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Abstract: A significant group of problems coming from the realm of combinatorial geometry can be approached fruitfully through the use of Algebraic Topology. From the first such application to Kneser''s problem in 1978 by Lovász [L. Lovász, Knester''s conjecture, chromatic number of distance graphs on the sphere, Acta. Sci. Math (Szeged) 45 (1983) 317–323] through the solution of the Lovász conjecture [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication; C. Schultz, A short proof of for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2], many methods from Algebraic Topology have been developed. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk–Ulam and Dold theorem to the integer/ideal-valued index theories. Recently equivariant obstruction theory has provided answers where the previous methods failed. For example, in papers [R.T. Živaljević, Equipartitions of measures in , arXiv: math.0412483, Trans. Amer. Math. Soc., submitted for publication] and [P. Blagojević, A. Dimitrijević Blagojević, Topology of partition of measures by fans and the second obstruction, arXiv: math.CO/0402400, 2004] obstruction theory was used to prove the existence of different mass partitions. In this paper we extract the essence of the equivariant obstruction theory in order to obtain an effective general position map scheme for analyzing the problem of existence of equivariant maps. The fact that this scheme is useful is demonstrated in this paper with three applications: [(A)] a “half-page” proof of the Lovász conjecture due to Babson and Kozlov [E. Babson, D. Kozlov, Proof of Lovasz conjecture, Annals of Mathematics (2) (2004), submitted for publication] (one of two key ingredients is Schultz'' map [C. Schultz, A short proof of for all n and a graph colouring theorem by Babson and Kozlov, 2005, arXiv: math.AT/0507346v2]), [(B)] a generalization of the result of V. Makeev [V.V. Makeev, Equipartitions of continuous mass distributions on the sphere an in the space, Zap. Nauchn. Sem. S.-Petersburg (POMI) 252 (1998) 187–196 (in Russian)] about the sphere measure partition by 3-planes (Section 3), and [(C)] the new , class of 3-fan 2-measures partitions (Section 3). These three results, sorted by complexity, share the spirit of analyzing equivariant maps from spheres to complements of arrangements of subspaces. [Copyright &y& Elsevier]

Abstract: The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination , with , , being nonzero complex scalars and , , being nonzero complex idempotent matrices such that two of them, and say, are disjoint, i.e., satisfy condition , is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of and to the commutativity condition . [Copyright &y& Elsevier]

Abstract: Since Cook introduced the notion of local influence in 1986 a great number of papers have appeared in statistics handling this topic in a wide variety of areas. However, until we know, there is no paper focussed on local influence in survey sampling. This paper assesses the local influence through the curvature of the perturbation-formed surface of the ratio estimator in a simple random sampling without replacement and the Horvitz–Thompson estimator in an inclusion proportional to size sampling design. Both estimators use auxiliary information (x) and can be obtained through a model-assisted reasoning where the ratio model with variance proportional to x plays an important role. Two perturbation schemes are considered, a model perturbation and an auxiliary variable perturbation. We illustrate the effectiveness of the method in order to identify influential points with two real examples. [Copyright &y& Elsevier]

Abstract: In this paper, our main purpose is to establish the existence of positive bounded entire solutions of second-order quasilinear elliptic systems. The main results of the present paper are new and extend the previously known results. [Copyright &y& Elsevier]

Abstract: In this paper, our main purpose is to establish the existence of singular positive radial solutions of second order quasilinear elliptic equations. The main results of the present paper are new and extend the previously known results. [Copyright &y& Elsevier]

Abstract: Proposed as a general framework, Liu and Yu [Generalization of matching extensions in graphs, Discrete Math. 231 (2001) 311–320.] introduced -graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let and d be non-negative integers such that and is even. If when deleting any n vertices from G, the remaining subgraph H of G contains a k-matching and each such k-matching can be extended to a defect-d matching in H, then G is called an -graph. Liu and Yu''s Papee''s paper, the recursive relations for distinct parameters and d were presented and the impact of adding or deleting an edge also was discussed for the case . In this paper, we continue the study begun by Liu and Yu and obtain new recursive results for -graphs in the general case . [Copyright &y& Elsevier]

