The computation of stability boundaries in state space for a class of biochemical engineering systems

The stability of a class of biochemical processes defined by a set of m biochemical reactions involving n components is analysed. The processes operate in a continuous mode and possess at least two stable equilibrium states: the normal operating point and a biological wash out state. Using a canonical state representation of the process dynamics the geometric structure of the operating point's stability boundary is characterized. Numerical algorithms are developed to evaluate this boundary and to visualize its extent in state space. The proposed technique is illustrated with a representative engineering example.     

Time substitution with application to data envelopment analysis

In this paper we analyze resource allocation distinguishing between the decision of when to begin allocation and over how many periods to apply the resources. We present analytical results for specific production technologies under different returns to scale assumptions, under capacity constraints and for production with technical change. Using a dynamic activity analysis framework we show how to compute in general optimal solutions for resource intensity use.     

Numerical solution of calcium-mediated dendritic branch model

The standard algorithms for spatial discretizations of calcium-mediated dendritic branch models via finite difference methods are quite accurate, but they are also extremely slow. To improve computational efficiency we apply spatial discretization using a spectral collocation method. Simulations using the spectral collocation method are compared to the finite difference approach using a model for calcium-mediated restructuring with spine pruning. We find that the spectral collocation method is about fifteen times more efficient to achieve similar accuracy than the finite difference approach even though spectral collocation requires more steps.     

Vector space bases associated to vanishing ideals of points

In this paper we discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we drastically improve the computational algebra approach to the reverse engineering of gene regulatory networks.     

Qualitative and numerical analysis of a class of prey-predator models

We consider a problem of the dynamics of prey-predator populations suggested by the content of a letter of the biologist Umberto D'Ancona to Vito Volterra. The main feature of the problem is the special type of competition between predators of the same species as well as of different species. Two classes of cases are investigated: a first class in which the behaviour of the predator is blind and the second one in which the behaviour is intelligent. A qualitative analysis of the dynamical systems under consideration is followed by a numerical analysis of the most significant cases.     

Bayesian analysis of multivariate t linear mixed models using a combination of IBF and Gibbs samplers

The multivariate linear mixed model (MLMM) has become the most widely used tool for analyzing multi-outcome longitudinal data. Although it offers great flexibility for modeling the between- and within-subject correlation among multi-outcome repeated measures, the underlying normality assumption is vulnerable to potential atypical observations. We present a fully Bayesian approach to the multivariate t linear mixed model (MtLMM), which is a robust extension of MLMM with the random effects and errors jointly distributed as a multivariate t distribution. Owing to the introduction of too many hidden variables in the model, the conventional Markov chain Monte Carlo (MCMC) method may converge painfully slowly and thus fails to provide valid inference. To alleviate this problem, a computationally efficient inverse Bayes formulas (IBF) sampler coupled with the Gibbs scheme, called the IBF-Gibbs sampler, is developed and shown to be effective in drawing samples from the target distributions. The issues related to model determination and Bayesian predictive inference for future values are also investigated. The proposed methodologies are illustrated with a real example from an AIDS clinical trial and a careful simulation study.     

MILP approach to pattern generation in logical analysis of data

Pattern generation methods for the Logical Analysis of Data (LAD) have been term-enumerative in nature. In this paper, we present a Mixed 0-1 Integer and Linear Programming (MILP) approach that can identify LAD patterns that are optimal with respect to various previously studied and new pattern selection preferences. Via art of formulation, the MILP-based method can generate optimal patterns that also satisfy user-specified requirements on prevalence, homogeneity and complexity. Considering that MILP problems with hundreds of 0-1 variables are easily solved nowadays, the proposed method presents an efficient way of generating useful patterns for LAD. With extensive experiments on benchmark datasets, we demonstrate the utility of the MILP-based pattern generation.     

Approximation to the mean curve in the LCS problem

The problem of sequence comparison via optimal alignments occurs naturally in many areas of applications. The simplest such technique is based on evaluating a score given by the length of a longest common subsequence divided by the average length of the original sequences. In this paper we investigate the expected value of this score when the input sequences are random and their length tends to infinity. The corresponding limit exists but is not known precisely. We derive a theoretical large deviation, convex analysis and Monte Carlo based method to compute a consistent sequence of upper bounds on the unknown limit. An empirical practical version of our method produces promising numerical results.     

Convergence to steady state of solutions of the euler equations,I

In this paper we analyze the convergence to steady state of solutions of the compressible and the incompressible isentropic Euler equations in two space dimensions. In the compressible case, the original equations do not converge. We replace the equation of continuity with an elliptic equation for the density, obtaining a new set of equations, which have the same steady solution. In the incompressible case, the equation of continuity is replaced by a Poisson equation for the pressure. In both cases, we linearize the equations around a steady solution and show that the unsteady solution of the linearized equations converges to the steady solution, if the steady solution is sufficiently smooth. In the proof we consider how the energy of the time dependent part developes with time, and find that it decrease exponentially.     

