Tchebycheff Systems Research Articles

Correction to: Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems

Correction to: Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems

Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems

This paper studies tight upper bounds for the mean and higher moments of the steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. These distributions are determined by having one mass at 0 or at the upper limit of support.

Extremal models for the [formula omitted] waiting-time tail-probability decay rate

We identify interarrival-time and service-time distributions that yield tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability in the GI∕GI∕K queue, given the first two moments of those underlying distributions plus alternative additional information. We apply Tchebycheff systems after providing a concise review.

Locally optimal designs for some binary dose-response models.

In this paper, we consider the problem of seeking locally optimal designs for nonlinear dose-response models with binary outcomes. Applying the theory of Tchebycheff Systems and other algebraic tools, we show that the locally D-, A-, and c-optimal designs for three binary dose-response models are minimally supported in finite, closed design intervals. The methods to obtain such designs are presented along with examples. The efficiencies of these designs are also discussed.

Optimal Sensor Scheduling in Batch Processes Using Convex Relaxations and Tchebycheff Systems Theory

In several industrial processes, measurements are limited due to time or cost constraints. Hence, scheduling of these measurements to obtain maximally informative and accurate estimates of the states becomes important. The focus of this contribution is to determine a schedule of measurements that maximizes the quality of the estimates at the end of a finite time horizon. This work finds applications in batch chemical processes. The estimate covariance matrix as obtained by applying standard Kalman filtering theory is approximated and the original problem is relaxed and reformulated as a cone program. For a certain class of systems, the approximated covariance matrix is parameterized in terms of a point belonging to a moment space induced by a Tchebycheff system. The theory of Tchebycheff systems is used to determine an improved upper bound on the guaranteed minimum number of measurements. A tractable solution methodology to obtain the optimal cost and a parsimonious discrete schedule using the theory of cone programming, generalized barrier functions and Tchebycheff systems is described.

Optimal designs for some selected nonlinear models

Some design aspects related to three complex nonlinear models are studied in this paper. For Klimpel׳s flotation recovery model, it is proved that regardless of model parameter and optimality criterion, any optimal design can be based on two design points and the right boundary is always a design point. For this model, an analytical solution for a D-optimal design is derived. For the 2-parameter chemical kinetics model, it is found that the locally D-optimal design is a saturated design. Under a certain situation, any optimal design under this model can be based on two design points. For the 2n-parameter compartment model, compared to the upper bound by Carathéodory׳s theorem, the upper bound of the maximal support size is significantly reduced by the analysis of related Tchebycheff systems. Some numerically calculated A-optimal designs for both Klimpel׳s flotation recovery model and 2-parameter chemical kinetic model are presented. For each of the three models discussed, the D-efficiency when the parameter misspecification happens is investigated. Based on two real examples from the mining industry, it is demonstrated how the estimation precision can be improved if optimal designs would be adopted. A simulation study is conducted to investigate the efficiencies of adaptive designs.

Markov's Moment Problem and the Range of Option Prices

A fundamental problem arising in Mathematical Finance is to construct arbitrage-free probability models from quoted prices of a finite number of futures and european-type derivatives, as has been achieved in (Davis and Hobson, 2007). This is not a parameter callibration procedure. If the data satisfy a certain condition, close to the no-arbitrage condition, then a no-arbitrage model can be constructed, including the probability space and associated processes. The problem can be cast within the framework of Markov's Moments Problems (MMP) (Krein & Nudelmann,1973). Moments apprear as prices corresponding to certain pay-off well defined functions. A central role in these problems is played by characterizing non-negative linear combinations of the functions as future pay-offs, and determining the present prices. Tchebycheff Systems (Karlin & Studden,1966) have been fruitfully applied for this purpose, in that their linear combinations exhibit managable properties of their zeroes together with their non-negativity. In the case of call-options, these nice properties are partially lost by the presence of 'interval zeroes', that is, whole intervals where their linear combinations may be zero. The full extent of the theory has to be adapted to this case by explicitly constructing its main objects and results. When so doing, the call-case reveals many properties which allow to establish the existence of a host of Markov transition- mean-preserving kernels, from which many probabilistic processes may be indicated and constructed. This material is not part of the classical literature, but owes its interpretation from the construction in Davis and Hobson's article.Additionally, quadratures of functions other than those originally present in the moments' definition, are possible, and precise lower and upper bounds may be given to them. Upper bounds for the integrals of convex functions are readily obtained, and form the nucleus of much that can be said in terms of Choquet's ordering of measure (P. A. Meyer,1966). Lower bounds for the same class of functions are more difficult to obtain exactly, since the sparseness of the strikes sets a limit to the precision we may aspire to exactly bound a large class of functions. Some steps in developing lower bounds for convex functions quadratures are indicated. Part of the MMP's Programme is to settle the question as to when is it possible to decare a full rectangle to consist of moments by testing just two of its vertices definining an off diagonal. This is another problem originally devised by Markov himself. This is done in the last section and has an interesting interpretions in terms of bid-ask prices.

