Loops and branches of coexistence states in a Lotka-Volterra competition model

A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate μ as a parameter, we describe the bifurcation diagram of the steady states, including stability, in terms of two real functions of μ. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers l and b, the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi-trivial steady states of the same type (they vanish at the same component), and at least b other bifurcating branches of positive stable steady states that connect semi-trivial steady states of different types.     

A combined approach for evaluating papers, authors and scientific journals

An integrated model for ranking scientific publications together with authors and journals recently presented in [Bini, Del Corso, Romani, ETNA 2008] is closely analyzed. The model, which relies on certain adjacency matrices H,K and F obtained from the relations of citation, authorship and publication, provides the ranking by means of the Perron vector of a stochastic matrix obtained by combining H,K and F. Some perturbation theorems concerning the Perron vector previously introduced by the authors are extended to more general cases and a counterexample to a property previously addressed by the authors is presented. The theoretical results confirm the consistency and effectiveness of our model. Some paradigmatic examples are reported together with some results obtained on a real set of data.     

On the irreducibility of a class of polynomials,III

This work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field $\text{L}$, where g(x) and f(x) are monic polynomials with integer coefficients in $\text{L}$, g is irreducible over $\text{L}$ and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over $\text{L}$ for all f having distinct roots in a given totally real number field.     

Evolution of a semilinear parabolic system for migration and selection in population genetics

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape (i.e., a bounded, open domain in R^d). The selection coefficients depend on position and may depend on the gene frequencies; the drift and diffusion coefficients may depend on position. Sufficient conditions are given for the global loss of an allele and for its protection from loss. A sufficient condition for the existence of at least one internal equilibrium is also offered, and the profile of any internal equilibrium in the zero-migration limit is obtained.     

Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

Comparison of turbulence models in simulating swirling pipe flows

Swirling flow is a common phenomenon in engineering applications. A numerical study of the swirling flow inside a straight pipe was carried out in the present work with the aid of the commercial CFD code fluent. Two-dimensional simulations were performed, and two turbulence models were used, namely, the RNG k–ε model and the Reynolds stress model. Results at various swirl numbers were obtained and compared with available experimental data to determine if the numerical method is valid when modeling swirling flows. It has been shown that the RNG k–ε model is in better agreement with experimental velocity profiles for low swirl, while the Reynolds stress model becomes more appropriate as the swirl is increased. However, both turbulence models predict an unrealistic decay of the turbulence quantities for the flows considered here, indicating the inadequacy of such models in simulating developing pipe flows with swirl.     

Bayesian inference in spherical linear models: robustness and conjugate analysis

The early work of Zellner on the multivariate Student-t linear model has been extended to Bayesian inference for linear models with dependent non-normal error terms, particularly through various papers by Osiewalski, Steel and coworkers. This article provides a full Bayesian analysis for a spherical linear model. The density generator of the spherical distribution is here allowed to depend both on the precision parameter φ and on the regression coefficients β. Another distinctive aspect of this paper is that proper priors for the precision parameter are discussed.The normal-chi-squared family of prior distributions is extended to a new class, which allows the posterior analysis to be carried out analytically. On the other hand, a direct joint modelling of the data vector and of the parameters leads to conjugate distributions for the regression and the precision parameters, both individually and jointly. It is shown that some model specifications lead to Bayes estimators that do not depend on the choice of the density generator, in agreement with previous results obtained in the literature under different assumptions. Finally, the distribution theory developed to tackle the main problem is useful on its own right.     

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C¹). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.     

Prior influence in linear regression when the number of covariates increases to infinity

It is becoming more typical in regression problems today to have the situation where “p>n”, that is, where the number of covariates is greater than the number of observations. Approaches to this problem include such strategies as model selection and dimension reduction, and, of course, a Bayesian approach. However, the discrepancy between p and n can be so large, especially in genomic data, that examining the limiting case where p→∞ can be a relevant calculation. Here we look at the effect of a prior distribution on the coefficients, and in particular characterize the conditions under which, as p→∞, the prior does not overwhelm the data. Specifically, we find that the prior variance on the growing number of covariates must approach zero at rate 1/p, otherwise the prior will overwhelm the data and the posterior distribution of the regression coefficient will equal the prior distribution.     