Semiparametric analysis of longitudinal zero-inflated count data

In this article, we consider a semiparametric zero-inflated Poisson mixed model that postulates a possible nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate in the longitudinal studies. A penalized log-likelihood function is proposed and Monte Carlo expectation-maximization algorithm is used to derive the estimates. Under some mild conditions, we establish the consistency and asymptotic normality of the resulting estimators. Simulation studies are carried out to investigate the finite sample performance of the proposed method. For illustration purposes, the method is applied to a data set from a pharmaceutical company where the variable of interest is the number of episodes of side effects after the patient has taken the treatments.     

Filippov solutions in the analysis of piecewise linear models describing gene regulatory networks

We study some properties of piecewise linear differential systems describing gene regulatory networks, where the dynamics are governed by sigmoid-type nonlinearities which are close to or coincide with the step functions. To overcome the difficulty of describing the dynamics of the system near singular stationary points (belonging to the discontinuity set of the system) we use the concept of Filippov solutions. It consists in replacing discontinuous differential equations with differential inclusions. The global existence and some other basic properties of the Filippov solutions such as continuous dependence on parameters are studied. We also study the uniqueness and non-uniqueness of the Filippov solutions in singular domains. The concept of Filippov stationary point is extensively exploited in the paper. We compare two ways of defining the singular stationary points: one is based on the Filippov theory and the other consists in replacing step functions with steep sigmoids and investigating the smooth systems thus obtained. The results are illustrated by a number of examples.     

A data envelopment approach to decision analysis

A decision aid to assist the development of a linear valuation function for multiple attribute problems is proposed, based on a linear programming formulation using a constraint set structured in a similar manner to data envelopment analysis (DEA). Value functions which favour each decision option are calculated, and efficient, potentially optimal, options identified. These are used to help a decision maker progressively to articulate preferences, indicators of his/her values, in an interactive, structurally flexible manner. As preference indications are provided, candidate value functions and hitherto efficient options inconsistent with his/her declarations are eliminated, thus proceeding towards an explicit value function and, if needed a corresponding complete option order.     

Analysis of correlated binary data under partially linear single-index logistic models

Clustered data arise commonly in practice and it is often of interest to estimate the mean response parameters as well as the association parameters. However, most research has been directed to address the mean response parameters with the association parameters relegated to a nuisance role. There is relatively little work concerning both the marginal and association structures, especially in the semiparametric framework. In this paper, our interest centers on the inference of both the marginal and association parameters. We develop a semiparametric method for clustered binary data and establish the theoretical results. The proposed methodology is investigated through various numerical studies.     

Kinetic (cellular) models of cell progression and competition with the immune system

This paper deals with the modelling of the immune response to the evolution of progressing (corrupted) endothelial cells, i.e. cancer cells. A mathematical model is proposed, on the basis of mathematical methods of the kinetic theory for a large system of interacting cells. Then a qualitative analysis is carried out to prove the existence of the solutions of the Cauchy problem related to the model and to show some results on the asymptotic behavior. Some computational simulations complete the analysis on the asymptotic behavior of the solutions, which depending on parameters and initial conditions shows either the prevalence of progressing cells or their depletion due to the action of the immune system.     

A stable approach to the equivariant Hopf theorem

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner [S. Waner, Equivariant RO(G)-graded bordism theories, Topology and its Applications 17 (1984) 1-26], the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterised in terms of congruences. This is first shown to be a stable problem, and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification

Given a row-stochastic matrix describing pairwise similarities between data objects, spectral clustering makes use of the eigenvectors of this matrix to perform dimensionality reduction for clustering in fewer dimensions. One example from this class of algorithms is the Robust Perron Cluster Analysis (PCCA+), which delivers a fuzzy clustering. Originally developed for clustering the state space of Markov chains, the method became popular as a versatile tool for general data classification problems. The robustness of PCCA+, however, cannot be explained by previous perturbation results, because the matrices in typical applications do not comply with the two main requirements: reversibility and nearly decomposability. We therefore demonstrate in this paper that PCCA+ always delivers an optimal fuzzy clustering for nearly uncoupled, not necessarily reversible, Markov chains with transition states.     

Cluster analysis of census data using the symbolic data approach

The aim of this paper is to investigate the economic specialization of the Italian local labor systems (sets of contiguous municipalities with a high degree of self-containment of daily commuter travel) by using the Symbolic Data approach, on the basis of data derived from the Census of Industrial and Service Activities. Specifically, the economic structure of a local labor system (LLS) is described by an interval-type variable, a special symbolic data type that allows for the fact that all municipalities within the same LLS do not have the same economic structure.