This paper proposed an edge detection scheme which is deduced from Fresnel diffraction. Analysis in this paper shows that Fresnel convolution kernel function performs well on edge enhancement when images are transformed into complex functions. Due to its mathematical complexity, the method is simplified into a linear convolution filter. The new edge detector is designed based on the simplified linear filter. Experimental results indicate that the new detector gives quantitative results equal to the Canny detector while it is more simple to be implemented. [Copyright &y& Elsevier]

In this paper we present a novel class of human model described by convolution surface attached to articulated kinematics skeletons. The human pose can be estimated from silhouette in monocular images. The contribution of this paper consists of three points: First, human model of convolution surface is presented and its shape is deformable when changing polynomial parameters and radius parameters. Second, convolution surface and curve correspondence theorem is presented to give a map between 3D pose and 2D contour. Third, we model the human silhouette with convolution curve in order to estimate joint parameters from monocular images and we also give an effective constraint function. Evaluation of this approach is performed on some video frames about a walking man. The experiment result shows that our method works well without self-occlusion. [Copyright &y& Elsevier]

Abstract: Jean-Yves Girard and John Reynolds independently discovered the second-order polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in second-order intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by well-behaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version. http://homepages.inf.ed.ac.uk/wadler [Copyright &y& Elsevier]

In this paper, we revisit fuzzy logic with evaluated syntax . An objection towards it is necessity to introduce logical constants (names of truth values) for all into its language which makes it uncountable. Hájek has shown in his book “Metamathematics of fuzzy logic” that when seeing as RPL logic where the provability degree is an additional concept then we may confine only to rational logical constants. In this paper, we show that the same result can be obtained directly in . The essential idea is to generalize slightly the concept of a complete fuzzy theory using the technique of . [Copyright &y& Elsevier]

Abstract: In recent papers, we have described a rigorous mathematical procedure for delineating the first-order elastic–plastic closures for polycrystalline microstructures where the crystallographic texture (also called the Orientation Distribution Function or the ODF) was assumed to be the main variable influencing the effective properties. These closures were based on elementary bounding theories, and delineate the complete set of theoretically feasible combinations of effective properties in the selected material system. In this paper, we demonstrate certain key properties of these elastic–plastic closures for the class of cubic-orthorhombic textures that, in turn, facilitate a fast computation of these closures with drastically reduced computational effort. Using the novel methods described herein, we have computed and presented numerous examples of property atlases covering a broad range of cubic materials. [Copyright &y& Elsevier]

Abstract: In this paper, on basis of other related works (see [R. Scott, Optimal L ∞ estimates for the finite element method on irregular meshes, Math. Comp. V 30 (1976) 681–697; Qun Lin, Qi-ding Zhu, The Preprocessing and Postprocessing for FEM, Shanghai Science Press, China, 1994; I. Babuska, A. Miller, The post-processing in the finite element method, Parts I–II, Internat. J. Numer. Methods Engrg. 20 (1984) 1085–1109, 1111–1129]), we will continue to study the theory of FEM for elliptic problems with singular righthand side. The main difference between this paper and other related works (see Scott, 1976; Lin and Zhu, 1994; Babuska and Miller, 1984) is that it will present some local estimates for FEM of elliptic problems with singular righthand side. Of course, some numerical examples will be given to investigate those error estimates. [Copyright &y& Elsevier]

Abstract: This paper presents the new convergence analysis for the Newton-like method proposed by [Xinyuan Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput. 112 (2000) 75–78.] Compared with the original version of the convergence analysis, the restriction imposed on f″(x) is removed thoroughly. In order to guarantee the quadratic convergence of the Newton-like method, it is suffices to suppose that f′(x ∗)≠0 and f′(x) is local Lipschitz near x ∗, where x ∗ is a solution of nonlinear equation f(x)=0. Moreover, some comments are given with examples for the Newton-like method, in comparison with Newton’s method. It can be concluded that the new algorithm is more feasible, effective than Newton’s method. In particularly, when it happens that, the derivative of the function f(x) at an iterate is singular or almost singular, the Newton-like method is vast superior to the classical Newton method. The numerical results of the paper strongly support the conclusion. [Copyright &y& Elsevier]

