Concept Of Monotonicity Research Articles

Maximum norm contractivity of discretization schemes for the heat equation

In this paper we present a necessary and sufficient condition for maximum norm contractivity of a large class of discretization methods for solving the heat equation. Specializing this result for the well-known Crank- Nicolson method we arrive at the criterion ▵ t /(▵ x) 2 ⩽ 3 2 , which is less restrictive than the weakest presently available sufficient condition ▵ t /(▵ x) 2 ⩽1. A comparison is made with other stability criteria and with sufficient conditions obtained by Spijker (1983, 1985) which are based on the concept of absolute monotonicity. The theoretical results are illustrated with several examples, among which the construction of some optimal explicit schemes.

Comparison theorems for algorithmic models

This paper discusses monotonicity concepts to compare iteration sequences and common fixed points of point-to-set mappings. Three particular applications are presented. A new proof of the Gronwall lemma is first introduced, and then the monotonicity of input-output systems is analysed. Matrices with nonnegative inverses are finally examined.

Monotone mobility matrices

The transition matrix of a discrete Markov chain is called monotone if each row stochastically dominates the row above it. Monotonicity is an ideal assumption to impose on a Markov chain model of mobility. Monotonicity is behaviorally weak yet mathematically strong. It is behaviorally weak in the sense that it is theoretically plausible and is empirically supported. It is mathematically strong in the sense that monotone Markov chains have a number of convenient mathematical properties. This paper reviews the convenient properties and applies the monotonicity concept to immobility measurement.

Order‐based statistics and monotonicity: A family of ordinal measures of association*

This paper reviews standards for measures of association. In particular, standards for order‐based measures are examined. The concept of monotonicity is shown to be ambiguous as it has been applied in this area, and it is clarified. The result is the specification of three — instead of the usual two — kinds of monotonic relations. These three monotonic models provide the basis for defining three non‐arbitrary measures of ordinal association. Eleven order‐based measures are reviewed in the light of measurement standards as extended by the clarification of monotonicity. Eight are shown either to embody non‐order‐based elements or to contain ad hoc characteristics. The remaining three are shown to be members of a single family of monotone based ordinal measures that can be applied where ever their particular monotone models are appropriate.

Two concepts of monotonicity in two-person bargaining theory

Monotonicity has been suggested as a principle of fairness and a principle of bargaining. It has been proposed as a substitute to the “independence of irrelevant alternatives” axiom in bargaining solutions and thus applied to the determination of wages and employment. In this paper we will discuss the Kalai-Smorodinsky solution and show that it does not correspond to the monotonicity notion which follows from the “principle of bargaining” idea.

A Concept of Monotonicity and Its Characterization for Closed Queueing Networks

We define a qualitative property, called monotonicity, for a closed queueing network. This property is defined both for a station in the network and for the network as a whole. Several results in this paper relate monotonic behavior to the known parameters of a network and allow us to state a priori, for many practical systems, whether the monotonicity property will apply for a network or a station. Monotonicity thus ensures that the system is well-behaved with respect to a number of parameters. We present diverse examples that demonstrate how knowledge of this property can be useful in network design and performance evaluation.

Monotonicity in Goal and Geometric Programming

The concept of monotonicity is applied to posynomial geometric programming establishing rules for constraint activity identification by inspection. The same concept applied to goal programming allows direct evaluation of deviational variables. Goal interpretation with dominance conditions obviates the use of deviational variables and permits easy parametric analysis. Examples for helical spring design and machining economics of a turning operation are presented.

Regional Monotonicity in Optimum Design

Regional monotonicity is introduced to identify dominant constraints and simplify non-monotonic expressions in optimum design problems. The technique is applied to obtain a simple solution for the design of an explosive-actuated cylinder, a problem formulated and solved by various methods in the literature. The concept of regional monotonicity provides an extension to the method of monotonicity analysis. Computational work is reduced to a simple plotting over a narrow interval. The design rules developed assist in quick parametric analysis.

A Stochastic Ordering Induced by a Concept of Positive Dependence and Monotonicity of Asymptotic Test Sizes

An ordering of distributions related to a concept of positive dependence is studied and stochastic monotonicity with respect to this ordering is established for a wide class of two-sample test statistics. Asymptotic conservativeness of test sizes under certain departures from independence between samples is discussed. For example, if the observations are paired and the joint density is positive semidefinite then tests such as Kolmogorov-Smirnov, $\chi^2$ and Cramer-von Mises, as well as a large class of linear rank tests, are shown to be asymptotically conservative.

Generalized inverse-positivity and splittings of M- matrices

The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A −1⩾0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A= sI− B, s>0, B⩾0, where the powers of ( 1 s )B converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere.

On the entropy production in chemically reacting Systems

The concept of monotonicity of entropy production is discussed regarding its application to chemically reacting systems. It is concluded that the concept cannot be generally applied even to elementary reactions in ideal homogeneous systems.

Approximations for M/G/s queues

For some parameters (mean waiting time, mean queue size, probability that a customer has to wait) of M/G/s queues with a constant or a variable arrival rate bounds are obtained. In the proofs the concept of system approximation and monotonicity is used.

A generalization of monotonicity applied to the DC analysis of nondifferentiable resistive networks

The dc analysis of an important class of nondifferentiable nonlinear resistive networks is considered which include both piecewise linear and continuously differentiable cases. A local \mu -functional is introduced to generalize a concept of monotonicity, and its useful properties are shown. The existence and uniqueness property of dc operating points is investigated in this class of resistive networks in terms of the local \mu -functional. Relations to previous concepts such as positive definiteness and monotone operators are clarified.

Axiomatisations of the average and a further generalisation of monotonic sequences

A bounded monotonic sequence is convergent. This paper shows that a bounded sequence which is g-monotonic (to be defined) also converges. The proof generalises one attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions: the first axiomatises the notion of “average“ and the second generalises the concept of monotonicity.

Realization of Arbitrary Logic Functions by Completely Monotonic Functions and Its Applications to Threshold Logic

Abstract—The concept of mutual monotonicity developed in the previous paper[1] is applied to the compound synthesis of an arbitrary logic function by a combination of completely monotonic functions or in most cases by threshold functions.

Shock-Producing Excitation in a Viscous Fluid

If Burgers' equation, vt+vvx=δvxx, is used as a descriptor of the propagation of waves of finite amplitude in a viscous fluid, one can give a characterization of the set of boundary functions that are shock-producing. The boundary conditions assumed are v(0,t)=a(t), vx(0,t)=b(t). Conditions for the appearance of shocks are given in terms of the quadratic form a2(t)−2δb(t) by means of the concept of complete monotonicity. The results obtained are an extension of those obtained by the author in “Mathematical Advances in the Theory of One Dimensional Flow,” in Symposium on Apollo Applications, Huntsville, Alabama, January 1966. [Work supported by the National Aeronautics and Space Administration.]