Bright soliton solution to a generalized Burgers–KdV equation with time-dependent coefficients

We consider a generalized Burgers–KdV type equation with time-dependent coefficients incorporating a generalized evolution term, the effects of third-order dispersion, dissipation, nonlinearity, nonlinear diffusion and reaction. The exact bright soliton solution for the considered model is obtained by using a solitary wave ansatz in the form of sech^s function. The physical parameters in the soliton solution are obtained as functions of the time varying coefficients and the dependent exponents. The dependent exponents and the temporal variations of the model coefficients satisfy certain parametric conditions as shown by the obtained soliton solution. This solution may be useful to explain some physical phenomena in genuinely nonlinear dynamical systems that are described by Burgers–KdV type models.     

Competitive diffusion process of repurchased products in knowledgeable manufacturing

This paper presents a diffusion model to explain the competitive diffusion of the repurchased products in knowledgeable manufacturing. The acute market competition accelerates the products’ improvement, which requires that the manufacturing enterprises be highly capable of rapid reaction by means of knowledgeable manufacturing. To forecast the diffusion behavior effectively enables the realization of knowledgeable manufacturing system (KMS) which targets T (time), Q (quality), C (cost), S (service), and E (environment). Various diffusion models have emerged since Bass model was firstly proposed in 1969. A nonlinear model of the repurchased competitive products is proposed on the basis of the product diffusion analysis. By taking the frequently purchased products as example, the stability of the model is examined in light of the qualitative theory of differential equations and proved by the approximate linearization method. As the qualitative analysis reveals, between the two frequently purchased products competing in the same market, one undoubtedly occupies a fixed market share while the other may finally be eliminated from the market. A special case of the problem is that both products are one-time-purchased. With the corresponding model given, the qualitative analysis shows that either of the products occupies a market share, the size of which is determined by the product’s competitive strength and the new product’s time-to-market. A system dynamics model is then established and simulated by vensim. The result is consistent with that of the qualitative analysis.     

Approximations of the resolvent for a non-self-adjoint diffusion operator with rapidly oscillating coefficients

A strongly inhomogeneous diffusion operatorwith drift depending on a small parameter ? is studied in the space L ²(? ⁿ ). The strong inhomogeneity consists in that the coefficients of the operator are ?-periodic and, in addition, the drift vector is of the order of ? ^?1. As ? → 0, approximations in the operator L ²-norm of order ? and ? ² are constructed for the resolvent of the operator. For each of these orders of approximation, an averaged diffusion operator is obtained. A spectral method based on the Bloch representation for an operator with periodic coefficients is used.     

Nonlinear reaction-diffusion models for interacting populations

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c_i(x, t), 1 ? i ? N (with c_i(x, t) representing the concentration of the ith species at position x in a set $\text{Ω}$ at time t ? 0), which exist for all t ? 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.     

On the ARCH model with random coefficients

In the usual ARCH model, the coefficients have a degenerate distribution, and it is thus constant over realizations. In this Note we introduce the ARCH model, involving coefficients that are independent random variables and may vary over realizations. Conditions for the existence of a stationary solution and conditions ensuring the existence of higher order moments are obtained. The covariance structures of such models are studied. To cite this article: A. Bibi, M. Bousseboua, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

JUMP DETECTION BY WAVELET IN NONLINEAR AUTOREGRESSIVE MODELS

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Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect

We introduce a new local sine transform that can completely localize image information both in the space domain and in the spatial frequency domain. This transform, which we shall call the polyharmonic local sine transform (PHLST), first segments an image into local pieces using the characteristic functions, then decomposes each piece into two components: the polyharmonic component and the residual. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace's equation, biharmonic equation, etc.) given the boundary values (the pixel values along the boundary created by the characteristic function). Subsequently this component is subtracted from the original local piece to obtain the residual. Since the boundary values of the residual vanish, its Fourier sine series expansion has quickly decaying coefficients. Consequently, PHLST can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. Combining this ability with the quickly decaying coefficients of the residuals, PHLST is also effective for image approximation, which we demonstrate using both synthetic and real images. In addition, we introduce the polyharmonic local Fourier transform (PHLFT) by replacing the Fourier sine series above by the complex Fourier series. With a slight sacrifice of the decay rate of the expansion coefficients, PHLFT allows one to compute local Fourier magnitudes and phases without revealing the edge effect (or Gibbs phenomenon), yet is invertible and useful for various filtering, analysis, and approximation purposes.     