Radial basis functions are a popular basis for interpolating scattered data during the image reconstruction process in graphic analysis. In this context, the solution of a linear system of equations represents the most time-consuming operation. In this paper an efficient preconditioning technique is proposed to solve these linear systems of equations. This algorithm consists of an iterative method which enforces at each iteration a projection of the residual onto a suitable subspace called coarse space. This constraint is ensured by solving an auxiliary problem at each iteration without regularisation. As increasing the number of the coarse space basis functions increases the computational cost of the algorithm, correct selection of coarse space basis is addressed in the paper. Numerical results illustrate the convergence properties of the proposed method with wavelet-like basis functions and regular distributed radial basis functions for image reconstruction. [Copyright &y& Elsevier]

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions. [Copyright &y& Elsevier]

The lack of a theory-based design methodology for mobile robot control programs means that control programs have to be developed through an empirical trial-and-error process. This can be costly, time consuming and error prone. In this paper we show how to develop a theory of robot–environment interaction, which would overcome the above problem. We show how we can model a mobile robot’s task (so-called “task identification”) using non-linear polynomial models (NARMAX), which can subsequently be formally analysed using established mathematical methods. This provides an understanding of the underlying phenomena governing the robot’s behaviour. Apart from the paper’s main objective of formally analysing robot–environment interaction, the task identification process has further benefits, such as the fast and convenient cross-platform transfer of robot control programs (“Robot Java”), parsimonious task representations (memory issues) and very fast control code execution times. [Copyright &y& Elsevier]

Abstract: In this paper, weak and strong convergence theorems of three-step iterations are established for asymptotically nonexpansive mappings in Banach spaces. The results obtained in this paper extend and improve the recent ones announced by Xu and Noor [B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444–453], Suthep Suantai [S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005) 506–517], Weerayuth Nilsrakoo and Satit Saejung [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.01.063] and others. [Copyright &y& Elsevier]

Abstract: A stochastic response surface method (SRSM) which has been previously proposed for problems dealing only with random variables is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modeled as random fields. The formalism of the extended SRSM is similar to the spectral stochastic finite element method (SSFEM) in the sense that both of them utilize Karhunen–Loeve (K–L) expansion to represent the input, and polynomial chaos expansion to represent the output. However, the coefficients in the polynomial chaos expansion are calculated using a probabilistic collocation approach in SRSM. This strategy helps us to decouple the finite element and stochastic computations, and the finite element code can be treated as a black box, as in the case of a commercial code. The collocation-based SRSM approach is compared in this paper with an existing analytical SSFEM approach, which uses a Galerkin-based weighted residual formulation, and with a black-box SSFEM approach, which uses Latin Hypercube sampling for the design of experiments. Numerical examples are used to illustrate the features of the extended SRSM and to compare its efficiency and accuracy with the existing analytical and black-box versions of SSFEM. [Copyright &y& Elsevier]

Matrix fields are becoming increasingly important in digital imaging. In order to perform shape analysis, enhancement or segmentation of such matrix fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the setting of matrices, in the literature sometimes referred to as tensors despite the fact that matrices are only rank two tensors. The goal of this paper is to introduce and explore two approaches to mathematical morphology for matrix-valued data: one is based on a partial ordering, the other utilises nonlinear partial differential equations (PDEs). We start by presenting definitions for the maximum and minimum of a set of symmetric matrices since these notions are the cornerstones of the morphological operations. Our first approach is based on the Loewner ordering for symmetric matrices, and is in contrast to the unsatisfactory component-wise techniques. The notions of maximum and minimum deduced from the Loewner ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. These properties are also shared by the dilation and erosion processes governed by a novel nonlinear system of PDEs we are proposing for our second approach to morphology on matrix data. These PDEs are a suitable counterpart of the nonlinear equations known from scalar continuous-scale morphology. Both approaches incorporate information simultaneously from all matrix channels rather than treating them independently. In experiments on artificial and real medical positive semidefinite matrix-valued images we contrast the resulting notions of erosion, dilation, opening, closing, top hats, morphological derivatives, and shock filters stemming from these two alternatives. Using a ball shaped structuring element we illustrate the properties and performance of our ordering- or PDE-driven morphological operators for matrix-valued data. [Copyright &y& Elsevier]

