Search

We consider network models for isentropic flow and evaluate the application of three different momentum-related coupling constants: pressure, momentum flux, and the Bernoulli invariant. Subsonic solutions to generalized Riemann problems with subsonic initial conditions are proved to be unique, and the region where such initial conditions yield subsonic solutions is identified. The proof is restricted to monotone coupling constants. We also prove that the three momentum-related coupling constants fulfill the monotonicity constraint. Further, we analyze the existence of entropic solutions using a commonly applied entropy condition. The condition states that nonphysical solutions are characterized by production of mechanical energy at a junction. Both pressure and momentum flux are seen to yield nonphysical solutions in a constructed test case consisting of three pipe sections of equal cross-sectional area that are connected at a junction. The Bernoulli invariant is proved to yield entropic solutions for all...

We consider a bistable differential-difference equation with inhomogeneous diffusion. Employing a piecewise linear nonlinearity, often referred to as McKean's caricature of the cubic, we construct front solutions which correspond, in the case of homogeneous diffusion, to monotone traveling front solutions or, in the case of propagation failure, to stationary front solutions. A general form for these fronts is given for essentially arbitrary inhomogeneous discrete diffusion, and conditions are given for the existence of solutions to the original discrete Nagumo equation. The specific case of one defect is considered in depth, giving a complete understanding of propagation failure and a grasp on changes in wave speed. Insight into the dynamic behavior of these front solutions as a function of the magnitude and relative position of the defects is obtained with the assistance of numerical results.

This paper develops a method of upper and lower solutions for a general system of second-order ordinary differential equations with two-point boundary conditions. Our motivation of study stems from a class of financial mathematics problems under regime-switching diffusion models. Two examples are double barrier option valuation and optimal selling rules in asset trading. We establish the existence of a unique $C^2$ solution of the two-point boundary value problem. We construct monotone sequences of upper and lower solutions that are shown to converge to the unique solution of the boundary value problem. This construction provides a feasible numerical method to compute approximate solutions. An important feature of the proposed numerical method is that the unique solution is bracketed by the upper and lower approximate solutions, which provide an interval estimate of the unique solution function. We apply the general results to a regime-switching mean-reverting model and improve related results already rep...

Sufficient conditions for the monotone decay of disturbances to oscillatory and steady Hagen–Poiseuille flow are rigorously derived within the framework of linear stability theory. The conditions hold both for axisymmetric and nonaxisymmetric disturbances, whereas the result for nonaxisymmetric disturbances to the oscillatory flow is of particular importance because in this case even numerical results are not available in the literature. Furthermore, the conditions provide explicit bounds on the range of parameters that must be examined in any prospective search for instability by numerical means. The derivation of the sufficient conditions on monotone decay is based on a detailed analysis of the spectrum of the Stokes operator.

We present a new model for traffic on a multilane freeway (with n lanes). Our basic descriptors are the car density $\rho$ (in cars/mile), taken across all lanes in the freeway, and the average car velocity u (in miles/hour). The flux of cars across all lanes is given by $\rho u = {\sum^n_{i=1}}\rho_i u_i$, where $\rho_i$ is the car density in the ith lane, and ui the velocity of cars in the ith lane. We shall track only $\rho$ and u and not what is going on in each individual lane. On such multilane freeways, one often observes distinct stable equilibrium relationships between car velocity and density. Prototypical situations involve two equilibria, $$ v=v_1(\rho) > v = v_2 (\rho), \ \ \ 0 \leq \rho < \rho_{\rm max,} where $v_1(\cdot)$ and $v_2(\cdot)$ are monotone decreasing and satisfy $v_1 (\rho_{\rm max})=v_2(\rho_{\rm max})=0$. The upper curve is typically stable for densities satisfying $0 \leq \rho \leq \rho_1$, whereas the lower curve is stable for densities satisfying $ \rho_2 \leq \rho \leq \rh...

A model of the chemostat involving two species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered. The model incorporates distributed time delay in the form of integral differential equations in order to describe the time involved in converting nutrient to biomass. The delays are included in the nutrient and species concentrations simultaneously. A general class of monotone increasing functions is used to describe nutrient uptake. Sufficient conditions based on biologically meaningful parameters in the model are given that predict competitive exclusion for certain parameter ranges and coexistence for others. We prove that the global asymptotic attractivity of steady states of the model is similar to that of the corresponding model without time delays. However, our results indicate that when the inherent delays are in fact large, ignoring them may result in incorrect predictions.

