Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space.

We develop a theory for sensitivity with respect to parameters in a convex subset of a topological vector space of dynamical systems in a Banach space. Specific motivating examples for probability measure dependent differential, partial differential and delay differential equations are given. Schemes that approximate the measures in the Prohorov sense are illustrated with numerical simulations for distributed delay differential equations.

Related papers:

title: Theory of Sensitivity in Dynamic Systems: An Introduction
abstract: The Principal Aspects of Sensitivity Theory.- Sensitivity Functions Generation.- Algebraic Properties of System Models Terminal Conditions Insensitivity.- Stability Robustness Analysis, Manoeuverability, and Sensitivity in the Large.- Sensitivity Reduction and Robustness.- The Theory of Sensitivity in  Selected Numerical Applications.

title: Introduction to System Sensitivity Theory
abstract: The author stresses that this book is written at a basic introductory level: accordingly, the book neither contains a rigorous treatment of the covered theory, nor is the covered theory chosen at a level other than the fundamental concepts of sensitivity theory. It is the reviewer's opinion that the author has accomplished his aims: thus he has written an introductory textbook on sensitivity theory which both covers the fundamentals of the theory and provides a guided entry to more advanced theory and to research. Indeed, it is the reviewer's personal experience  that the existence of this textbook, enfolding a unified set of definitions and a unified set of terminologies, can both facilitate subsequent research in sensitivity and enable it to be conducted in a smoother less disjointed way than in the past. Relative to the importance of sensitivity theory, it is to be recalled that in system identification, and therefore in the control domain, there exists

title: Design Sensitivity Analysis of Structural Systems
abstract: This volume presents a treatment of structural design sensitivity analysis. Both the mathematical and engineering foundations of design sensitivity analysis of structural systems are examined with a broad range of applications to structural components and structural systems. This work considers finite element models, design distribution of material described by functions and shape, and built-up structures that are composed of interconnected components. Each chapter includes problem formulation, examples, method development, illustrations, and theoretical foundations.

title: Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations
abstract: PRELIMINARIES.  Sensitivity and Large-Scale Systems. Nonlinear Solid Mechanics: Continuous and Semi-Discretized Formulation. Concepts of Sensitivity Analysis for Linear Systems.THE SENSITIVITY OF NONLINEAR SYSTEMS. The Basic Concepts of Nonlinear Quasi-Static Problems at Regular States. Inelastic Systems. Shape Sensitivity. Buckling and Post-Buckling. Nonlinear Dynamics. Metal Forming Using the Flow Approach. Nonlinear Thermal Systems.

title: NONLINEAR MODELS FOR REPEATED MEASUREMENT DATA.
abstract: Marie Davidian and David M. Giltiman, Chapman and Hall, Great Britain, 1995. No. of pages: xv+359.  0-412-98431-9

title: Model selection and multimodel inference : a practical information-theoretic approach
abstract: The second edition of this book is unique in that it focuses on methods for making formal statistical inference from all the models in an a priori set (Multi-Model Inference). A philosophy is presented for model-based data analysis and a general strategy outlined for the analysis of empirical data. The book invites increased attention on a priori science hypotheses and modeling. Kullback-Leibler Information represents a fundamental quantity in science and is Hirotugu Akaike's basis for model selection. The maximized log-likelihood function can be bias-corrected as an estimator of expected, relative Kullback-Leibler information. This leads to Akaike's Information Criterion (AIC) and various extensions. These methods are relatively simple and easy to use in practice, but based on deep statistical theory. The information theoretic approaches provide a unified and rigorous theory, an extension of likelihood theory, an important application of information theory, and are objective and practical to employ across a very wide class of empirical problems. The book presents several new ways to incorporate model selection uncertainty into parameter estimates and estimates of precision. An array of challenging examples is given to illustrate various technical issues. This is an applied book written primarily for biologists and statisticians wanting to make inferences from multiple models and is suitable as a graduate text or as a reference for professional analysts.