Abstract: In this paper, we consider completely generalized nonlinear quasi-variational-like inclusions in Banach spaces and propose an Ishikawa type iterative algorithm for computing their approximate solutions by applying the new notion of -proximal mapping given in [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308]. We prove that the approximate solutions obtained by the proposed algorithm converge to the exact solution of our quasi-variational-like inclusions. The results presented in this paper extend and improve the corresponding results of [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308; X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; N.J. Huang, Generalized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996) 25–29; A. Hassouni, A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185(3) (1994) 706–712]. Some special cases are also discussed. [Copyright &y& Elsevier]

Abstract: The paper is dealing with the problem of finding the densest packings of equal circles in the unit square. Recently, a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem. With this method it became possible to solve the previously open problems of packing 28, 29, and 30 circles in the numerical sense: tight guaranteed enclosures were given for all the optimal solutions and for the optimum value. The present paper completes the optimality proofs for these cases by determining all the optimal solutions in the geometric sense. Namely, it is proved that the currently best-known packing structures result in optimal packings, and moreover, apart from symmetric configurations and the movement of well-identified free circles, these are the only optimal packings. The required statements are verified with mathematical rigor using interval arithmetic tools. [Copyright &y& Elsevier]

Abstract: Digital planarity is defined by digitizing Euclidean planes in the three-dimensional digital space of voxels; voxels are given either in the grid-point or the grid-cube model. The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as supporting or separating Euclidean planes, characterizations in arithmetic geometry, periodicity, connectivity, and algorithmic solutions. The paper provides a uniform presentation, which further extends and details a recent book chapter in [R. Klette, A. Rosenfeld, Digital Geometry—Geometric Methods for Digital Picture Analysis, Morgan Kaufmann, San Francisco, 2004]. [Copyright &y& Elsevier]

Abstract: The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positive-definite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision. [Copyright &y& Elsevier]

Abstract: The three-phase composite model (TPM) is analyzed from the viewpoint of the asymptotic homogenization method. It is shown that such an asymptotic process can be constructed that the three-phase model will represent a zero-order approximation. Therefore the present paper presents a mathematical asymptotic justification of the TPM technique. Further more, it is shown that the TPM approach provides a sufficiently accurate solution of the unit cell local problem arising in the asymptotic homogenization for periodic composite materials with cubic inclusions with arbitrary properties. The same conclusion is proved for the unidirectional fiber-reinforced composites in the cases of parallelepiped inclusions and rectangular periodic structure. The multi-point Padé approximants are used in the present paper in order to evaluate the accuracy of three-phase models in the cases of small and large inclusions, and to derive the estimations for the effective properties of composite materials. [Copyright &y& Elsevier]

Abstract: A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59–73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such that is a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such that is a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that is a tree. [Copyright &y& Elsevier]

Abstract: We investigate the free complex of a convex geometry. Edelman and Reiner showed that the free complex of a convex geometry is contractible. Moreover, their paper stated a conjecture about the topology of free complexes. This paper proves their conjecture. [Copyright &y& Elsevier]

Abstract: In recent years, much interest was devoted to the Urysohn space and its isometry group; this paper is a contribution to this field of research. We mostly concern ourselves with the properties of isometries of , showing for instance that any Polish metric space is isometric to the set of fixed points of some isometry φ. We conclude the paper by studying a question of Urysohn, proving that compact homogeneity is the strongest homogeneity property possible in . [Copyright &y& Elsevier]

