Nonlinear dynamics of open marine population with space-limited recruitment: The case of mortality control

In this paper we develop a nonlinear extension for the open marine population model which has been proposed by Roughgarden et al. [Ecology 66 (1985) 54-67]. To avoid the negative population density, which is a drawback of the original model, we introduce a nonlinear mechanism that the mortality rate depends on the size of area occupied by the adult population. Then we give a rigorous mathematical framework to analyse the model equation, and we show sufficient conditions for stability and instability of the steady state. Our instability result suggests, as was proposed by Roughgarden et al., that there exists a sustained oscillation of the population density.     

The onto-semiotic approach to research in mathematics education

In this paper we synthesize the theoretical model about mathematical cognition and instruction that we have been developing in the past years, which provides conceptual and methodological tools to pose and deal with research problems in mathematics education. Following Steiner’s Theory of Mathematics Education Programme, this theoretical framework is based on elements taken from diverse disciplines such as anthropology, semiotics and ecology. We also assume complementary elements from different theoretical models used in mathematics education to develop a unified approach to didactic phenomena that takes into account their epistemological, cognitive, socio cultural and instructional dimensions.     

A temporal database forecasting algebra

Though forecasting methods are used in numerous fields, we have seen no work on providing a general theoretical framework to build forecast operators into temporal databases, producing an algebra that extends the relational algebra. In this paper, we first develop a formal definition of a forecast operator as a function that satisfies a suite of forecast axioms. Based on this definition, we propose three families of forecast operators called deterministic, probabilistic, and possible worlds forecast operators. Additional properties of coherence, orthogonality, monotonicity, and fact preservation are identified that these operators may satisfy (but are not required to). We show how deterministic forecast operators can always be encoded as probabilistic forecast operators, and how both deterministic and probabilistic forecast operators can be expressed as possible worlds forecast operators. Issues related to the complexity of these operators are studied, showing the relative computational tradeoffs of these types of forecast operators. We explore the integration of different forecast operators with standard relational operators in temporal databases—including extensions of the relational algebra for the probabilistic and possible worlds cases—and propose several policies for answering forecast queries. Instances where these different forecast policies are equivalent have been identified, and can form the basis of query optimization in forecasting. These policies are evaluated empirically using a prototype implementation of a forecast query answering system and several forecast operators.     

On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.     

Fonctions constructibles et intégration motivique I

We introduce a direct image formalism for constructible motivic functions. One deduces a very general version of motivic integration for which a change of variables theorem is proved. These constructions are generalized to the relative framework, in which we develop a relative version of motivic integration. To cite this article: R. Cluckers, F. Loeser, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Bayesian time series regression with nonparametric modeling of autocorrelation

Series models have several functions: comprehending the functional dependence of variable of interest on covariates, forecasting the dependent variable for future values of covariates and estimating variance disintegration, co-integration and steady-state relations. Although the regression function in a time series model has been extensively modeled both parametrically and nonparametrically, modeling of the error autocorrelation is mainly restricted to the parametric setup. A proper modeling of autocorrelation not only helps to reduce the bias in regression function estimate, but also enriches forecasting via a better forecast of the error term. In this article, we present a nonparametric modeling of autocorrelation function under a Bayesian framework. Moving into the frequency domain from the time domain, we introduce a Gaussian process prior to the log of the spectral density, which is then updated by using a Whittle approximation for the likelihood function (Whittle likelihood). The posterior computation is simplified due to the fact that Whittle likelihood is approximated by the likelihood of a normal mixture distribution with log-spectral density as a location shift parameter, where the mixture is of only five components with known means, variances, and mixture probabilities. The problem then becomes conjugate conditional on the mixture components, and a Gibbs sampler is used to initiate the unknown mixture components as latent variables. We present a simulation study for performance comparison, and apply our method to the two real data examples.     

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

The primary objective of this work is a detailed theoretical and computational study of the elasticity imaging inverse problem for tumor identification within the human body. Apart from this inverse problem's important and interesting application, it also poses noteworthy mathematical challenges since the underlying mathematical model is a system of elasticity involving incompressibility. This gives rise to the “locking” effect and special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we introduce a general computational scheme for handling parameter identification in saddle point problems along with the introduction and analysis of a new energy output least-squares objective functionals. We also present a treatment of the identification of discontinuous elasticity coefficients using the total variation regularization method. General formulas for the computation of the coefficient-to-solution map and a complete convergence analysis are given for the continuous problem as well as for its discrete analogue. Discrete formulas and implementation issues are discussed in detail and numerical examples for smooth and discontinuous coefficients are given.     

