An extension of the FlowSort sorting method to deal with imprecision

We propose an extension of the FlowSort sorting method to the case when there is imprecision on the input data. Within multicriteria decision aid, a lot of attention has been paid to sorting problems where a set of actions has to be assigned to completely ordered categories. However, few methods suit when the data or the parameters of the model are not precisely defined. In this paper, instead of reducing the imprecise data to single values, we consider that the sorting parameters or the data are defined by intervals. We analyse the properties usually required for a sorting method and illustrate this extension on a practical example.     

An alternative to the Bathe algorithm

This paper presents a new composite sub-steps algorithm for solving reliable numerical responses in structural dynamics. The newly developed algorithm is a two sub-steps, second-order accurate and unconditionally stable implicit algorithm with the same numerical properties as the Bathe algorithm. The detailed analysis of the stability and numerical accuracy is presented for the new algorithm, which shows that its numerical characteristics are identical to those of the Bathe algorithm. Hence, the new sub-steps scheme could be considered as an alternative to the Bathe algorithm. Meanwhile, the new algorithm possesses the following properties: (a) it produces the same accurate solutions as the Bathe algorithm for solving linear and nonlinear problems; (b) it does not involve any artificial parameters and additional variables, such as the Lagrange multipliers; (c) The identical effective stiffness matrices can be obtained inside two sub-steps; (d) it is a self-starting algorithm. Some numerical experiments are given to show the superiority of the new algorithm and the Bathe algorithm over the dissipative CH-α algorithm and the non-dissipative trapezoidal rule.     

An alternative to the Brauer set

We derive an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set. It is accompanied by non-singularity conditions.     

An alternative to the Ritz pairs with the EN subspace method

In this note we define EN subspaces by using the Eirola-Nevanlinna algorithm for solving a linear system. We compare this construction with the Arnoldi method for generating Krylov subspaces and computing eigenvalue approximations. Further, we compute Ritz pairs by restricting the updated preconditionerH _k of the EN algorithm to the generated EN subspaces.     

An alternative approach to unitoidness

In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287-290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sum

eⁱ⁰I_p⊕e^iπI_q⊕0I_n-r.     

An ordering of dependence for contingency tables

A majorization ordering is defined on matrices with the same row and column sums. This ordering is used as an ordering of dependence for contingency tables. Results are derived for maximal and minimal matrices with respect to the majorization ordering. This theory can be used to maximize and minimize Schur concave functions defined over matrices, when there are row and column sum constraints; in this paper, it is applied to the algorithm of Mehta and Patel (1983) for finding the P-value of Fisher's exact test.     

An alternative approach to integro-differential equations

A class of integro-differential equations with operator-valued integral kernels, including equations of neutral type is studied. We show that this class is covered by the framework of evolutionary equations and we derive sufficient conditions on the kernels within this framework in order to show well-posedness. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An alternative approach to the structure theory of PI-rings

We expose a rather simple and direct approach to the structure theory of prime PI-rings (“Posner’s theorem”), based on fundamental properties of the extended centroid of a prime ring.     

An alternative approach to nonlinear filtering

The structure of a nonlinear filter with observation process having continuous and discontinuous components is considered. The approach is based on the so-called “Bayes” formula for conditional expectations. “Fubini” type theorems for stochastic integrals are given and used to obtain the representations of an optimal estimate and of the conditional likelihood ratio. A linear unnormalized filtering equation for controlled system process is derived.     

An alternative to traditional computational tomography

An alternate method to standard computational tomography is suggested and investigated. In the proof of the fundamental theorem one makes use of an approach that is related to an idea of Yu. V. Linnik.The investigated alternate method has been communicated by me in 1978 to D. A. Novik, who introduced me to the pioneer works on computational tomography and drew my attention to this topic. In 1988 this investigation has been communicated at the All-Union conference "Probabilistic and Statistical Methods, and Their Application in Physics and Engineering" (Alma-Ata, May 1988) (see also [9]).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 271–278, 1990.     

How to deal with unlabeled random graphs

Let U(n,M) be a graph chosen at random from the family of all unlabeled graphs with n vertices and M edges. In the paper we study the asymptotic behavior of U(n,M) when n → ∞. In particular, we show how properties of U(n,M) could be derived from analogous properties of a labeled random graph.     

An alternative approach to solving the Black–Scholes equation with time-varying parameters

In this note we provide a simple derivation of an explicit formula for the price of an option on a dividend-paying equity when the parameters in the Black–Scholes partial differential equation (PDE) are time dependent. With the aid of general transformations, the option value is expressed as a product of the Black–Scholes price for an option on a non-dividend-paying equity with constant parameters, the ratio of the strike price in the time-varying case to the strike price in the constant-parameter case, and a modified discount factor containing a parametrised time variable.     