Abstract: In many statistical problems, the variances of the populations cannot be assumed to be equal. These inhomogeneity problems are often more difficult to handle than the corresponding homogeneity problems. In sequential estimation, this often means that only first order sequential procedures are available in the statistics literature for inhomogeneity problems. The purpose of this paper is to illustrate by using the classical Behrens–Fisher problem how to construct a second order sequential procedure using the batch sampling idea of Hall [Ann. Statist. 9 (1981) 1229–1238]; the cost of assuming variance inhomogeneity even when the two variances are equal turns out to be very limited. The approach of this paper can readily be applied to many other inhomogeneous problems. [Copyright &y& Elsevier]

Abstract: Evaluating the efficiency of organizational units continues to be a difficult problem to solve, especially when the multiplicity of inputs (resources, costs) and outputs (services, products) associated with these units is considered. This paper considers a previous article published by Wang and Luo [Y. Wang, Y. Luo, DEA efficiency assessment using ideal and anti-ideal decision-making units, Applied Mathematics and Computation 173 (2) (2006) 902–915] in the journal of Applied Mathematics and Computation where data envelopment analysis (DEA) and the technique for order preference by similarity to ideal solution (TOPSIS) are integrated in ranking of decision-making units (DMUs). Wang and Luo (2006) contribute to a very interesting topic by showing that the TOPSIS idea can be combined to DEA for a comprehensive ranking of DMUs. However, this paper finds that their approach is problematic in employing the negative ideal point (NIP) for DEA computation. Their ideal point based models rely on using conflicting efficiency concepts. We slightly revised the models so that a DEA analysis using TOPSIS idea can be performed. Numerical demonstration reveals that our approach produces a ranking result that is consistent with that in previous studies. [Copyright &y& Elsevier]

Abstract: This paper shows that Hedstrom''s nonreflecting boundary condition works for some nonsimple-wave solutions in the case where the solution is . Hedstrom''s boundary condition is a pioneering work in the discipline of characteristic nonreflecting boundary conditions and Hedstrom gave a theoretical assurance of the nonreflectivity for simple-wave solutions. In this paper, we extend Hedstrom''s theorem for solutions, that is, we prove that the Hedstrom boundary condition eliminates the reflection if the solution is and there do not exist ingoing waves generated by the nonlinear interactions near the boundary. [Copyright &y& Elsevier]

In this paper, we present a Newton-type method for a class of mathematical programs with complementarity constraints. Under the MPEC-LICQ, we use the definition of B-stationary point to construct a constrained equations model, and apply the Newton method to solve the problem. At the end of this paper, numerical results are reported to show our method's validity. [Copyright &y& Elsevier]

Abstract: This paper revisits a well-known synthesis problem in iterative learning control, where the objective is to optimize a performance criterion over a class of causal iterations. The approach taken here adopts an infinite-time setting and looks at limit behavior. The first part of the paper considers iterations without current-cycle-feedback (CCF) term. A notion of admissibility is introduced to distinguish between pairs of operators that define a robustly converging iteration and pairs that do not. The set of admissible pairs is partitioned into disjoint equivalence classes. Different members of an equivalence class are shown to correspond to different realizations of a (stabilizing) feedback controller. Conversely, every stabilizing controller is shown to allow for a (non-unique) factorization in terms of admissible pairs. Class representatives are introduced to remove redundancy. The smaller set of representative pairs is shown to have a trivial parameterization that coincides with the Youla parameterization of all stabilizing controllers (stable plant case). The second part of the paper considers the general family of CCF-iterations. Results derived in the non-CCF case carry over, with the exception that the set of equivalent controllers now forms but a subset of all stabilizing controllers. Necessary and sufficient conditions for full generalization are given. [Copyright &y& Elsevier]

Abstract: Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the imprecise profit of the inventory model when the demand quantity and unit cost are imprecise. Since imprecise parameters contained in the inventory model lie in intervals, the profit value calculated from the model should be in a range as well. We formulate a pair of two-level mathematical programming model to derive the upper bound and lower bound of the profit. Based on the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs. Solving the corresponding pair of geometric programs produces the interval of profit. An example illustrates the idea in this paper. That example also shows that the interval profit contains more information for determining inventory policy. [Copyright &y& Elsevier]