On semi-infinite spectral elements for Poisson problems with re-entrant boundary singularities

A spectral element method is described which enables Poisson problems defined in irregular infinite domains to be solved as a set of coupled problems over semi-infinite rectangular regions. Two choices of trial functions are considered, namely the eigenfunctions of the differential operator and Chebyshev polynomials. The coefficients in the series expansions are obtained by imposing weak C¹ matching conditions across element interfaces. Singularities at re-entrant corners are treated by a post-processing technique which makes use of the known asymptotic behaviour of the solution at the singular point. Accurate approximations are obtained with few degrees of freedom.     

Membership functions,some mathematical programming models and production scheduling

The authors have studied in [5] alternative real variable models based on the function $\text{d(x) =}\text{x}\text{(α + x)}\text{, α >0}$, for certain integer or mixed-interger programming problems. Mainly, we have shown that there exists a vector α > 0 such that the solution to the problem min $\text{σ}{}_1\text{(x, α) = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}\text{(ga}{}_{\mathrm{i}}\text{+x}{}_{\mathrm{i}}\text{)}\text{, Ax = b, x ? 0}$, is a solution to the problem min σxσ⁺, Ax = b, x ? 0, where σxσ⁺ denotes the cardinal of x, i.e. the number of strictly positive components of x, thus obtaining a new model for solving in real numbers a Generalized Lattice Point Problem (Cabot, [3]).The function d(x) has been introduced by use as a general tool for solving integer or mixed-integer problems due to its property of having almost everywhere almost discrete values. In the meantime we noticed that this function may represent a membership function of a fuzzy set.In this paper, we study in detail the features of this membership function and show that Cabot's results [3] may be derived in this more general setting using the complementary function $\text{s(x) =}\text{1 ? x}\text{(α + x)}\text{=}\text{α}\text{(α+x)}$.At the same time, in the paper there are some production scheduling models within the framework of fuzzy-sets theory. To this end, a nonconvex production model is presented and it is shown that the value of the objective function $\text{μ}{}_2\text{=}\text{1 ? σ}{}_1\text{n}$ for a production programming model whose deman and/or resource vector components are parametrized, may be considered as a grade of membership of the solution of the parametrized model to the feasible set of the nonparametrized production programming model.Consequently, we get a nonconvex production programming model whose convex envelope is linear with coefficients which are in an inverse proportior to the magnitude of the nonparametrized demand or resource vector components. This result agrees with the intuitive idea that a high level of demand or resource allows a greater interval of variation in the production process than a lower level of demand or resource.     

A Mixed Linear Quadratic Optimal Control Problem with a Controlled Time Horizon

A mixed linear quadratic (MLQ) optimal control problem is considered. The controlled stochastic system consists of two diffusion processes which are in different time horizons. There are two control actions: a standard control action \(u(\cdot )\) enters the drift and diffusion coefficients of both state equations, and a stopping time \(\tau \) , a possible later time after the first part of the state starts, at which the second part of the state is initialized with initial condition depending on the first state. A motivation of MLQ problem from a two-stage project management is presented. It turns out that solving an MLQ problem is equivalent to sequentially solve a random-duration linear quadratic (RLQ) problem and an optimal time (OT) problem associated with Riccati equations. In particular, the optimal cost functional can be represented via two coupled stochastic Riccati equations. Some optimality conditions for MLQ problem is therefore obtained using the equivalence among MLQ, RLQ and OT problems. In case of seeking the optimal time in the family of deterministic times (even through somewhat restrictive, such seeking is still reasonable from practical standpoint), we give a more explicit characterization of optimal actions.