Abstract: This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone econometric function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural quantile functions using data on Vietnam veteran status and earnings.

Abstract: The majorize-minimize (MM) optimization technique has received considerable attention in signal and image processing applications, as well as in statistics literature. At each iteration of an MM algorithm, one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of iterates that reduces the cost function monotonically. A well-known special case of MM methods are expectation-maximization algorithms. In this paper, we expand on previous analyses of MM, due to Fessler and Hero, that allowed the tangent majorants to be constructed in iteration-dependent ways. Also, this paper overcomes an error in one of those earlier analyses. There are three main aspects in which our analysis builds upon previous work. First, our treatment relaxes many assumptions related to the structure of the cost function, feasible set, and tangent majorants. For example, the cost function can be nonconvex and the feasible set for the problem can be any convex set. Second, we propose convergence conditions, based on upper curvature bounds, that can be easier to verify than more standard continuity conditions. Furthermore, these conditions allow for considerable design freedom in the iteration-dependent behavior of the algorithm. Finally, we give an original characterization of the local region of convergence of MM algorithms based on connected (e.g., convex) tangent majorants. For such algorithms, cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods. This expanded treatment widens the scope of the MM algorithm designs that can be considered for signal and image processing applications, allows us to verify the convergent behavior of previously published algorithms, and gives a fuller understanding overall of how these algorithms behave.

Abstract: We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

Abstract: This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: R k ,[0,1[ k ,N0‐the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation invariant positive definite and conditionally negative definite functions on euclidean spaces via completely monotonic functions and Bernstein functions. It is therefore important to be able to decide complete monotonicity of a given function. We combine complete monotonicity with complex analysis via the relation to Stieltjes functions and Pick functions and we give a survey of the many interesting relations between these classes of functions and completely monotonic functions, logarithmically completely monotonic functions and Bernstein functions. In Section 6 it is proved that logx ( x) and 0 (x) are logarithmically completely monotonic (where ( x) = 0 (x)/( x)), and these results are new as far as we know. We end with a list of completely monotonic functions related to the Gamma function.

Abstract: In this paper, we propose a solution to the problem of adaptive control and parameter estimation in systems with unstable target dynamics. Models of uncertainties are allowed to be nonlinearly parameterized, and required to be smooth and monotonic functions of linear functionals of the parameters. The mere assumption of existence of nonlinear operator gains for the target dynamics is sufficient to guarantee that system solutions are bounded, reach a neighborhood of the target set, and the mismatches between the modeled uncertainties and their compensator converge to zero. With respect to parameter convergence, a standard persistent excitation condition suffices to ensure that it is exponential. When a weaker, nonlinear version of persistent excitation is satisfied, asymptotic convergence is guaranteed. The spectrum of possible applications ranges from tyre-road slip control to asynchronous message transmission in spiking neural oscillators.

Abstract: This paper deals with nonparametric maximum likelihood estimation for Gaussian locally stationary processes. Our nonparametric MLE is constructed by minimizing a frequency domain likelihood over a class of functions. The asymptotic behavior of the resulting estimator is studied. The results depend on the richness of the class of functions. Both sieve estimation and global estimation are considered. Our results apply, in particular, to estimation under shape constraints. As an example, autoregressive model fitting with a monotonic variance function is discussed in detail, including algorithmic considerations. A key technical tool is the time-varying empirical spectral process indexed by functions. For this process, a Bernstein-type exponential inequality and a central limit theorem are derived. These results for empirical spectral processes are of independent interest.

Abstract: The notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex, lower semicontinuous, and proper function and any (not necessarily symmetric) continuous linear positive operator. It was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing Fitzpatrick functions. In this paper, we study Fitzpatrick functions of continuous linear monotone operators defined on a Hilbert space. A novel characterization of skew operators is presented. A result by Brezis and Haraux is reproved using the Fitzpatrick function. We investigate the Fitzpatrick function of the sum of two operators, and we show that a known upper bound is actually exact in finite-dimensional and more general settings. Cyclic monotonicity properties are also analyzed, and closed forms of the Fitzpatrick functions of all orders are provided for all rotators in the Euclidean plane.