Abstract: A Cramer rule for solving a kind of restricted matrix equationsis presented in [G. Wang, Jie Sun, A Cramer rule for solution of the general restricted matrix equation, Appl. Math. Comput. 154 (2004) 415–422]. This paper gives a more condensed Cramer rule for the solution of the matrix equations. The results in [G. Wang, Z.L. Xu, Solving a kind of restricted matrix equations and Cramer rule, Appl. Math. Comput. 162 (2005) 329–338, Chao Gu, G.R. Wang, Z.L. Xu, PCR Algorithm for the Parallel Computation of the Solution of a class of Singular linear Systems, Appl. Math. Comput. 176 (2006) 237–244] are partially the special cases in our paper. [Copyright &y& Elsevier]

Abstract: This paper deals with multialgebras. An important tool in the theory of multialgebras is the fundamental relation, which brings us into the class of the universal algebras. In this paper we will present the construction of the direct limit of a direct system of multialgebras, some basic properties of this construction, and we will see that the fundamental algebra of the direct limit of multialgebras is the direct limit of the fundamental algebras of the given multialgebras. [Copyright &y& Elsevier]

Abstract: This paper reports the effort of deducing the initial strategy distributions (ISDs) of agents in mix-game models that is used to predict a real financial time series generated from a target financial market. Using mix-games to predict Shanghai Index, we find that the time series of prediction accurate rates is sensitive to the ISDs of agents in group 2 who play a minority game, but less sensitive to the ISDs of agents in group 1 who play a majority game. And agents in group 2 tend to cluster in full strategy space (FSS) if the real financial time series has obvious tendency (upward or downward), otherwise they tend to scatter in FSS. We also find that the ISDs and the number of agents in group 1 influence the level of prediction accurate rates. Finally, this paper gives suggestion about further research. [Copyright &y& Elsevier]

Abstract: A graph H of order n is said to be k-placeable into a graph G of order n, if G contains k edge-disjoint copies of H. It is well known that any non-star tree T of order n is 2-placeable into the complete graph . In the paper by Kheddouci et al. [Packing two copies of a tree into its fourth power, Discrete Math. 213 (2000) 169–178], it is proved that any non-star tree T is 2-placeable into . In this paper, we prove that any non-star tree T is 2-placeable into . [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to build sequences of suitably smooth approximate solutions to the Saint-Venant model that preserve the mathematical structure discovered in [D. Bresch, B. Desjardins, Comm. Math. Phys. 238 (1–2) (2003) 211–223]. The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. Extension of this mollifying procedure to the case of compressible Navier–Stokes equations is also provided. Using the recent paper written by the authors, this provides global existence results of weak solutions for the barotropic Navier–Stokes equations and for compressible Navier–Stokes equations with heat conduction using a particular cold pressure term close to vacuum. [Copyright &y& Elsevier]

Abstract: A rank-niche evolution strategy (RNES) algorithm has been developed in this paper to solve unconstrained multiobjective optimization problems. A required number of Pareto-optimal solutions can be generated by the algorithm in a single run. In addition to the operations of recombination, mutation and selection used in original evolution strategy (ES), an external elite set which contains a given number of non-dominated elites is updated and trimmed by a clustering technique to maintain a uniformly distributed Pareto front. The fitness function for each individual contains the information of rank and crowding status. The selection operation using this fitness function considers the superiority and distribution simultaneously. Eight test problems illustrated in other papers are used to test RNES. For some test problems the Pareto-optimal solutions obtained by RNES are better than those obtained by GA-based algorithms. [Copyright &y& Elsevier]