We show for the p-system with viscosity-capillarity that all heteroclinic phase transitions lying sufficiently close to the Maxwell line are (linearly) dynamically stable under localized perturbations, independent of the form of the flux. On the other hand, we show that nearby homoclinic phase transitions (involving two or more changes in phase) are exponentially unstable. These results together imply that stationary and slowly moving phase transitions are dynamically stable if and only if they are monotone, in analogy with results obtained by Carr, Gurtin, and Slemrod in the variational setting [Arch. Rational Mech. Anal., 86 (1984), pp. 317--351]. This research is significant as the first proof of stability of a strong viscous shock wave for systems. Our analysis uses the technical framework developed in [R. Gardner and K. Zumbrun, Comm. Pure Appl. Math., 51 (1998), pp. 797--855; K. Zumbrun and P. Howard, Indiana Univ. Math. J., 47 (1998), pp. 741--871] together with a nonstandard energy estimate.

A model of the simple chemostat which allows for growth on the wall (or other marked surface) is presented as three nonlinear ordinary differential equations. The organisms which are attached to the wall do not wash out of the chemostat. This destroys the basic reduction of the chemostat equations to a monotone system, a technique which has been useful in the analysis of many chemostat-like equations. The adherence to and shearing from the wall eliminates the boundary equilibria. For a reasonably general model, the basic properties of invariance, dissipation, and uniform persistence are established. For two important special cases, global asymptotic results are obtained. Finally, a perturbation technique allows the special results to be extended to provide the rest point as a global attractor for nearby growth functions.

The shape-from-shading (SFS) equation relating u(y,r), the unknown (angular) height of a surface, to I(y,r), the known synthetic aperture radar (SAR) intensity data from the surface, is I = \frac{u_r^2}{\sqrt{1+u_r^2+u_y^2}}, where y and r are axial and radial cylindrical coordinates. Unlike the more common eikonal SFS equation which relates surface height in Cartesian coordinates to optical/photographic intensity data, the above radar equation can be transformed into Hamilton--Jacobi Cauchy form: ur+g(I,uy)=0. We explore the case where I is a discontinuous function, which occurs commonly in radar data. By considering sequences of continuous intensity functions that converge to I, we obtain corresponding sequences of viscosity solutions. We prove that these sequences must converge. We also establish conditions that guarantee that these sequences converge to a common limit, which we define as the solution to the radar equation. Finally, we establish and demonstrate that when this common limit exists, monot...

Two different approaches are employed to investigate the global attractivity of delayed Hopfield neural network models. Without assuming the monotonicity and differentiability of the activation functions, Liapunov functionals and functions (combined with the Razumikhin technique) are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays. In the case of monotone and smooth activation functions, the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model. Such criteria depend on the magnitude of delays and show that self-inhibitory connections can contribute to the global convergence.

A simple model for the bursting electrical activity of individual pancreatic $\beta$-cells is introduced. Using the model, a continuum model for collections (islets) of large numbers of such cells is then analyzed, using asymptotic methods, linear stability techniques, and monotone methods. When the coupling strength D is large, fast and slow subsystems for the continuum model are shown to be the same as those for the single cell if the slow variables are replaced by their spatial average. It is demonstrated that fast variables can synchronize while the slow variables simultaneously exhibit nonsynchronous behavior. Last, spatial inhomogeneities in slow subsystem parameters are shown to inactivate the islet. It is demonstrated that if a sufficient fraction of cells are inactive, the average value of the slow variables can be decreased significantly.

In this paper, we consider a competition model between n species in a chemostat that incorporates both monotone and nonmonotone general response functions and distinct removal rates. We show that only the species with the lowest break-even concentration survives, provided that the variation of distinct removal rates relative to the flow rate of the chemostat can be controlled by either the difference between the two lowest break-even concentrations or by a parameter based on the structure of response functions. LaSalle's extension theorem of the Lyapunov stability theory and fluctuation lemma are the main tools.

This paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat. The model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations. In the case where only two species are engaged in competition, it is shown that competitive exclusion holds for any monotone growth response functions. Sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n-species case. In regard to the delay effects on the qualitative outcome of competition, it is demonstrated that when the delays are relatively small, the predictions of the model are identical with the predictions given by corresponding models without time delays. However, including large delays in the model may alter the predicted outcome of competition. The techniques used also work when different removal rates are permitted, an...

