The influence of domain interpretations on computational models

Computational models are usually defined over specific domains. For example, Turing machines are defined over strings, and the recursive functions over the natural numbers. Nevertheless, one often uses one computational model to compute functions over another domain, in which case, one is obliged to employ a representation, mapping elements of one domain into the other. For instance, Turing machines (or modern computers) are understood as computing numerical functions, by interpreting strings as numbers, via a binary or decimal representation, say.We ask: Is the choice of the domain interpretation important? Clearly, complexity is influenced, but does the representation also affect computability? Can it be that the same model computes strictly more functions via one representation than another? We show that the answer is “yes”, and further analyze the influence of domain interpretation on the extensionality of computational models (that is, on the set of functions computed by the model).We introduce the notion of interpretation-completeness for computational models that are basically unaffected by the choice of domain interpretation, and prove that Turing machines and the recursive functions are interpretation-complete, while two-counter machines are incomplete. We continue by examining issues based on model extensionality that are influenced by the domain interpretation. We suggest a notion for comparing computational power of models operating over arbitrary domains, as well as an interpretation of the Church-Turing Thesis over arbitrary domains.     

Hierarchical sets in mathematical programming modeling languages

The mathematical notation commonly applied for the formulation of mathematical programming models is extended to include hierarchical structures. The proposed notation is related to hierarchical set concepts in the languages UIMP, AMPL, and LPL. With the proposed notation it is possible to aggregate and disaggregate over hierarchical structures. In addition, views are introduced to permit the use of hierarchical substructures and to create new hierarchies out of existing ones. The proposed notation for hierarchical sets and views is illustrated by applying it to the representation and estimation of social accounting matrices (SAMs).     

Rotation Techniques in Asymmetric Multidimensional Scaling

Multidimensional scaling (MDS) is a set of techniques, used especially in behavioral and social sciences, that enable a researcher to visualize proximity data in a multidimensional space. This article focuses on a particular class of MDS models proposed to deal with proximities which describe asymmetric relationships (i.e., trade indices for a set of countries, brand switching data, occupational mobility tables, and so on). They are based on the decomposition of the relationships into a symmetric and a skew-symmetric part. In this way the objects are represented as points in a multidimensional space and the intensity of their relationships as scalar products (symmetry) or triangle areas (skew-symmetry). These models are seen as special cases of a general model and their rotational indeterminacy is investigated. The aim is to propose a rotation method that makes easier the visual inspection of the graphical representation, highlighting the simple structure of the data. In particular an orthomax-like family of rotation methods and a general algorithm are proposed. Advantages of the proposal are illustrated by analysis of import-export data.     

Bilinear markovian representation and bilinear models

An extension of the linear Markovian repsentation called the bilinear Markovian representation is introduced, and is shown to provide representations of all-diagonal bilinear time series models. Some properties of the bilinear Markovian representation are also given.     

Parity representations of posets

Parity representations, introduced in this paper, comprise a new method of representation of posets that yields insight into the combinatorics of the poset of all intervals of a poset. Results here generalize some results previously obtained for the face lattices of binary partition polytopes.     

IMAGE RESTORATION UNDER CAUCHY NOISE WITH SPARSE REPRESENTATION PRIOR AND TOTAL GENERALIZED VARIATION

This article introduces a novel variational model for restoring images degraded by Cauchy noise and/or blurring.The model integrates a nonconvex data-fidelity term with two regularization terms,a sparse representation prior over dictionary learning and total generalized variation(TGV)regularization.The sparse representation prior exploiting patch information enables the preservation of fine features and textural patterns,while adequately denoising in homogeneous regions and contributing natural visual quality.TGV regularization further assists in effectively denoising in smooth regions while retaining edges.By adopting the penalty method and an alternating minimization approach,we present an efficient iterative algorithm to solve the proposed model.Numerical results establish the superiority of the proposed model over other existing models in regard to visual quality and certain image quality assessments.     

因素空间理论与知识表示的数学框架（X）——基于因素空间的神经元模型

本文是文［１—９］的继续，该文及后续论文仍将系统地研究因素空间理论及其在知识表示中的应用。该文研究基于因素空间的神经元模型：首先讨论了因素空间的神经元机理，然后讨论了几种典型的神经元模型；特别提出了基于Ｗｅｂｅｒ—Ｆｅｃｈｎｅｒ法则的神经元模型以及基于变权的神经元模型。     

Schatten–von Neumann norm inequalities for two-wavelet localization operators associated with β-Stockwell transforms

In this article we define two-wavelet localization operators corresponding to an irreducible and square-integrable representation of a locally compact Hausdorff group on a Hilbert space. The group structure admitting an irreducible and square-integrable representation which is related to β-Stockwell transform, that we shall use in this article β?∈?R have been introduced in Boggiatto et al. [P. Boggiatto, C. Fernandez, and A. Galbis, A group representation related to the Stockwell transform, Indiana Univ. Math. J. 58(5) (2009), pp. 2277–2296]. The Schatten–von Neumann norm inequalities of these two-wavelet localization operators are established. The traces and the trace class norm inequalities of the trace class two-wavelet localization operators are given.     

Variable resolution triangulations

A comprehensive study of multiresolution decompositions of planar domains into triangles is given. A general model is introduced, called a Multi-Triangulation (MT), which is based on a collection of fragments of triangulations arranged into a directed acyclic graph. Different decompositions of a domain can be obtained by combining different fragments of the model. Theoretical results on the expressive power of the MT are given. An efficient algorithm is proposed that can extract a triangulation from the MT, whose level of detail is variable over the domain according to a given threshold function. The algorithm works in linear time, and the extracted representation has minimum size among all possible triangulations that can be built from triangles in the MT, and that satisfy the given level of detail. Major applications of these results are in real-time rendering of complex surfaces, such as topographic surfaces in flight simulation.     