title: Elements of large-sample theory
abstract: This introductory book on the most useful parts of large-sample theory is designed to be accessible to scientists outside statistics and certainly to master’s-level statistics students who ignore most of measure theory. According to the author, “the subject of this book, first-order large- sample theory, constitutes a coherent body of concepts and results that are central to both theoretical and applied statistics.” All of the other existing books published on the subject over the last 20 years, from Ibragimov and Has’minskii in 1979 to the most recent by Van der Waart in 1998 have a common prerequisite in mathematical sophistication (measure theory in particular) that do not make the concepts available to a wide audience.

title: Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: an HIV Model, as an Example
abstract: HIV transmission models have become very complex. The behavior of some of these models may only be explored by uncertainty and sensitivity analyses, because the structural complexity of the model are coupled with a high degree of uncertainty in estimating the values of the input parameters. Uncertainty analysis may be used to assess the variability (prediction imprecision) in the outcome variable that is due to the uncertainty in estimating the input values. A sensitivity analysis can extend an uncertainty analysis by identifying which parameters are important in contributing to the prediction imprecision (i.e., how do changes in the values of the input parameters alter the value of the outcome variable). In this paper an uncertainty and a sensitivity analysis are described and applied; both analyses are based upon the Latin Hypercube Sampling (LHS) scheme, which is an extremely efficient sampling design proposed by McKay, Conover & Beckman (1979). The methods described in this paper have not previously been applied to deterministic models of disease transmission, although these models have many characteristics in common with the risk assessment models that the strategies were designed to investigate. The utility of the LHS uncertainty and the LHS/PRC (Latin Hypercube Sampling/Partial Rank Correlation) sensitivity analysis techniques are illustrated by analyzing a complex deterministic model of HIV transmission.

title: Sensitivity Analysis of Discrete Structural Systems
abstract: Methods for calculating sensitivity derivatives for discrete structural systems are surveyed, primarily covering literature published during the past two decades. Methods are described for calculating derivatives of static displacements and stresses, eigenvalues and eigenvectors, transient structural response, and derivatives of optimum structural designs with respect to problem parameters. The survey is focused on publications developed in nonstructural fields such as electronics, controls, and physical chemistry which are directly applicable to structural problems. Most notable among the nonstructural-based methods are the adjoint variable technique from control theory, and the Green's function and FAST methods from physical chemistry.

title: A PDE Sensitivity Equation Method for Optimal Aerodynamic Design
abstract: The use of gradient-based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design. A typical procedure uses a simulation scheme to evaluate the objective function (from the approximate states) and its gradient, then passes this information to an optimization algorithm. Once the simulation scheme (CFD flow solver) has been selected and used to provide approximate function evaluations, there are several possible approaches to the problem of computing gradients. One popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients. Although this black-box approach has many advantages in shape optimization problems, one must compute mesh sensitivities in order to compute the design sensitivity. In this paper, we present an alternative approach using the PDE sensitivity equation to develop algorithms for computing gradients. This approach has the advantage that mesh sensitivities need not be computed. Moreover, when it is possible to use the CFD scheme for both the forward problem and the sensitivity equation, then there are computational advantages. An apparent disadvantage of this approach is that it does not always produce consistent derivatives. However, for a proper combination of discretization schemes, one can showasymptotic consistencyunder mesh refinement, which is often sufficient to guarantee convergence of the optimal design algorithm. In particular, we show that when asymptotically consistent schemes are combined with a trust-region optimization algorithm, the resulting optimal design method converges. We denote this approach as thesensitivity equation method.The sensitivity equation method is presented, convergence results are given, and the approach is illustrated on two optimal design problems involving shocks.

title: On Efficient Solutions to the Continuous Sensitivity Equation Using Automatic Differentiation
abstract: Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.