The existence and stability of stable rotating solutions (spiral waves) in a discrete system of coupled phase models is proven. Monotone methods are used to obtain existence and qualitative features of the solutions. An application of Fenichel's results for singular perturbation implies the existence for a one variable model for excitability. Numerical comparisons with the phase model and a realistic membrane model are made. A numerically computed Hopf bifurcation shows the existence of the “wobbling core” observed in spatially continuous models.

A simple model for the time-dependent behavior of semiconductor inverter circuits is used for an analysis of the dynamics of inverter rings. VLSI applications of inverter rings include flip-flops and ring oscillators. The analysis concentrates on the construction of periodic solutions in the form of rotating waves. Existence of periodic solutions is established using a Poincare–Bendixson theorem for monotone cyclic feedback systems. Then three different approaches are used to construct these solutions. First, rotating waves are computed explicitly for an idealized model problem. Second, a Hopf bifurcation theorem for problems with symmetries is employed to study the local behavior of the periodic solutions. Third, software for numerical bifurcation and path following is used to obtain global bifurcation diagrams connecting the different analytical results.

A model of exploitative competition of n species in a chemostat for a single, essential, nonreproducing, growth-limiting resource is considered. S. B. Hsu [SIAM J. Appl. Math., 34 (1978), pp. 760–763] applies LaSalle’s extension theorem of Lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by Michaelis–Menten dynamics. G. J. Butler and G. S. K. Wolkowicz [SIAM J. Appl. Math., 45 (1985), pp. 138–151], on the other hand, allow more general response functions (including monotone and nonmonotone functions), but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate, and hence can be ignored. By means of Lyapunov stability theory, the global dynamics of the model for a large class of response functions are studied, including both monotone and nonmonotone functions (though it is not as general as the class studied by Butler and Wolkowicz) and the results in ...

This paper presents some fundamental results on nonlinear networks with uncoupled, monotone increasing, but not necessarily surjective characteristics.Necessary and sufficient conditions are obtained for the existence and uniqueness of solutions for strictly increasing resistive networks and a general class of increasing resistive networks. They are also necessary and sufficient for the existence of solutions for increasing resistive networks. These conditions are sufficient for the existence of solutions for eventually strictly increasing resistive networks, and for a class of eventually increasing resistive networks. The conditions are circuit-theoretic and can readily be used as a criterion in design. The dependence of solutions on the inputs is studied and also a bounded-input bounded-solution result is presented. Existence and uniqueness results for monotone RLC networks are obtained by viewing them as combinations of three one-element-kind subnetworks. Finally, two algorithms are given for testing t...

We consider a discrete review, single product, dynamic inventory model. New conditions are found for the optimality of an $(s,S)$-policy which generalize those of Scarf (1960) and Veinott (1966). Moreover, we obtain as a special case a result very similar to that of Porteus (1971) without any assumption on the probability distribution of demand. The analysis is based on a general concept of convexity which includes convex, functions in the usual sense, and monotone functions as special cases. The paper builds on ideas of Veinott.

A model of the chemostat involving n microorganisms competing for a single essential, growth-limiting substrate is considered. Instead of assuming the familiar Michaelis-Menten kinetics for nutrient uptake, a general class of functions is used which includes all monotone increasing uptake functions, but which also allows uptake functions that describe inhibition by the substrate at high concentrations.The qualitative behaviour of this generalized model is determined analytically. It is shown that the behaviour depends intimately upon certain parameters. Provided that all the parameters are distinct (which is a biologically reaonable assumption), at most one competitor survives. The substrate and the surviving competitor (if one exists), approach limiting values. Thus there is competitive exclusion. However, unlike the standard model, in certain cases the outcome is initial condition dependent.

The neutron flux in a nuclear reactor considered as a one-region, bare and homogeneous system with no delayed neutrons is described by a semilinear parabolic equation in a cylindrical region. A generalization of this equation subject to the first initial boundary condition in a region which can be noncylindrical is studied. Global existence of maximal and minimal solutions is established by using iteration schemes of the chord type, which give respectively one monotone nonincreasing sequence of upper bounds and one monotone nondecreasing sequence of lower bounds. Conditions under which uniqueness of the solution occurs are also given. With this, the above two sequences are shown respectively to converge uniformly and geometrically to the solution. It is also proved that the iteration scheme of the Picard type gives an alternating sequence consisting of two monotone subsequences of upper and lower bounds. This sequence is shown to converge uniformly and geometrically to the solution. Finally, the quasiline...