Abstract: Safety factor is a universally utilized concept in several branches of engineering. On one hand, most engineers, as it were, neglect uncertainty, but on the other hand, the allowable stress level was introduced long time ago as a ratio of the yield stress to the so-called safety factor to provide the region for the safe utilization of the structure. Thus the uncertainty is introduced into practice by the “back door”. This observation led to a considerable literature dedicated to the probabilistic interpretation of the safety factor. The present paper deals with the novel aspect of elucidation of the concept of safety factor through the theory of fuzzy sets, apparently for the first time in the literature. The aim of the paper is to present the safety factor that is uniformly employed by engineers, but in the new light. The safety factor in the fuzzy setting is introduced. The ideas are illustrated on two strength of material problems; simple examples are chosen so as to allow for clearer illustration of ideas. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of real-valued random-walk processes, observed over random epochs of time, that forms a delayed renewal process. The present model does not restrict this class to a merely monotone random walk, which is easier to analyze and find explicit form functional. The objective is to find the first passage of the process exiting a rectangular set and registering the value of the process at this time, thus generalizing past models where either the observed process was monotone or the first passage time reduced to the moment of the first drop. The joint transformation of the named random characteristics of the process are derived in a closed form. The paper concludes with examples, including numerical examples, demonstrating the use of the results as well as practical applications to finance. [Copyright &y& Elsevier]

Abstract: By axially connecting a constant-section circular tube to two tapered circular tubes, a so-called circular tube with tapered ends structural member is obtained. To get the element stiffness matrix of a tube with tapered ends, a transfer matrix method is introduced in this paper. Transfer matrices for the constant-section tube and tapered tubes are derived firstly, and then the element stiffness matrix for a tube with tapered ends is obtained on the basis of the relationship of the transfer matrices. Two typical examples are given to verify the element stiffness matrix. And the stiffness matrix for a tube with tapered ends can be treated as a modification of the stiffness matrix for a constant-section tube. Stiffness modified coefficients are introduced to represent the modification, and a table of stiffness modified coefficients for tubes of frequently used dimensions is given at the end of the paper. [Copyright &y& Elsevier]

Abstract: A primary shortcoming of existing techniques for hree-dimensional (3D) model matching is the reliance on global information of the model’s structure. Models are matched in their entirety, depending on overall topology and geometry information. A currently open challenge is how to perform partial matching. Partial matching is important for finding similarities across part models with different global shape properties and for the segmentation and matching of data acquired from 3D scanners. This paper presents a Scale–Space feature extraction technique based on recursive decomposition of polyhedral surfaces into surface patches. The experimental results presented in this paper suggest that this technique can potentially be used to perform matching based on local model structure. In our previous work, Scale–Space decomposition has been used successfully to extract features from mechanical artifacts. Scale–Space techniques can be parameterized to generate decompositions that correspond to manufacturing, assembly or surface features relevant to mechanical design. One application of these technique is to support matching and content-based retrieval of solid models. This paper shows how a Scale–Space technique can extract features that are invariant with respect to the global structure of the model as well as small perturbations that 3D laser scanning process introduce. In order to accomplish this, we introduce a new distance function defined on triangles instead of points. We believe this technique offers a new way to control the feature decomposition process, which results in the extraction of features that are more meaningful from an engineering viewpoint. The new technique is computationally practical for use in indexing large models. Examples are provided that demonstrate effective feature extraction on 3D laser scanned models. In addition, a simple sub-graph isomorphism algorithm was used to show that the feature adjacency graphs, obtained through feature extraction, are meaningful descriptors of 3D CAD objects. All of the data used in the experiments for this work is freely available at: http://www.designrepository.org/datasets/. [Copyright &y& Elsevier]

Abstract: In this paper we continue the study of bi-conformal vector fields started in [A. García-Parrado, J.M.M. Senovilla, Class. Quant. Grav. 21 (2004) 2153–2177]. These are vector fields defined on a pseudo-Riemannian manifold by the differential conditions , where are orthogonal and complementary projectors with respect to the metric tensor . In a previous paper we explained how the analysis of these differential conditions enabled us to derive local geometric characterizations of the most relevant cases of conformally separable (also called double twisted) pseudo-Riemannian manifolds. In this paper we carry on this analysis further and provide local invariant characterizations of conformally separable pseudo-Riemannian manifolds with conformally flat leaf metrics. These characterizations are rather similar to that existing for conformally flat pseudo-Riemannian manifolds but instead of the Weyl tensor, we must demand the vanishing of certain four rank tensors constructed from the curvature of an affine, non-metric, connection (bi-conformal connection). We also speculate with possible applications to finding results for the existence of foliations by conformally flat hypersurfaces in any pseudo-Riemannian manifold. [Copyright &y& Elsevier]

