgebras. We show that any effect algebra E which satisfies the Riesz interpolation property (RIP) and the so-called difference-meet property (DMP) can be covered by blocks, maximal subsets of mutually strongly compatible elements of E, which are always MV-subalegbras. This result generalizes that of Rieanová who proved the same result for lattice-ordered effect algebras. We show that for effect algebras with only (RIP) the result in question can fail.     

Preparation of Daidzein microparticles through liquid antisolvent precipitation under ultrasonication

In this study, daidzein microparticles (DMP) were prepared using an improved ultrasound-assisted antisolvent precipitation method. Preliminary experiments were conducted using six single-factor experiments, and principal component analysis (PCA) was adopted to obtain the three staple elements of the ultrasonic power, solution concentration, and nozzle diameter. The response surface Box-Behnken (BBD) design was used to optimize the level of the above factors. The optimal preparation conditions of the DMP were obtained as follows: the flow rate was 4 mL/min, the concentration of the daidzein solution was 16 mg/mL, the ratio of antisolvent to solvent (liquid-to-liquid ratio) was 9, the nozzle diameter was 300 μm, the ultrasonic power was 180 W (665 W/L), and the system speed was 760 r/min. The minimum average particle size of DMP was 181 ± 2 nm. The properties of daidzein particles before and after preparation were analyzed via scanning electron microscopy, X-ray diffraction analysis, Differential scanning calorimetry and Fourier transform infrared spectroscopy, no obvious change in its chemical structure was observed, but crystallinity was reduced. Compared with daidzein powder, DMP has a higher solubility and stronger antioxidant capacity. The above results indicate that the improved method of ultrasonication combined with antisolvent can reduce the size of daidzein particles and has a great potential in practical production.     

Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems

The maximum principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the discrete maximum principle (DMP), is one of the most difficult properties to achieve in numerical methods, especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic. Violation of the DMP may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one. In this work, we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference (MFD) method on two- and three-dimensional meshes. These conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element. Efficient solutions are devised for meshes consisting of simplexes, parallelograms and parallelepipeds, and orthogonal locally refined elements as those used in the AMR methodology. On simplicial meshes, it turns out that the MFD method coincides with the mixed-hybrid finite element methods based on the low-order Raviart–Thomas vector space. Thus, in this case we recover the well-established conventional angle conditions of such approximations. Instead, in the other cases a suitable design of the MFD method allows us to formulate a monotone discretization for which the existence of a DMP can be theoretically proved. Moreover, on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the Multi-Point Flux Approximation (MPFA) methods. Numerical experiments confirm the effectiveness of the considered monotonicity conditions.     

Integrable deformations of integrable Hamiltonian systems

We give new effective necessary conditions for the integrability of nonhomogeneous potentials which are sums of two homogeneous terms. These conditions were deduced from an analysis of the differential Galois group of variational equations. We apply the obtained result to polynomial perturbations of linear oscillator with two degrees of freedom.     

A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains

We present the first space–time hybridizable discontinuous Galerkin (HDG) finite element method for the incompressible Navier–Stokes and Oseen equations. Major advantages of a space–time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher-order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space–time elements. Our formulation is related to the HDG formulation for incompressible flows introduced recently in, e.g., [N.C. Nguyen, J. Peraire, B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Eng. 199 (2010) 582–597]. However, ours is inspired in typical DG formulations for compressible flows which allow for a more straightforward implementation. Another difference is the use of polynomials of fixed total degree with space–time hexahedral and quadrilateral elements, instead of simplicial elements. We present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time.     

Analytic Expression of Arbitrary Matrix Elements for Boson Exponential Quadratic Polynomial Operators

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.     

Preparation and characterization of smart polymer brush-modified magnetic nanoparticles for biomedicine application

This paper investigates synthesis and characterization of smart polymer brush-modified magnetic nanoparticles (SPB-MNPs). Magnetic nanoparticles (MNPs) modified with poly(poly(ethylene glycol) monomethacrylate (poly(PEGMA)) brush were first functioned with activated pyridyldithio. Then, MNPs functioned with pyridyldithio (MNPs-PEG-PDT) were conjugated with 4-diamino-6-mercaptopyrimidine (DMP) to form SPB-MNPs via stimuli-responsive disulfide linkage. Fourier transform infrared spectra (FTIR) was used to monitor the preparation process of MNPs-PEG-PDT. MNPs-PEG-PDT is very highly reactive toward DMP. The disulfide linkage of SPB-MNPs can be cleaved by reduced glutathione (GHS). The concentration of GHS plays an important role in controlling the cleaved efficiency. The optimum concentration of GHS to release DMP is in the millimolar range.     

