New model and heuristic solution approach for one-dimensional cutting stock problem with usable leftovers

In the one-dimensional cutting stock problem with usable leftovers (1DCSPUL), items of the current order are cut from stock bars to minimize material cost. Here, stock bars include both standard ones bought commercially and old leftovers generated in processing previous orders, and cutting patterns often include new leftovers that are usable in processing subsequent orders. Leftovers of the same length are considered to be of the same type. The number of types of leftovers should be limited to simplify the cutting process and reduce the storage area. This paper presents an integer programming model for the 1DCSPUL with limited leftover types and describes a heuristic algorithm based on a column-generation procedure to solve it. Computational results show that the proposed approach is more effective than several published algorithms in reducing trim loss, especially when the number of types of leftovers is limited.     

The slab theorem for minimal surfaces in $$\mathbb {E}(-1,\tau )$$

Unlike \(\mathbb {R}^{3}\), the homogeneous spaces \(\mathbb {E}(-1,\tau )\) have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantee that a minimal surface in \(\mathbb {E}(-1,\tau )\) is such a graph. More specifically, we introduce the definition of a generalized slab in \(\mathbb {E}(-1,\tau )\) and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph.     

High-order finite elements in numerical electromagnetism: degrees of freedom and generators in duality

Explicit generators for high-order (r>1) scalar and vector finite element spaces generally used in numerical electromagnetism are presented and classical degrees of freedom, the so-called moments, revisited. Properties of these generators on simplicial meshes are investigated, and a general technique to restore duality between moments and generators is proposed. Algebraic and exponential optimal h- and r-error rates are numerically validated for high-order edge elements on the problem of Maxwell’s eigenvalues in a square domain.     

Generalizing the ENO-DB2<Emphasis Type="Italic">p</Emphasis> transform using the inverse wavelet transform

The essentially non-oscillatory (ENO)-wavelet transform developed by Chan and Zhou (SIAM J. Numer. Anal. 40(4), 1369–1404, 2002) is based on a combination of the Daubechies-2p wavelet transform and the ENO technique. It uses extrapolation methods to compute the scaling coefficients without differencing function values across jumps and obtains a multiresolution framework (essentially) free of edge artifacts. In this work, we present a different way to compute the ENO-DB2p wavelet transform of Chan and Zhou which allows us to simplify the process and easily generalize it to other families of orthonormal wavelets.     

A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal <Emphasis Type="Italic">B</Emphasis>-spline

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ _m as the weight functions on [0, m] (m ∈ ?) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.     

A note on Taylor boundary conditions for accurate image restoration

In recent years, several efforts were made in order to introduce boundary conditions for deblurring problems that allow to get accurate reconstructions. This resulted in the birth of Reflective, Anti-Reflective and Mean boundary conditions, which are all based on the idea of guaranteeing the continuity of the signal/image outside the boundary. Here we propose new boundary conditions that are obtained by suitably combining Taylor series and finite difference approximations. Moreover, we show that also Anti-Reflective and Mean boundary conditions can be attributed to the same framework. Numerical results show that, in case of low levels of noise and blurs able to perform a suitable smoothing effect on the original image (e.g. Gaussian blur), the proposed boundary conditions lead to a significant improvement of the restoration accuracy with respect to those available in the literature.     

Difference operators for partitions under the Littlewood decomposition

Let n be a positive integer. Let \(\delta _3(n)\) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum

$$\begin{aligned} D_3(n):=\sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$

satisfies the relation

$$\begin{aligned} 3D_3(n)+\delta _3(n)={\sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$

In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas.

     

Generic substitution policy,an incentive approach

Generic substitution policy has been adopted in several countries in order to control health expenditures. Using a model based on incentives, this work aims to analyze the response of doctors and pharmaceutical companies to the implementation of this policy. It is shown that after the implementation of GSP, the effort of doctor’s convincing the patient to take generics increase or decrease depending on his level of concern for patient well-being; pharmaceutical companies decrease the amount of detailing and the market share of generics tends to increase.