Abstract: The structure of a Frobenius manifold encodes the geometry associated with a flat pencil of metrics. However, as shown in the authors’ earlier work [L. David, I.A.B. Strachan, Compatible metrics on manifolds and non-local bi-Hamiltoninan structures, Int. Math. Res. Notices 66 (2004) 3533–3557], much of the structure comes from the compatibility property of the pencil rather than from the flatness of the pencil itself. In this paper conformally flat pencils of metrics are studied and examples, based on a modification of the Saito construction, are developed. [Copyright &y& Elsevier]

Abstract: Constructive Zermelo–Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice. The paper builds on a self-interpretation of CZF (developed in [M. Rathjen, The formulae-as-classes interpretation of constructive set theory, in: Proof Technology and Computation (Proceedings of the International Summer School Marktoberdorf 2003) IOS Press, Amsterdam, 2004 (in press)]) that provides an “inner” model of CZF which also validates the so-called -axiom of choice, . The crucial technical step taken in the present paper is to investigate the absoluteness properties of this model under the hypothesis . It is also shown that CZF plus the -axiom of choice possesses the disjunction property, the numerical existence property and the existence property for an important group of formulae. [Copyright &y& Elsevier]

Abstract: The paper deals with the analysis of some of G.I. Marchuk''s and R.V. Petrov''s models of immune response of organism [Mathematical Models in Immunology, Nauka Publ, Moscow, 1985 (in Russian)]. The models are described by using the systems of functional-differential equations. In addition to the problems of stability the statement of problems of control for a simplest immunological model with temperature reaction of organism is also presented in the paper. Besides, the condition of asymptotic stability as a whole for the Marchuk–Petrov''s model of immunophysiological reaction of defense and the properties of controllability and stabilization are analyzed as well. [Copyright &y& Elsevier]

Abstract: This paper describes the NERC Soil Biodiversity Programme''s integrated approach to data, information management and analysis, from field sampling through to modelling the soil ecosystem. It discusses the handling of meta-data from sampling in the field, construction of an integrated database to provide the framework whereby project data and meta-data can be linked and analysed together, and the development of the web-based data discovery and delivery system that fronts the database. The paper then describes how such data can be synthesised into mathematical models, concentrating initially on the Stable Isotope Modelling (SIM) framework before moving on to discuss models incorporating temporal and spatial dynamics. The paper concludes with a discussion of the future challenges facing soil modelling, and the potential of the technologies developed by the programme in helping to meet these challenges. [Copyright &y& Elsevier]

Abstract: After a certain number of hours of running, no two mechanical components are completely the same due to normal wear or foreign object damage. A nominal CAD model from a component designer is different from its corresponding worn one and therefore cannot be directly used for tool path generation for build up and machining repair processes. This is the main reason that most repair process used for complex geometry parts, such as gas turbine blades, is currently carried out manually and is called the “Black Art”. This paper proposes a defects-free model-based repair strategy to generate correct tool paths for build up process and machining process adaptive to each worn component through the reverse engineering application. Based on 3D scanning data, a polygonal modelling approach is introduced in this paper to rapidly restore worn parts for direct use of welding, machining and inspection processes. With this nominal model, this paper presents the procedure to accurately define and extract repair error, repair volume and repair patch geometry for the tool path generation, which is adaptive to each individual part. The tool paths are transferred to a CNC machine for the repairing trials. Further research work is performed on repair geometry extraction algorithm and repair module development within the reverse engineering environment. [Copyright &y& Elsevier]

Abstract: Given complex-valued matrices A, B and C of appropriate dimensions, this paper investigates certain invariance properties of the product AXC with respect to the choice of X, where X is a generalized inverse of B. Different types of generalized inverses are taken into account. The purpose of the paper is three-fold: First, to review known results scattered in the literature, second, to demonstrate the connection between invariance properties and the concept of extremal ranks of matrices, and third, to add new results related to the topic. [Copyright &y& Elsevier]