Nonlinear obstacle problems with double phase in the borderline case

In this paper we study a double phase problem with an irregular obstacle. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm, which can be regarded as a borderline case of the double phase functional with $(p,q)$-growth. We obtain an optimal global Calderón–Zygmund type estimate for the obstacle problem with double phase in the borderline case.     

Sommerfeld-type integrals for discrete diffraction problems

Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem).     

The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities

The study explored the impact of Please Go Bring Me-COnceptual Model-based Problem Solving (PGBM-COMPS) computer tutoring system on multiplicative reasoning and problem solving of students with learning disabilities. The PGBM-COMPS program focused on enhancing the multiplicative reasoning and problem solving through nurturing fundamental mathematical ideas and moving students above and beyond the concrete level of operation. This is achieved by taking advantages of the constructivist approach from mathematics education and explicit conceptual model-based problem solving approach from special education. Participants were three elementary students with learning disabilities (LD). A mixed method design was employed to investigate the effect of the PGBM-COMPS program on enhancing students’ multiplicative reasoning and problem solving. It was found that the PGBM-COMPS program significantly improved participating students’ problem solving performance not only on researcher developed criterion tests but also on a norm-referenced standardized test. Qualitative and quantities data from this study indicate that, in addition to nurturing fundamental concept of composite units, it is necessary to help students to understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills.     

理想弹塑性固体Riemann问题的解结构与初值条件

针对理想弹塑性固体材料的一维Riemann问题，在不考虑真空的情况下，讨论其所有可能存在的解结构，给出每一种解结构下对应的初值条件且证明该系列初值条件的完备性，即任意给定的物理量初值均有且只有一种解结构与之对应.基于该理论，在设计精确或近似理想弹塑性Riemann问题求解器时，可以依据初值条件对任意物理量初值直接判断其对应的解结构，从而提高求解器的精度和效率.数值试验验证了该系列初值条件的正确性和有效性.     

约束Fractional Tikhonov正则化的模迭代方法

反问题是现在数学物理研究中的一个热点问题,而反问题求解面临的一个本质性困难是不适定性。求解不适定问题的普遍方法是:用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解,这种方法称为正则化方法.如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容.当前,最为流行的正则化方法有基于变分原理的Tikhonov正则化及其改进方法,此类方法是求解不适定问题的较为有效的方法,在各类反问题的研究中被广泛采用,并得到深入研究.     

A benchmark of optimally folded protein structures using integer programming and the 3D-HP-SC model

The Protein Structure Prediction (PSP) problem comprises, among other issues, forecasting the three-dimensional native structure of proteins using only their primary structure information. Most computational studies in this area use synthetic data instead of real biological data. However, the closer to the real-world, the more the impact of results and their applicability. This work presents 17 real protein sequences extracted from the Protein Data Bank for a benchmark to the PSP problem using the tri-dimensional Hydrophobic-Polar with Side-Chains model (3D-HP-SC). The native structure of these proteins was found by maximizing the number of hydrophobic contacts between the side-chains of amino acids. The problem was treated as an optimization problem and solved by means of an Integer Programming approach. Although the method optimally solves the problem, the processing time has an exponential trend. Therefore, due to computational limitations, the method is a proof-of-concept and it is not applicable to large sequences. For unknown sequences, an upper bound of the number of hydrophobic contacts (using this model) can be found, due to a linear relationship with the number of hydrophobic residues. The comparison between the predicted and the biological structures showed that the highest similarity between them was found with distance thresholds around 5.2–8.2 Å. Both the dataset and the programs developed will be freely available to foster further research in the area.     

Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method (CVBEFM) for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers. To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative, a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures. To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables, a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure. The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers, even for extremely large wavenumbers such as k = 10 000.     

Space-time finite element methods stabilized using bubble function spaces

ABSTRACT

In this paper, a stabilized space-time finite element method for solving linear parabolic evolution problems is analyzed. The proposed method is developed on a base of a space-time variational setting, that helps on the simultaneous and unified discretization in space and in time by finite element techniques. Stabilization terms are constructed by means of classical bubble spaces. Stability of the discrete problem with respect to an associated mesh dependent norm is proved, and a priori discretization error estimates are presented. Numerical examples confirm the theoretical estimates.