A theory of rolling horizon decision making

In this paper, we develop a theoretical framework for the common business practice of rolling horizon decision making. The main idea of our approach is that the usefulness of rolling horizon methods is, to a great extent, implied by the fact that forecasting the future is a costly activity. We, therefore, consider a general, discrete-time, stochastic dynamic optimization problem in which the decision maker has the possibility to obtain information on the uncertain future at given cost. For this non-standard optimization problem with optimal stopping decisions, we develop a dynamic programming formulation. We treat both finite and infinite horizon cases. We also provide a careful interpretation of the dynamic programming equations and illustrate our results by a simple numerical example. Various generalizations are shown to be captured by straightforward modifications of our model.This research is supported in part by NSERC Grant A4619, SSHRC Grant 410-87-0524, and Manufacturing Research Corporation of Ontario. Comments and suggestions from Qing Zhang and the referees are gratefully acknowledged.     

Non-consistent approximations of self-adjoint eigenproblems: application to the supercell method

In this article, we introduce a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator. We focus in particular on the discrete eigenvalues laying in spectral gaps. We first provide a priori error estimates on the eigenvalues and eigenvectors in the absence of spectral pollution. We then show that the supercell method for perturbed periodic Schrödinger operators falls into the scope of our study. We prove that this method is spectral pollution free, and we derive optimal convergence rates for the planewave discretization method, taking numerical integration errors into account. Some numerical illustrations are provided.     

Sub-Riemannian calculus on hypersurfaces in Carnot groups

We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.     

Dimension splitting for evolution equations

In this paper, we are concerned with splitting methods for the time integration of abstract evolution equations. We introduce an analytic framework which allows us to prove optimal convergence orders for various splitting methods, including the Lie and Peaceman–Rachford splittings. Our setting is applicable for a wide variety of linear equations and their dimension splittings. In particular, we analyze parabolic problems with Dirichlet boundary conditions, as well as degenerate equations on bounded domains. We further illustrate our theoretical results with a set of numerical experiments. This work was supported by the Austrian Science Fund under grant M961-N13.     

Price formation model with zero-Neumann boundary condition

ABSTRACT

This paper concerns the mathematical analysis of a mathematical model for price formation. We take a large number of rational buyers and vendors in the market who are trading the same good into consideration. Each buyer or vendor will choose his optimal strategy to buy or sell goods. Since markets seldom stabilize, our model mimics the real market behavior. We introduce three models. All of them are modifications of the original J.-M. Lasry and P. L. Lions evolution model. In the first modified model, a random term is added to mimic the randomness of trading in the real market. This reflects markets with low volatility, where it might be difficulty to buy or sell goods at specific price. In the second model, we use cumulative density function instead of density function. We give numerical simulations on these two models in order to have a general picture on the solution. In the third model, we add a term associated with the parameter R to destabilize the original Larsy–Lions model and study oscillations and wave solutions depending on different values of R. We also study existence and uniqueness of the solution. Moreover, Several plots are given to demonstrate these results corresponding to the theoretical prediction.     

Phase-type distribution of customer relationship with Markovian response and marketing expenditure decision on the customer lifetime value

In recent years, more and more companies have adopted relationship marketing (RM). At the core of RM is the development and maintenance of long-term relationships with valuable customers. In RM, the customer lifetime value (CLV) is the discounted profit streams of a customer across the entire customer life cycle. The CLV plays a key role in customer acquisition and retention decisions. In this paper, we present a general mathematical framework for RM, and introduce a Markov chain model which is appropriate in modeling RM because of its flexibility and probabilistic nature. We also develop the life distribution of the customer relationship, which is a phase-type distribution since it is the distribution of the first arrival stopping state. And we obtain the expectation of the CLV, which is an important statistic for good decision-making. Finally, we illustrate how to find the optimal remarketing policy numerically. The framework developed for RM systems in this paper should be seen as a practical approach to RM where one can directly apply the results of phase-type distribution and expectation of CLV to marketing decisions.     

Mathematic education in a cultural setting

The paper develops some proposals for curriculum development in mathematics based on an analysis of the intercultural transmission of mathematical knowledge. By introducing a concept of culture which calls for an analysis of individual and social behaviours, we are lead to recognize ‘ethnomathematics’ as a form of structured knowledge and to recognize ‘matheracy’ as a characteristic behaviour of human beings. Upon these two concepts, we introduce a theoretical framework for curriculum development in mathematics.