Monotone equivariant cluster methods are characterized within a certain order theoretic model for the subject. These methods are shown necessarily to be graph theoretic in nature, and the results used to construct a number of new agglomerative monotone equivariant techniques—some of them proving useful in connection with applications to biology, geology, and sociology. In that all of the techniques are intermediate between single linkage and complete linkage clustering, a characterization of the latter method is provided to be considered along with the well known characterization of the former method. Some of the difficulties inherent in obtaining a numerical measure of the optimality of a monotone equivariant cluster method are also examined.

A model of two species competing for a single nutrient in the chemostat is proposed, incorporating general monotone uptake functions with discrete time delays. After transforming the model, an anal...

We consider the nonlinear singular integral magnetic field equation $R{\bf{M}} = h{\bf{M}} + A{\bf{M}} = {\bf{H}}_a $, in the Hilbert space of vector-functions ${\bf{L}}^2 ( \Omega )$, where ${\bf{M}}$ is the magnetization vector, $( {h{\bf{M}}} )(x) = g [ {{\bf{M}}( x ),x]}$ is the total field, and $({\bf {AM}})(x) = ( - 1/(4\pi )){\operatorname{grad}}\,{\operatorname{div}}\smallint _\Omega ({\bf M}(y)/r)dy$. We prove that: (i) A is bounded, with $\parallel A\parallel = 1$; (ii) A is self-adjoint; (iii) A is positively semidefinite, with $( {A{\bf{M}},{\bf{M}}} )\geqq 0$.Uniqueness is proved in case h is strictly monotone; existence of $R^{ - 1} $ and its continuity are proved in case h is strongly monotone, continuous and bounded. In this case the Galerkin method (and, if magnetic-material is also isotropic, the Ritz method) is shown to yield a numerical solution of the equation.

Many $0 - 1$ programming problems can be reduced, at least in theory, to monotone problems in the same variables, i.e., ones whose set of feasible solutions is monotone. A class of special monotone sets, called master sets, is introduced. To each monotone set S one can associate a family F of master sets $S^ * $, so that all the facets of (the convex hull of) S can be obtained (by means of a straightforward transformation) from those of any $S^ * \in F$.

This paper develops a formalism within which the application of dynamic programming to discrete, deterministic problems is rigorously studied. The two central concepts underlying this development are discrete decision process and sequential decision process. Discrete decision processes provide a convenient means of problem statement, while monotone sequential decision processes (which are finite automata with a certain cost structure superimposed) correspond naturally to dynamic programming algorithms. The representations of discrete decision processes by monotone sequential decision processes are characterized, and this characterization is used in the deviation of dynamic programming algorithms for a variety of problems.

A priori estimates are obtained for solutions of the displacement problem in isotropic linear viscoelasticity theory under the assumption of monotone nonincreasing positive relaxation moduli. In these estimates, integral measures of the displacement and the displacement gradient are bounded by surface integrals involving the Dirichlet data and a tangential derivative of the Dirichlet data and by volume integrals of the body force. Coefficients appearing in these estimates are explicitly determined. Integral estimates are then used to derive a pointwise bound, in the case of zero body force.

Several theorems are proved concerning the existence and uniqueness of solutions of the equation $AF( x ) + Bx = c$, in which A and B are $n \times n$ matrices of real numbers, c is a real n-vector, and $F( x ) \equiv [ f_1 ( x_1 ), \cdots ,f_n ( x_n ) ]^T $ is a “diagonal” nonlinear mapping of real n-dimensional Euclidean space $E^n $ into itself. For the case in which the functions $f_k $ are continuous, strictly monotone increasing mappings of $E^1 $ into $E^1 $ (but not necessarily onto$E^1 $) and for a large and important class of pairs of matrices $( A,B )$, we give necessary and sufficient conditions for the existence of the global inverse of the mapping $AF( \cdot ) + B$. For not-necessarily-monotone $f_k $ (tunnel-diode type functions, for example) we give conditions which guarantee the existence of at least one solution of $AF( x ) + Bx = c$ for each $c \in E^n $.Several significant network-theoretic implications of the results are discussed. For example, we present necessary and sufficient cond...

By confining Lagrange multipliers to a class of two-parameter continuous monotone nondecreasing functionals, a new necessary and sufficient saddle-point optimality criterion may be obtained for nonconvex programming problems. This device also allows the solution of a nonconvex dual problem under general conditions. New necessary and sufficient conditions are given, as part of this development, for an extended form of stability in nonlinear programming. The theory is applied to the solution of finite- and infinite-dimensional programming problems.