A systematic study of the degradation of dimethyl phthalate using a high-frequency ultrasonic process

A comprehensive study of the sonochemical degradation of dimethyl phthalate (DMP) was carried out using high-frequency ultrasonic processes. The effects of various operating parameters were investigated, including ultrasonic frequency, power density, initial DMP concentration, solution pH and the presence of hydrogen peroxide. In general, a frequency of 400 kHz was the optimum for achieving the highest DMP degradation rate. The degradation rate was directly proportional to the power density and inversely related to the initial DMP concentration. It was interesting to find that faster removal rate was observed under weakly acidic condition, while hydrolysis effect dominated in extreme-basic condition. The addition of hydrogen peroxide can increase the radical generation to some extent. Furthermore, both hydroxylation of the aromatic ring and oxidation of the aliphatic chain appear to be the major mechanism of DMP degradation by sonolysis based on LC/ESI-MS analysis. Among the principle reaction intermediates identified, tri- and tetra-hydroxylated derivatives of DMP, as well as hydroxylated monomethyl phthalates and hydroxylated phthalic acid were reported for the first time in this study. Reaction pathways for DMP sonolysis are proposed based on the detected intermediates.     

Hidden supersymmetry in quantum bosonic systems

We show that some simple well-studied quantum mechanical systems without fermion (spin) degrees of freedom display, surprisingly, a hidden supersymmetry. The list includes the bound state Aharonov-Bohm, the Dirac delta and the Pöschl-Teller potential problems, in which the unbroken and broken N = 2 supersymmetry of linear and nonlinear (polynomial) forms is revealed.     

First integrals and Darboux polynomials of natural polynomial Hamiltonian systems

In this Letter we study some aspects of the relationship between the existence of Darboux polynomials and additional polynomial first integrals in natural polynomial Hamiltonian systems with a finite number of degrees of freedom. More precisely, first we improve results of the paper of Maciejewski and Przybylska [A.J. Maciejewski, M. Przybylska, Phys. Lett. A 326 (2004) 219]; and after we answer two open questions presented in that paper.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Calibration of a structured-light projection system: Development to large dimension objects

In this study, performances of the profilometry method based on structured-light projection are analyzed in the case of a large dimension object and a short distance from the acquisition system. This work is based on the calibration procedure introduced by Brèque et al. After the evaluation of Brèque's pinhole model, a new calibration procedure based on polynomial transformations is proposed. The proposed work is founded on a systematic study of different degrees of a polynomial form. Random and systematic errors are calculated to show accuracies for polynomial degrees equal to 1, 2, 3 and 4. Different sources of errors are presented and discussed. The developed calibration is applied to a human body and a human-size dummy.     

High-order adaptive methods for parabolic systems

We consider the adaptive solution of parabolic partial differential systems in one and two space dimensions by finite element procedures that automatically refine and coarsen computational meshes, vary the degree of the piecewise polynomial basis and, in one dimension, move the computational mesh. Two-dimensional meshes of triangular, quadrilateral, or a mixture of triangular and quadrilateral elements are generated using a finite quadtree procedure that is also used for data management. A posteriori estimates, used to control adaptive enrichment, are generated from the hierarchical polynomial basis. Temporal integration, within a method-of-lines framework, uses either backward difference methods or a variant of the singly implicit Runge-Kutta (SIRK) methods. A high-level user interface facilitates use of the adaptive software.     

Removal of ghost images by using tilted element optical systems with polynomial surfaces for aberration compensation

A novel solution to problematic ghost images is implemented by using tilted lens elements with polynomial surfaces. Tilting the lens surfaces sends reflections out of the imaging path. The nonrotationally symmetric polynomial surfaces correct aberrations caused by tilts. The complex lens surfaces are fabricated by using gray-scale lithographic patterning of hybrid solgel glass.