title: A Petrov Galerkin finite-element method for interface problems arising in sensitivity computations
abstract: Continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design, to fast algorithms in computational fluid dynamics to the quantification of uncertainty. In order to make use of these methods for interface problems, one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces. In this paper we develop a discontinuous Petrov Galerkin finite-element scheme for solving the sensitivity equation resulting from a 1D interface problem. The 1D example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities. In particular, the sensitivity boundary value problem must be formulated in a very weak sense, and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes. Numerical examples are presented to illustrate the benefits of this approach.

title: A note on the use of transformations in sensitivity computations for elliptic systems
abstract: Sensitivity analysis is an important tool in engineering with applications ranging from optimal design to computational fluid dynamics. There are many algorithms available for computing sensitivities. These range from automatic differentiation techniques to variational methods. In this paper we describe two variational methods for computing sensitivities. We use a 1D model problem as a platform for describing and comparing these methods. In particular, the regularity of the sensitivity equations obtained from each method is addressed. Numerical approximations to each sensitivity are calculated using finite element schemes and numerical comparisons are presented.

title: Design Sensitivity Analysis: Computational Issues on Sensitivity Equation Methods
abstract: This book provides an understandable introduction to one approach to design sensitivity computation and illustrates some of the important mathematical and computational issues inherent in using the sensitivity equation method (SEM) for partial differential equations. The authors use basic models to illustrate the computational issues that one might encounter when applying the SEM in a laboratory or research setting, while providing an overview of applications and computational issues regarding sensitivity calculations performed by way of continuous sensitivity equation methods (CSEM). Here they focus on the construction and analysis of algorithms for computing sensitivities. For readers already acquainted with the concept of a sensitivity equation, the authors include mathematical background for a deeper understanding of their approach. Finally, the book explores the use of SEMs for applications in the area of computational fluid dynamics, demonstrating that the early examples readers encounter in the book can be observed in the context of a more realistic physical setting. Several colour figures are included.

title: Modeling and Imaging Techniques with Potential for Application in Bioterrorism
abstract: In this paper we present a survey of several recent and emerging ideas and e orts on modeling and system interrogation in the presence of uncertainty that we feel have signi cant potential for applications related to bioterrorism. The rst focuses on physiologically based pharmacokinetic (PBPK) type models and the e ects of drugs, toxins and viruses on tissue, organs, individuals and populations wherein both intraand inter-individual variability are present when one attempts to determine kinetic rates, susceptibility, eÆcacy of toxins, antitoxins, etc., in aggregate populations.  combining deterministic and stochastic concepts are necessary to formulate and computationally solve the associated estimation problems. Similar issues arise in the HIV infectious models we also present below. A second e ort concerns the use of remote electromagnetic interrogation pulses linked to dielectric properties of materials to carry out macroscopic structural imaging of bulk packages (drugs, explosives, etc.) as well as test for presence and levels of toxic chemical compounds in tissue. These techniques also may be useful in functional imaging (e.g., of brain and CNS activity levels) to determine levels of threat in potential adversaries via changes in dielectric properties and conductivity. The PBPK and cellular level virus infectious models we discuss are special examples of a much wider class of population models that one might utilize to investigate potential agents for use in attacks, such as viruses, bacteria, fungi and other chemical, biochemical or radiological agents. These include general epidemiological models such as SIR infectious

title: A parameter sensitivity methodology in the context of HIV delay equation models
abstract: A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.

title: Incorporation of variability into the modeling of viral delays in HIV infection dynamics.
abstract: We consider classes of functional differential equation models which arise in attempts to describe temporal delays in HIV pathogenesis. In particular, we develop methods for incorporating arbitrary variability (i.e., general probability distributions) for these delays into systems that cannot readily be reduced to a finite number of coupled ordinary differential equations (as is done in the method of stages). We discuss modeling from first principles, introduce several classes of non-linear models (including discrete and distributed delays) and present a discussion of theoretical and computational approaches. We then use the resulting methodology to carry out simulations and perform parameter estimation calculations, fitting the models to a set of experimental data. Results obtained confirm the statistical significance of the presence of delays and the importance of including delays in validating mathematical models with experimental data. We also show that the models are quite sensitive to the mean of the distribution which describes the delay in viral production, whereas the variance of this distribution has relatively little impact.

