Legendre polynomial kernel estimation of a density function with censored observations and an application to clinical trials

Let f(x), x ∈ ?^M, M ≥ 1, be a density function on ?^M, and X₁, …., X_n a sample of independent random vectors with this common density. For a rectangle B in ?^M, suppose that the X's are censored outside B, that is, the value X_k is observed only if X_k ∈ B. The restriction of f(x) to x ∈ B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M ≥ 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter‐plane onto the strip {(u, v): 0 ≤ u ≤ 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author [2]. The extrapolator has the following form: For m ≥ 1 and n ≥ 1, let K_{m, n}(x) be the classical kernel estimator of f(x), x ∈ B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E|K_{m, n}(x) ? f(x)| for x ∈ C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f(x), x ∈ C, if f is sufficiently smooth and if m and n both tend to ∞ in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc.     

On the graph of a function over a prime field whose small powers have bounded degree

Let f be a function from a finite field with a prime number p of elements, to . In this article we consider those functions f(X) for which there is a positive integer with the property that f(X)ⁱ, when considered as an element of , has degree at most p−2−n+i, for all i=1,…,n. We prove that every line is incident with at most t−1 points of the graph of f, or at least n+4−t points, where t is a positive integer satisfying n>(p−1)/t+t−3 if n is even and n>(p−3)/t+t−2 if n is odd. With the additional hypothesis that there are t−1 lines that are incident with at least t points of the graph of f, we prove that the graph of f is contained in these t−1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t−1 and prove the conjecture for t=2 and t=3. These results apply to functions that determine less than directions. In particular, the proof of the conjecture for t=2 and t=3 gives new proofs of the result of Lovász and Schrijver [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2(p−1)/3 directions.     

Canonical connections on paracontact manifolds

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A -homothetic transformation is determined as a special gauge transformation. The η-Einstein manifold are defined, it is proved that their scalar curvature is a constant, and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with -homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.      

Conformal paracontact curvature and the local flatness theorem

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.     

Index 2 DAEs in Plasticity Theory

The equations of rate‐independent elastoplasticity form a differential‐algebraic equation (DAE) with discontinuities. For the numerical solution, implicit Runge‐Kutta methods are applied and combined with the return mapping strategy of computational plasticity. It turns out that the convergence order depends crucially on the switching point detection. Further, it is shown that algebraically stable Runge‐Kutta methods preserve the contractivity of the elastoplastic flow.     

Convergence of Generic Infinite Products of Order-Preserving Mappings

In this paper we establish several results concerning the asymptotic behavior of (random) infinite products of generic sequences of order-preserving mappings on intervals of an ordered Banach space. In addition to weak ergodic theorems we also obtain convergence to a unique common fixed point.     

Scalable Formation of Concentrated Monodisperse Lignin Nanoparticles by Recirculation-Enhanced Flash Nanoprecipitation

A highly controllable and scalable process for fabrication of large amounts of concentrated lignin nanoparticles (LNPs) is reported. These lignin core nanoparticles are formed through flash nanoprecipitation, however, scaling up of the fabrication process requires fundamental understanding of their operational formation mechanism and surface properties. It is shown how a semicontinuous synthesis system with a recirculation loop makes it possible to produce flash precipitated lignin nanoparticles in large amounts for practical applications. The roles of the process parameters, including flow rates and lignin concentration, are investigated and analyzed. The results indicate that the LNPs are formed by a process of continuous burst nucleation at the point of mixing without diffusive growth, which yields nanoparticles of highly uniform size following a modified LaMer nucleation and growth mechanism. This mechanism makes possible facile process control and scale-up. Effective control of the resulting nanoparticle size is achieved through the initial concentration of lignin in the injected solution. The impressive capability to produce suspensions of any predesigned multimodal distribution is demonstrated. The resulting nanofabrication technique can produce large volumes of concentrated LNP suspensions of high stability and tightly controlled size distributions for biological or agricultural applications.     

Asymptotic Behavior of Solutions of Evolution Equations and the Construction of Holomorphic Retractions

We first study the asymptotic behavior of nonlinear semigroups with holomorphic generators, and then use our results, inter alia, to construct holomorphic retractions onto the fixed point sets of holomorphic self-mappings of bounded convex domains in a complex Banach space.     

DAEs and PDEs in elastic multibody systems

Elastic multibody systems arise in the simulation of vehicles, robots, air- and spacecrafts. They feature a mixed structure with differential-algebraic equations (DAEs) governing the gross motion and partial differential equations (PDEs) describing the elastic deformation of particular bodies. We introduce a general modelling framework for this new application field and discuss numerical simulation techniques from several points of view. Due to different time scales, singular perturbation theory and model reduction play an important role. A slider crank mechanism with a 2D FE grid for the elastic connecting rod illustrates the techniques. This revised version was published online in June 2006 with corrections to the Cover Date.