Abstract: Compared to the correlation-consistent basis sets, it is not known if polarization-consistent pc-n basis sets, which were initially developed for HF and DFT calculations, can provide a monotonic and faster convergence toward the basis-set limit for results at correlated levels as well as better accuracy for a similar number of basis functions. It is also not known whether the pc-n basis sets can compute second derivatives of energy, such as nuclear magnetic shielding tensors, efficiently. To address these questions, the pc-n (n = 1−4), cc-pVxZ, and/or aug-cc-pVxZ (x = D, T, Q, 5, and 6) basis sets were used to compute the molecular and/or spectroscopic parameters of H2, H2O, and NH3 at the RHF, B3-LYP, MP2, and/or CCSD(T) levels of theory. The results show that compared to the cc-pVxZ and/or aug-cc-pVxZ basis sets the pc-n basis sets yield faster convergence toward the basis-set limit but equivalent molecular and/or spectroscopic parameters in the basis-set limit at the RHF, DFT, MP2, and CCSD(T) levels. ...

Abstract: We find an explicit formula for the invariant density of a generalized β-map. This allows us to also find an explicit formula for the invariant density of Chebyshev map and discuss the monotonicity of the asymptotic average for such maps. Our results are based on a generalization of works of Parry (Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416; 15 (1964), 95–105).

Abstract: This chapter presents nonlinear variational inequalities of elliptic type that arise in various mathematical modelings of nonlinear phenomena such as reaction–diffusion problems, free boundary problems, and finance problems. The approach to these problems is based on the theory of nonlinear mappings of the monotone type. It introduces the concepts of maximal monotonicity, semi-monotonicity, pseudo-monotonicity and those of type M in abstract settings, and gives some existence results for abstract functional equations of the form: (1) f ∈ Au + Bu , where A and B are multivalued (nonlinear) mappings from a real reflexive Banach space X into its dual space X*. The solvability of (1) is reduced to investigate the range of A + B , and the regular approximation theory is especially important for solving many concrete problems. This chapter also studies the convergence of monotone mappings in the graph sense and convex functions.

Abstract: The problem of constructing globally convergent, reduced-order observers for general nonlinear systems is addressed. It is shown, by means of the Immersion and Invariance (I&I) methodology, that an asymptotic estimate of the unknown states can be obtained by rendering invariant and attractive an opportune manifold in the extended state space. Current results on nonlinear observer design require that the nonlinearities appearing in the system equations are either linear functions of the unmeasured states or monotonic functions of a linear combination of the states. In this paper we relax these two assumptions by allowing for a wider class of nonlinearities to appear in the system equations. The proposed approach is applied on several examples including a perspective vision system and a general two-degrees-of-freedom (2DOF) mechanical system.

Abstract: Knowing when a process has changed would simplify the search for and identification of the special cause. In this paper, we propose a maximum-likelihood estimator for the change point of the process fraction non-conforming without requiring knowledge of the exact change type a priori. Instead, we assume the type of change present belongs to a family of monotonic changes. We compare the proposed change-point estimator to the maximum-likelihood estimator for the process change point derived under a simple step change assumption. We do this for a number of monotonic change types and following a signal from a binomial cumulative sum (CUSUM) control chart. We conclude that it is better to use the proposed change point estimator when the type of change present is only known to be monotonic. The results show that the proposed estimator provides process engineers with an accurate and useful estimate of the time of the process change regardless of the type of monotonic change that may be present. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract: A five-parameter family of planar vector fields, which models the dynamics of certain populations of predators and their prey, is discussed. The family is a variation of the classical Volterra-Lotka system by taking into account group defense strategy, competition between prey and competition between predators. Also we initiate computer-assisted research on time-periodic perturbations, which model seasonal dependence. We are interested in persistent features. For the planar autonomous model this amounts to structurally stable phase portraits. We focus on the attractors, where it turns out that multi-stability occurs. Further, the bifurcations between the various domains of structural stability are investigated. It is possible to fix the values of two of the parameters and study the bifurcations in terms of the remaining three. Here we find several codimension 3 bifurcations that form organizing centres for the global bifurcation set. Studying the time-periodic system, our main interest is the chaotic dynamics. We plot several numerical examples of strange attractors.