title: Well-posedness in Maxwell systems with distributions of polarization relaxation parameters
abstract: We present existence, uniqueness and continuous dependence (with respect to probability distributions on polarization parameters) of solutions in Maxwell systems. This provides a theoretical and computational foundation for associated inverse problems.

title: Identification of Hysteretic Control Influence Operators Representing Smart Actuators Part I: Formulation
abstract: A large class of emerging actuation devices and materials exhibit strong hysteresis characteristics during their routine operation. For example, when piezoceramic actuators are operated under the influence of strong electric fields, it is known that the resulting input–output behavior is hysteretic. Likewise, when shape memory alloys are resistively heated to induce phase transformations, the input–output response at the structural level is also known to be strongly hysteretic. This paper investigates the mathematical issues that arise in identifying a class of hysteresis operators that have been employed for modeling both piezoceramic actuation and shape memory alloy actuation. Specifically, the identification of a class of distributed hysteresis operators that arise in the control influence operator of a class of second order evolution equations is investigated. In Part I of this paper we introduce distributed,hysteretic control influence operators derived from smoothed Preisach operators and generalized hysteresis operators derived from results of Krasnoselskii and Pokrovskii. For these classes, the identification problem in which we seek to characterize the hysteretic control influence operator can be expressed as an ouput least square minimization over probability measures defined on a compact subset of a closed half-plane. In Part II of this paper, consistent and convergent approximation methods for identification of the measure characterizing the hysteresis are derived.

title: Hysteresis and Phase Transitions
abstract: 1. Some Mathematical Tools.- 1.1 Measure and Integration.- 1.2 Function Spaces.- 1.3 Nonlinear Equations.- 1.4 Ordinary Differential Equations.- 2. Hysteresis Operators.- 2.1 Basic Examples.- 2.2 General Hysteresis Operators.- 2.3 The Play Operator.- 2.4 Hysteresis Operators of Preisach Type.- 2.5 Hysteresis Potentials and Energy Dissipation.- 2.6 Hysteresis Counting and Damage.- 2.7 Characterization of Preisach Type Operators.- 2.8 Hysteresis Loops in the Prandtl Model.- 2.9 Hysteresis Loops in the Preisach Model.- 2.10 Composition of Preisach Type Operators.- 2.11 Inverse and Implicit Hysteresis Operators.- 2.12 Hysteresis Count and Damage, Part II.- 3. Hysteresis and Differential Equations.- 3.1 Hysteresis in Ordinary Differential Equations.- 3.2 Auxiliary Imbedding  3.3 The Heat Equation with Hysteresis.- 3.4 A Convexity Inequality.- 3.5 The Wave Equation with Hysteresis.- 4. Phase Transitions and Hysteresis.- 4.1 Thermodynamic Notions and Relations.- 4.2 Phase Transitions and Order Parameters.- 4.3 Landau and Devonshire Free Energies.- 4.4 Ginzburg Theory and Phase Field Models.- 5. Hysteresis Effects in Shape Memory Alloys.- 5.1 Phenomenology and Falk's Model.- 5.2 Well-Posedness for Falk's Model.- 5.3 Numerical Approximation.- 5.4 Complementary Remarks.- 6. Phase Field Models With Non-Conserving Kinetics.- 6.1 Auxiliary Results from Linear Elliptic and Parabolic Theory.- 6.2 Well-Posedness of the Caginalp Model.- 6.3 Well-Posedness of the Penrose-Fife Model.- 6.4 Complementary Remarks.- 7. Phase Field Models With Conserved Order Parameters.- 7.1 Well-Posedness of the Caginalp Model.- 7.2 Well-Posedness of the Penrose-Fife Model.- 8. Phase Transitions in Eutectoid Carbon Steels.- 8.1 Phenomenology of the Phase Transitions.- 8.2 The Mathematical Model.- 8.3 Well-Posedness of the Model.- 8.4 The Jominy Test: A Numerical Study.

