This article concerns the existence of non-oscillatory solutions for non-linear, nonhomogeneous differential equations. In these equations, the unknown function depends on distributed arguments that can be retarded or advanced and have variable delays. Using contraction mappings, we show the existence of solutions, and estimate their norms. Then our results are extended to dynamic equations on ...

We present an oscillation criterion for first order delay dynamic equations on time scales, which contains well-known criteria for delay differential equations and delay difference equations as special cases. We illustrate our results by applying them to various kinds of time scales.

Evaluation routines are essential for any application that uses triangular B-spline surfaces. This paper describes an algorithm to efficiently evaluate triangular B-spline surfaces with arbitrary many variables. The novelty of the algorithm is its generality: there is no restriction on the degree of the B-spline surfaces or on the dimension of the domain. Constructing an evaluation graph allows...

We consider orthogonal polynomials {pn,N(x)}n=0 on the real line with respect to a weight w(x) = e (x) and in particular the asymptotic behaviour of the coefficients an,N and bn,N in the three term recurrence xπn,N (x) = πn+1,N (x)+bn,Nπn,N (x)+an,Nπn−1,N (x). For one-cut regular V we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrenc...

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term x (t) + a(t)f(x(q(t))) = e(t) on a time scale T. Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.

In this paper, we obtain a direct result and Voronovskaya type asymptotic formula for the exponential-type operators in simultaneous approximation with polynomial growth. In the end, we obtain the recurrence formulae for the central moments, direct results and Voronovskaya type asymptotic formulae for the various mixed summation-integral type operators. 2000 AMS Subject Classification. 41A28, 4...

Consider a time series with missing observations but a known final point. Using control theory ideas we estimate/predict these missing observations. We obtain recurrence equations which minimize sum of squares of a control sequence. An advantage of this method is in easily computable formulae and flexibility of its application to different structures of missing data.

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation p t xΔ t γ Δ q t f x τ t 0, on a time scale T, where γ is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous rd-continuous functions on T. Our results improve and extend ...

This paper is concerned with oscillation of the second-order halflinear delay dynamic equation (r(t)(x)) + p(t)x(τ(t)) = 0, on a time scale T, where γ ≥ 1 is the quotient of odd positive integers, p(t), and τ : T → T are positive rd-continuous functions on T, r(t) is positive and (delta) differentiable, τ(t) ≤ t, and limt→∞ τ(t) = ∞. We establish some new sufficient conditions which ensure that...

Background: Prostate specific antigen (PSA) is an important biomarker to monitor patients after treated with radiation therapy (RT). The aim of this study is to evaluate the relationship between the PSA data and prostate cancer recurrence using the joint modeling. Methods: This historical cohort study was performed on 422 prostate cancer patients. Inclusion criteria included: patients with loc...