Curricular space is conceptualized as a three‐dimensional space with components, contents, methods and objectives considered solidarily. This relies upon an epistemology of action, based on an integration of episteme, techne and praxis. This allows for an approach in which theory and practice are in a dialectical relationship. Special reference is made to the problems of mathematical education for culturally differentiated groups, and in particular to the situation in third‐world countries.     

Modeling empowered by information and communication technologies

In this article, we describe our work with mathematical modeling (MM) at different educational levels and discuss how the use of information and communication technologies (ICTs) empowered such work. Characteristics of two trends in research which have influenced our work are presented: one is a Brazilian perspective of MM, and the other is the use of ICTs in mathematics classrooms seen through the lens of the theoretical construct “humans-with-media”. We introduce some key questions regarding the notion of mathematical model and the phases of the modeling process that were paramount for us. Finally, we describe and analyze two experiences using modeling in different educational contexts, and present some evidence of the empowering role of ICTs in such contexts.     

Algorithm engineering for optimal alignment of protein structure distance matrices

Protein structural alignment is an important problem in computational biology. In this paper, we present first successes on provably optimal pairwise alignment of protein inter-residue distance matrices, using the popular dali scoring function. We introduce the structural alignment problem formally, which enables us to express a variety of scoring functions used in previous work as special cases in a unified framework. Further, we propose the first mathematical model for computing optimal structural alignments based on dense inter-residue distance matrices. We therefore reformulate the problem as a special graph problem and give a tight integer linear programming model. We then present algorithm engineering techniques to handle the huge integer linear programs of real-life distance matrix alignment problems. Applying these techniques, we can compute provably optimal dali alignments for the very first time.     

The exponentiated generalized power Lindley distribution: Properties and applications

In this paper, we introduce a new extension of the power Lindley distribution, called the exponentiated generalized power Lindley distribution. Several mathematical properties of the new model such as the shapes of the density and hazard rate functions, the quantile function, moments, mean deviations, Bonferroni and Lorenz curves and order statistics are derived.Moreover, we discuss the parameter estimation of the new distribution using the maximum likelihood and diagonally weighted least squares methods. A simulation study is performed to evaluate the estimators. We use two real data sets to illustrate the applicability of the new model. Empirical findings show that the proposed model provides better fits than some other well-known extensions of Lindley distributions.     

Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school

This paper presents a study aimed at investigating the didactic potentiality of the combined use of two different kinds of artefacts for the purpose of constructing and conceptualizing mathematical meanings related to the notion of axial symmetry. In our view, the process of meanings construction can be fostered by the use of adequate artefacts, but it requires a teaching/learning model, which explicitly takes care of the evolution of meanings, from those personal, emerging through the activities, to the mathematical ones, aims of the teaching intervention. The main hypothesis of this study is that a potential synergy may occur between the use of different artefacts, synergy that can foster the integration of different and complementary meanings providing a rich support to the development of the expected mathematical meaning. The Theory of Semiotic Mediation offers the theoretical framework suitable to design the teaching sequence and to analyze the collected data. Specifically, the construct of semiotic potential provides the tool for describing the potentialities of the two artefacts, while that of didactic cycle offers a model for the organization of the different activities. The paper reports on a teaching sequence and its implementation in a teaching experiment, involving pupils at fourth grade level. We describe them, within the chosen theoretical framework, and provide the analysis of key episodes of the teaching sequence. We show evidence supporting our main hypothesis about the combined use of an artefact that can be manipulated (paper and pin), and a digital artefact (Dynamic Geometry Environment) in the development of the notion of axial symmetry and its properties: the combined, intentional and controlled use of the two artefacts may develop a synergy, so that each activity enhances the potential of the other.     

A decomposition approach for facility location and relocation problem with uncertain number of future facilities

In this paper, we discuss two challenges of long term facility location problem that occur simultaneously; future demand change and uncertain number of future facilities. We introduce a mathematical model that minimizes the initial and expected future weighted travel distance of customers. Our model allows relocation for the future instances by closing some of the facilities that were located initially and opening new ones, without exceeding a given budget. We present an integer programming formulation of the problem and develop a decomposition algorithm that can produce near optimal solutions in a fast manner. We compare the performance of our mathematical model against another method adapted from the literature and perform sensitivity analysis. We present numerical results that compare the performance of the proposed decomposition algorithm against the exact algorithm for the problem.