Monotone polygon

About: Monotone polygon is a research topic. Over the lifetime, 14573 publications have been published within this topic receiving 270286 citations.

Monotone Regression Splines in Action

Abstract: Piecewise polynomials or splines extend the advantages of polynomials to include greater flexibility, local effects of parameter changes and the possibility of imposing useful constraints on estimated functions. Among these constraints is monotonicity, which can be an important property in many curve estimation problems. This paper shows the virtues of monotone splines through a number of statistical applications, including response variable transformation in nonlinear regression, transformation of variables in multiple regression, principal components and canonical correlation, and the use of monotone splines to model a dose-response function and to perform item analysis. Computational and inferential issues are discussed and illustrated.

Classification of solutions for an integral equation

Abstract: Let n be a positive integer and let 0 < α < n. Consider the integral equation We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn. This solves an open problem posed by Lieb 12. The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well-known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.

4. Monotone Dynamical Systems

Abstract: This chapter surveys a restricted but useful class of dynamical systems, namely, those enjoying a comparisonprinciple with respect to a closed order relation on the state space.Such systems, variously called monotone, order-preserving orincreasing, occur in many biological, chemical, physical and economicmodels.