title: Mathematical models of hysteresis.
abstract: A new approach to the scalar Preisach model of hysteresis, which emphasizes its phenomenological nature and mathematical generality, is described. The theorem, which gives the necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by the scalar Preisach model, is reported. The significance of this theorem is that it establishes the limits of applicability of Preisach's model regardless of the physical nature of hysteresis. Then, the vector Preisach models are formulated and some basic properties of these models are briefly summarized. Numerical implementations of Preisach's models are discussed and some computational results are given.

title: Multiscale Considerations in Modeling of Nonlinear Elastomers
abstract: We present a survey of results from an extended project focused on the understanding of the dynamic behavior of elastomers or filled rubbers. This entailed experimental, modeling, computational and theoretical efforts. Of particular emphasis are the nonlinear and hysteretic aspects of dynamic deformations.

title: Nonlinear Reptation in Molecular Based Hysteresis Models for Polymers
abstract: We extend the linear stick-slip models of Doi-Edwards and Johnson-Stacer to nonlinear tube reptation models. We then show that such models, when combined with probabilistic formulations allowing distributions of relaxation times, provide a good description of dynamic experiments with highly filled rubber in tensile deformations. A connection to other applications including dielectric polarization and reptation in other viscoelastic materials (e.g., living tissue) is noted.

title: A Probabilistic Multiscale Approach to Hysteresis in Shear Wave Propagation in Biotissue
abstract: Motivated by a problem involving wave propagation through viscoelastic biotissue, we present a theoretical framework for treating hysteresis as multiscale phenomena which must be averaged across distributions of internal variables. The resulting systems entail probability measure-dependent partial differential equations for which we establish well-posedness in a framework that leads readily to computationally useful approximations.

title: Functions of Relaxed Controls
abstract: Mathematical control theory problems involving solutions of certain partial differential equations, nonadditive set functions, or other functionals - approximation and existence theorems

title: Hereditary Control Problems: Numerical Methods Based on Averaging Approximations
abstract: An approximation scheme involving approximation of linear functional differential equations by systems of high order ordinary differential equations is formulated and convergence is established in the context of known results from linear semigroup theory. Applications to optimal control problems are discussed and a summary of numerical results is given. The paper is concluded with a brief survey of previous literature on this class of approximations for systems with delays.

title: Electromagnetic material interrogation using conductive interfaces and acoustic wavefronts
abstract: Preface 1.  Introduction and physical modeling 3. Wellposedness 4. Computational methods for dielectrics with supraconductive boundary 5. Computational methods for general polarization models 6. Computational methods for acoustically backed dielectrics 7. Concluding summary and remarks of potential applications Bibliography Index.

title: Dielectric properties of tissues and biological materials: a critical review.
abstract: We critically review bulk electrical properties of tissues and other biological materials, from DC to 20 GHz, with emphasis on the underlying mechanisms responsible for the properties. We summarize the classical principles behind dielectric relaxation and critically review recent developments in this field. Special topics include a summary of the significant recent advances in theories of counterion polarization effects, dielectric properties of cancer vs. normal tissues, properties of low-water-content tissues, and macroscopic field-coupling considerations. Finally, the dielectric properties of tissues are summarized as empirical correlations with tissue water content in other compositional variables; in addition, a comprehensive table is presented of dielectric properties. The bulk electrical properties of tissues are needed for many bioengineering applications of electric fields or currents, and they provide insight into the basic mechanisms that govern the interaction of electric fields with tissue.

title: Spline approximations for functional differential equations
abstract: We develop an approximation framework for linear hereditary systems which includes as special cases approximation schemes employing splines of arbitrary order. Numerical results for first- and third-order spline-based methods are presented and compared with results obtained using a previously developed scheme based on averaging ideas.