Abstract: Recall [7, 11, 14] that a function f is called completely monotonic on an interval I if f has derivatives of all orders on I and 0 ≤ (−1)f (x) < ∞ for all k ≥ 0 on I. The well known Bernstein’s Theorem [14, p. 161] states that f ∈ C[(0,∞)] if and only if f(x) = ∫∞ 0 e−xs dμ(s), where μ is a nonnegative measure on [0,∞) such that the integral converges for all x > 0. The class of completely monotonic functions on I is denoted by C[I]. For more information on C[I], please refer to [5, 6, 7, 8, 9, 10, 11, 14] and the references therein. By using the convolution theorem of Laplace transforms, the increasingly monotonicity of x ∣∣ψ(i)(x + 1)∣∣ is presented in [9, 10]: The function xα∣∣ψ(i)(x + 1)∣∣ is strictly increasing in (0,∞) if and only if α ≥ i, where ψ(x), the logarithmic derivative of the classical Euler’s gamma function Γ(x), is called psi function and ψ(x) for i ∈ N are called polygamma functions. In [3], in order to show the subadditive property of the function ψ(a + e), it was proved that the function xψ′(x+a) is strictly increasing on [0,∞) for a ≥ 1. In [2], it was also showed, using the convolution theorem of Laplace transforms, that the function x ∣∣ψ(k)(x)∣∣ for k ≥ 1 is strictly decreasing in (0,∞) if and only if c ≤ k and is strictly increasing in (0,∞) if and only if c ≥ k+1. In [4], the monotonicity of the more general function x ∣∣ψ(i)(x+ β)∣∣ was studied without using the convolution theorem of Laplace transforms and, except the above results, the following conclusions are obtained: For i ∈ N, α > 0 and β ≥ 0, (1) the function x ∣∣ψ(i)(x+ β)∣∣ is strictly increasing in (0,∞) if (α, β) ∈ {α ≥

Abstract: Common approaches to monotonic regression focus on the case of a unidimensional covariate and continuous response variable. Here a general approach is proposed that allows for additive structures where one or more variables have monotone influence on the response variable. In addition the approach allows for response variables from an exponential family, including binary and Poisson distributed response variables. Flexibility of the smooth estimate is gained by expanding the unknown function in monotonic basis functions. For the estimation of coefficients and the selection of basis functions a likelihood-based boosting algorithm is proposed which is simple to implement. Stopping criteria and inference are based on AIC-type measures. The method is applied to several datasets.

Abstract: We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be matrix multiplication and the Cartesian product of two lists. The type checking problem for the type system is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic. Furthermore, a method that infers polynomial size dependencies for a non-trivial class of function definitions is suggested.

Abstract: Discrete-time analogues of predator–prey models with monotonic and nonmonotonic functional responses are introduced, respectively. The discrete-time analogues are considered to be numerical discretizations of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models with monotonic functional responses. Moreover, it is the first time that multiplicity of periodic solutions are studied when modeled with nonmonotonic functional responses. Unlike other types of functional responses, nonmonotone functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation Lx = λ Nx are inapplicable to our case and some new arguments are employed for the first time.

Abstract: A parameterized family of continuous functions which was considered by the first author is re-visited in the case when they are monotonically increasing. We prove that the functions are not only continuous and strictly increasing but also singular, i.e., their derivatives are zero almost everywhere.