On Residue Symbols and the Mullineux Conjecture

This paper is concerned with properties of the Mullineux map, which plays a rôle in p-modular representation theory of symmetric groups. We introduce the residue symbol for a p-regular partitions, a variation of the Mullineux symbol, which makes the detection and removal of good nodes (as introduced by Kleshchev) in the partition easy to describe. Applications of this idea include a short proof of the combinatorial conjecture to which the Mullineux conjecture had been reduced by Kleshchev.     

On Local Coefficients for Non-generic Representations of Some Classical Groups

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal. The local coefficients are related to the Plancherel measures, and their zeroes are shown to be among the poles of the standard intertwining operators.     

The OWA-based consensus operator under linguistic representation models using position indexes

When using linguistic approaches to solve decision problems, we need linguistic representation models. The symbolic model, the 2-tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes. Together with these three linguistic models, the corresponding ordered weighted averaging operators, such as the linguistic ordered weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extended ordered weighted averaging operator, have been developed, respectively. In this paper, we analyze the internal relationship among these operators, and propose a consensus operator under the continuous linguistic model (or the 2-tuple fuzzy linguistic representation model). The proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures. Some desired properties of the consensus operator are also presented. In particular, the consensus operator provides an alternative consensus model for group decision making. This consensus model preserves the original preference information given by the decision makers as much as possible, and supports consensus process automatically, without moderator.     

Fuzzy coefficient volatility (FCV) models with applications

Recently, Carlsson and Fuller [C. Carlsson, R. Fuller, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001) 315–326] have introduced possibilistic mean, variance and covariance of fuzzy numbers and Fuller and Majlender [R. Fuller, P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems 136 (2003) 363–374] have introduced the notion of crisp weighted possibilistic moments of fuzzy numbers. In this paper, we propose a class of FCV (Fuzzy Coefficient Volatility) models and study the moment properties. The method used here is very similar to the method used in Appadoo et al. [S.S. Appadoo, M. Ghahramani, A. Thavaneswaran, Moment properties of some time series models, Math. Sci. 30 (1) (2005) 50–63]. The proposed models incorporate fuzziness, subjectivity, arbitrariness and uncertainty observed in most financial time series. The usual forecasting method does not incorporate parameter variability. Fuzzy numbers are used to model the parameters to incorporate parameter variability.     

模糊数的一种新运算方法及其若干应用

首先给出了模糊数的一种新的函数表示定理;基于该表示定理 ,提出了模糊数的一种新运算方法并将其直接应用到模糊数理论中若干问题的讨论,包括:模糊数的差问题,绝对值问题以及模糊数的确界问题 .     

The middle-parametric representation of fuzzy numbers and applications to fuzzy interpolation

In this paper we introduce the middle-parametric representation of a fuzzy number presenting some of the advantages in the use of this representation. A special attention is focused on the subset of symmetric fuzzy numbers presenting the special properties of their arithmetic. The approach on symmetric fuzzy numbers is sustained by the applications of these kinds of fuzzy numbers in fuzzy linear programming and by the presence of the symmetric Gaussian type fuzzy numbers in the theory of errors. As potential applications of the middle-parametric representation, some fuzzy interpolation problems are considered.     

On residuals in generalized Johnson SB regressions

An important challenge in the class of generalized Johnson S_B regression models is to define residuals which are capable of identifying departures from the model assumptions, as well as to assess the overall goodness-of-fit of the model. On this regard, we propose a new residual for this class of models, and numerically evaluate its behaviour relative to the deviance residual initially proposed for this class of regression models. Monte Carlo simulation experiments and empirical applications using real and simulated data are provided. Overall, the results favour the residual we propose.     

Fuzzy multi-criteria decision-making based on positive and negative extreme solutions

To encompass decision data vagueness, many researchers generalized multi-criteria decision-making (MCDM) methods in certain environment into fuzzy multi-criteria decision-making (FMCDM) methods under fuzzy environment. In these FMCDM methods, ranking fuzzy numbers based on fuzzy pair-wise comparison is normally essential, but the comparison is a complexity work. To avoid fuzzy pair-wise comparison, we propose a FMCDM method based on positive and negative extreme solutions of alternatives. In the proposed method, two extreme solutions of alternatives are obtained by MAX and MIN operations of fuzzy TOPSIS. Then weakness and strength matrices between alternatives and extreme solutions are derived by a difference function revised from fuzzy preference relation of Lee, and multiplied with weight matrix to be weighted weakness and strength indices. The two weighted indices are respectively transferred into positive and negative indices, and then the two indices integrated into a total performance index. Finally, alternatives can be sorted according to their related performance indices, and FMCDM problems are easily solved, not by fuzzy pair-wise comparison.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

Analysis on the minimal representation of O(p,q) I. Realization via conformal geometry

This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov–Kostant. We begin by applying methods from conformal geometry of pseudo-Riemannian manifolds to a general construction of an infinite-dimensional representation of the conformal group on the solution space of the Yamabe equation. By functoriality of the constructions, we obtain different models of the unitary representation, as well as giving new proofs of unitarity and irreducibility. The results in this paper play a basic role in the subsequent papers, where we give explicit branching formulae, and prove unitarization in the various models.     

Triangular domain extension of algebraic trigonometric Bézier-like basis

In computer aided geometric design (CAGD), Bézier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bézier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bézier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bézier-like surfaces. Finally, some applications of the proposed basis are shown.