title: Estimation Techniques for Distributed Parameter Systems
abstract: The research detailed in this monograph was originally motivated by our interest in control problems involving partial and delay differential equations. Our attempts to apply control theory techniques to such prob lems in several areas of science convinced us that in the need for better and more detailed models of distributed/ continuum processes in biology and mechanics lay a rich, interesting, and challenging class of fundamen tal questions. These questions, which involve science and mathematics, are typical of those arising in inverse or parameter estimation problems. Our efforts on inverse problems for distributed parameter systems, which are infinite dimensional in the most common realizations, began about seven years ago at a time when rapid advances in computing capabilities and availability held promise for significant progress in the development of a practically useful as well as theoretically sound methodology for such problems. Much of the research reported in our presentation was not begun when we outlined the plans for this monograph some years ago. By publishing this monograph now, when only a part of the originally intended topics are covered (see Chapter VII in this respect), we hope to stimulate the research and interest of others in an area of scientific en deavor which has exceeded even our optimistic expectations with respect to excitement, opportunity, and stimulation. The computer revolution alluded to above and the development of new codes allow one to solve rather routinely certain estimation problems that would have been out of the question ten years ago."

title: Taura syndrome virus (TSV) lesion development and the disease cycle in the Pacific white shrimp Penaeus vannamei
abstract: The Taura syndrome virus (TSV) disease cycle was redefined through histological and gene probe analysis of experimentally infected specific pathogen-free (SPF) Penaeus vannamei sampled at timed intervals. The cycle consists of 3 overlapping, but clinically and histologically distinct, phases: a -7 d peracute to acute phase, a -5 d transition phase (previously termed the chronic or recovery phase), and a definitive chronic phase. The acute phase is characterized by the rapid development of severe, multifocal to diffuse cuticular epithelial necrosis and high mortalities. Using in situ hybridization analysis, infected pre-lytic cuticular epithelial cells display very strong TSV-positive probe signals, and a total of 3 stages of acute phase necrosis are described. Surviving P. vannamei then enter the transition phase, which is distinguished histologically by multifocal melanized lesions within regions of the cuticular epithelium (resolving acute phase lesions), focal active acute phase lesions, and the onset of lymphoid organ (LO) spheroid development. Gene probe analysis of transitionally infected shrimp reveals probe-positive foci of active acute phase lesions, a diffuse probe signal within the walls of morphologically normal L 0 tubules and/or focal probe signals within developing L 0 spheroids. Shrimp surviving this stage enter the chronic phase infection after ecdysis. The defining characteristics of the chronic phase include the cessation of mortalities, the resumption of normal behavioral patterns, the con~plete absence of v~sible melanized lesions and acute phase histological lesions of the cuticular epithelium, and marked L 0 hypertrophy directly resulting from the rapid development of numerous L0 spheroids, some of which are TSV positive by in situ hybridization analysis.

title: Estimation of Dynamic Rate Parameters in Insect Populations Undergoing Sublethal Exposure to Pesticides
abstract: With newer, more environmentally friendly and, subsequently less lethal, pesticides in use, evaluating efficacy of a pesticide now requires more than simply counting deaths after treatment. A discrete, age-structured matrix model that incorporates a species’ life history traits (such as birth rate, death rate and fecundity) has previously been used by ecologists. This model will be presented and discussed along with an alternative continuous, age-structured model which offers significant advantage in considering sublethal damage. We use this continuous model to estimate time-dependent mortality parameters in an ordinary least-squares technique. Confidence intervals are given and results from tests for statistical significance of added parameters are presented.

title: Modeling and estimating uncertainty in parameter estimation
abstract: In this paper we discuss questions related to reliability or variability of estimated parameters in deterministic least-squares problems. By viewing the parameters for the inverse problem as realizations for a random variable we are able to use standard results from probability theory to formulate a tractable probabilistic framework to treat this uncertainty. We discuss method stability and approximate problems and are able to show convergence of solutions of the approximate problems to those of the original problem. The efficacy of our approach is demonstrated in numerical examples involving estimation of constant parameters in differential equations.