Likelihood ratio tests for positivity in polynomial regressions

A polynomial that is nonnegative over a given interval is called a positive polynomial. The set of such positive polynomials forms a closed convex cone K. In this paper, we consider the likelihood ratio test for the hypothesis of positivity that the estimand polynomial regression curve is a positive polynomial. By considering hierarchical hypotheses including the hypothesis of positivity, we define nested likelihood ratio tests, and derive their null distributions as mixtures of chi-square distributions by using the volume-of-tubes method. The mixing probabilities are obtained by utilizing the parameterizations for the cone K and its dual provided in the framework of Tchebycheff systems for polynomials of degree at most 4. For polynomials of degree greater than 4, the upper and lower bounds for the null distributions are provided. Moreover, we propose associated simultaneous confidence bounds for polynomial regression curves. Regarding computation, we demonstrate that symmetric cone programming is useful to obtain the test statistics. As an illustrative example, we conduct data analysis on growth curves of two groups. We examine the hypothesis that the growth rate (the derivative of growth curve) of one group is always higher than the other.

Optimal Plant Friendly Input Design for System Identification

The primary objective in solving optimal input design problems is to obtain maximally informative inputs to be used as perturbation signals in system identification experiments. In plant-friendly identification, the designer has to respect constraints on experiment time, input and output amplitudes or input move sizes. This work focuses on plant friendly input design with constraints on input move size and output power. We present a convex relaxation to the problem of designing an informative input subject to input move size and output power constraints. The problem is finitely parametrized using ideas from Tchebycheff systems and reformulated as a SemiDefinite Programme.

Total Positivity: an application to positive linear operators and to their limiting semigroups

Some shape-preserving properties of positive linear operators, involving higher order convexity and Lipschitz classes, are investigated from the point of view of weak Tchebycheff systems and total positivity in the sense of Karlin [8]. The same properties are shown to be fulfilled by the strongly continuous semigroup \((T(t))_{t\geq 0}\), if any, generated by the iterates of the relevant operators, in the spirit of Altomare's theory.

On a class of weak Tchebycheff systems

In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the form**requiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.

Gaussian interval quadrature formula

We prove the existence of a Gaussian quadrature formula for Tchebycheff systems, based on integrals over non-overlapping subintervals of arbitrary fixed lengths and the uniqueness of this formula in the case the subintervals have equal lengths.

Total Positivity and Convexity Preservation

In this paper we define convexity and rational convexity preservation of systems of functions and we show that total positivity and rational convexity preservation are equivalent. We also characterize certain convexity preserving systems in terms of weak Tchebycheff systems. Curve intersections and curvatures of Bézier curves are also studied.

Strictly Totally Positive Systems

We combine methods of linear algebra and analysis to obtain new results on splicing, domain extension, and integral representation of Tchebycheff and weak Tchebycheff systems.

Tchebycheff Systems for Probabilistic Analysis

SYNOPTIC ABSTRACTWe introduce the theory of Tchebycheff systems and highlight theory relevant to probability applications—especially approximation of stochastic models. We restate and interpret definitions and results of Karlin and Studden (1966) and other sources so that their probabilistic significance is readily apparent. We also provide two original theorems and show how they can be used to identify Tchebycheff systems useful in obtaining queueing-approximation error bounds. Several applications are described in sufficient detail to illustrate the usefulness of Tchebycheff systems.

Representations of weak tchebycheff systems

In this note, representations of weak Tchebycheff systems of real valued functions or analytic functions are shown.

On the solvability of bivariate Hermite-Birkhoff interpolation problems

In a recent paper by Hack (1987), certain bivariate polynomial Hermite-Birkhoff interpolation problems are considered and sufficient conditions for unique solvability are obtained by applying a general result on interpolation with tensor-functionals. In this paper, using a matricial approach and with the guideline of a result due to Stenger (1968), these conditions are shown to be also necessary and expressed in terms of the unisolvence of certain univariate interpolation problems. The results are valid not only for the polynomial case but also for interpolation with bivariate functions built from univariate extended complete Tchebycheff (ECT) systems.

Extensions of endpoint equivalent and periodic Tchebycheff systems

Let IZ > 0 be an integer. Given an ordered set A of cardinality at least II + 2, a sequence of real functions (yi}r= 0 defined on A will be called a Tchebycheff system (T-system) on A, provided that, for every sequence 0 0, *.., t,} of points from A such that to y1+ 2) when the functions are periodic with period equal to the length of the set A and are a T-system on A, the set A containing either its infimum, II, or its supremum, I,. Similar to periodicity but not totally coincident is the concept of “endpoint equivalence.” A real function f defined on a real, nonempty set A will be called endpoint equivalent provided that, for all sequences (xn}, { y,} in A, such that x, + I, and yn -+ I,, the limits lim f (x,), lim f (y) exist (finite or infinite), and are equal. A T-system defined on such a set A will be called endpoint equivalent if the functions in it are endpoint equivalent. Although the functions considered should be real valued, one can often permit A to be a subset of the extended real number system. The advantage of this will be seen in what follows.

Uniform approximation by generalized splines with free knots

Recently we have examined best approximation by certain classes of generalized splines with fixed knots. The purpose of this paper is to study best approximation by similar classes of generalized splines, but with free knots. Our main results concern segment approximation and approximation by continuously composed Tchebycheff systems

Generalized polynomials of minimal norm

We consider the problem of finding optimal generalized polynomials of minimal L p norm (1 ⩽ p < ∞). As an application of our results we obtain Gaussian quadrature formulas for extended Tchebycheff systems.