Abstract: In wireless networks, monotonic, strictly subhomogeneous functions have been used to analyze power control algorithms. We provide an alternative analysis based on the observation that such functions are shrinking with respect to certain metrics. These metrics are then used to analyze problems involving two classes of non-monotonic functions. The first class consists of normalized interference functions that can be used to compute the maximum achievable signal-to-interference ratio under power constraints. The second class consists of absolutely subhomogeneous functions that are used to approximate power control dynamics with outages. The eigensolutions of monotonic, strictly subhomogeneous functions are characterized as part of this development.

Abstract: We present characterizations of some generalized convexity properties of functions with the help of a general subdifferential. We stress the case of lower semicontinuous functions. We also study the important case of marginal functions and we provide representation results.

Abstract: This paper defines invariant utility functions to continuous monotonic transformations. We also define transformation invariance as the condition in which the certain equivalent of a lottery follows a continuous monotonic transformation that is applied to its outcomes. We show that invariant utility functions uniquely satisfy transformation invariance, and we illustrate how knowledge of an invariance criterion determines the functional form of the utility function. This formulation extends the widely used notions of invariance to shift and scale transformations on the outcomes of a lottery to more general monotonic transformations. Moreover, we interpret any continuous and strictly monotonic utility function as an invariant utility function to a composite monotonic transformation. Furthermore, we show how this composite transformation uniquely characterizes the utility function up to a linear transformation. We derive the invariance formulations that lead to the assignment of hyperbolic absolute risk-averse (HARA) utility functions, linear plus exponential utility functions, and a two-parameter power-logarithmic utility function that generalizes the logarithmic utility function. We work through several examples to illustrate the approach.

Abstract: The widely used “sequential order” rules in the generalized two-dimensional (2D) correlation spectroscopy were adopted from the mechanical perturbation-based 2D infrared, where dynamic spectral intensity variation must be a simple sinusoid. 2D correlation analysis is fundamentally a form of parametrization of the integrated or overall relationship between two variable quantities. In generalized 2D correlation spectroscopy, however, the dynamic spectral intensity variations are generally nonperiodic and monotonic, and spectral intensity changes are largely instantaneous. The sequential orders in generalized situations are therefore localized. It is naturally necessary and important to testify whether the analysis result obtained by using the sequential order rules is consistent with the local sequential order of events, which reflects the real sequential order in generalized situations. Unfortunately, this test was not done yet. In this report, the sequential order rules have been tested in the generalized...

Abstract: We address power minimization of earliest deadline first and rate monotonic schedules by voltage and frequency scaling. We prove that the problems are NP-hard, and present (1+∈) fully polynomial time approximation techniques that generate solutions which are guaranteed to be within a specified quality bound (QB= ∈) (say within 1% of the optimal). We demonstrate that our techniques can match optimal solutions when QB is set at 1%, out perform existing approaches [1] even when QB is set at 10%, generate solutions that are quite close to optimal ( 5%) for large 100 node task sets.

Abstract: A straightforward generalization of the classical inverse of a real function based on reflections leads to several insuperable difficulties. We introduce a new type of inverse w.r.t. monotone bijections $\phi$ that is determined by the direction of the base vectors of the real Euclidean plane. Inverting a monotone function in the real plane does not necessarily result in a function. Given an increasing real function $f$, Schweizer and Sklar geometrically construct a set of inverse functions. We will largely extend their construction to our new concept of $\phi$-inverses, also incorporating decreasing functions $f$. Furthermore, the geometrical and algebraical aspects of our approach are elaborated comprehensively. Special attention goes to the symmetry of a monotone function $f$ w.r.t. some monotone bijection $\phi$.

Abstract: This correspondence investigates the asymptotic performance of the discrete-time and continuous-time, time-varying, minimum-variance, fixed-interval smoothers. Comparison theorems are generalized to provide sufficient conditions for the monotonic convergence of the underlying Riccati equations. Under these conditions, the energy of the estimation errors asymptotically approach a lower bound and attain lscr2 /L2 stability