An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.     

An optimization algorithm for the pile driver problem

In this paper we present the analysis of an algorithm of Uzawa type to compute solutions of the quasi variational inequality $$\begin{gathered} (QVI)\left( {\frac{{\partial ^2 u}}{{\partial t^2 }},\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + \left( {\frac{{\partial u}}{{\partial x}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \left( {\frac{{\partial ^2 u}}{{\partial x\partial t}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \hfill \\ + \left[ {u(1,t) + \frac{{\partial u}}{{\partial t}}(1,t)} \right]\left[ {\upsilon (1) - \frac{{\partial u}}{{\partial t}}(1,t)} \right] + J(u;\upsilon ) - J\left( {u;\frac{{\partial u}}{{\partial t}}} \right) \geqslant \hfill \\ \geqslant \left( {f,\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + F(t)\left[ {\upsilon (0) - \frac{{\partial u}}{{\partial t}}(0,t)} \right],t > 0,\forall \upsilon \in H^1 (0,1), \hfill \\ \end{gathered} $$ which is a model for the dynamics of a pile driven into the ground under the action of a pile hammer. In (QVI) (...) is the scalar product inL ²(0, 1) andJ(u;.) is a convex functional onH ¹(0, 1), for eachu, describing the soil-pile friction effect.     

An optimising approach to alternative clustering schemes

Clustering objects into groups is usually done using a statistical heuristic or an optimisation. The method depends on the size of the problem and its purpose. There may exist a number of partitions which do not differ significantly but some of which may be preferable (or equally good) when aspects of the problem not formally contained in the model are considered in the interpretation of the result. To decide between a number of good partitions they must first be enumerated and this may be done by using a number of different heuristics. In this paper an alternative method is described which uses an integer linear programming model having the number and size distribution of groups as objectives and the criteria for group membership as constraints. The model is applied to three problems each having a different measure of dissimilarity between objects and so different membership criteria. In each case a number of optimal solutions are found and expressed in two parts: a core of groups, the membership of which does not change, and the remaining objects which augment the core. The core is found to contain over three quarters of the objects and so provides a stable base for cluster definition.     

An alternative approach to the uniqueness of unconditional basis of for

We give an alternative and much simpler proof of the uniqueness of unconditional basis (up to equivalence and permutation) in the quasi-Banach spaces ℓ_p(c₀) for 0<p<1 and its complemented subspaces with unconditional basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its unconditional basis.     

An alternative to the Kelvin decomposition for plane anisotropic elasticity

In this paper, we propose an alternative tensorial decomposition to the Kelvin's one (introduced by Kelvin in 1856) for plane anisotropic elasticity using the polar formalism (introduced by Verchery in 1979). In the first part of the paper, a parallel between the two approaches is proposed. Thanks to it, some new results are found; namely, the projectors introduced have a direct interpretation in terms of material symmetry and are intrinsic for any type of symmetry considered, that is, they do not depend on any elastic modulus for any type of symmetry, unlike in the Kelvin decomposition. The introduction of what we call, in the paper, the polar projectors, stresses and strains gives a new insight into the polar formalism. The results proposed in this paper will hopefully be useful in some cases, for example, in the modeling of anisotropic damage evolution in solids. Copyright © 2013 John Wiley & Sons, Ltd.     

Vehicle routing problems with alternative paths: An application to on-demand transportation

The class of vehicle routing problems involves the optimization of freight or passenger transportation activities. These problems are generally treated via the representation of the road network as a weighted complete graph. Each arc of the graph represents the shortest route for a possible origin–destination connection. Several attributes can be defined for one arc (travel time, travel cost, etc.), but the shortest route modeled by this arc is computed according to a single criterion, generally travel time. Consequently, some alternative routes proposing a different compromise between the attributes of the arcs are discarded from the solution space. We propose to consider these alternative routes and to evaluate their impact on solution algorithms and solution values through a multigraph representation of the road network. We point out the difficulties brought by this representation for general vehicle routing problems, which drives us to introduce the so-called fixed sequence arc selection problem (FSASP). We propose a dynamic programming solution method for this problem. In the context of an on-demand transportation (ODT) problem, we then propose a simple insertion algorithm based on iterative FSASP solving and a branch-and-price exact method. Computational experiments on modified instances from the literature and on realistic data issued from an ODT system in the French Doubs Central area underline the cost savings brought by the proposed methods using the multigraph model.