Support-type properties of generalized convex functions

Chebyshev systems induce in a natural way a concept of convexity. The functions convex in this sense behave in many aspects similarly to ordinary convex functions. In this paper support-type properties are investigated. Using osculatory interpolation, the existence of support-like functions is established for functions convex with respect to Chebyshev systems. Unique supports are determined. A characterization of the generalized convexity via support properties is presented.     

Polynomials with graphs of maximum length

In this paper we find the pairs of functionsg, G (g<G) such that the maximum length of the graph of polynomials of given degree contained betweeng andG is attained on one of the two snakes generated by the functionsg andG. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 61–69, January, 1999.     

Bifractional inequalities and convex cones

In this paper we give a systematic study of a class of linear inequalities related to convex cones in linear spaces. In particular, Chebyshev and Andersson type inequalities are discussed. Some classical and new inequalities are derived from the results.     

Optimal Trajectories and Guidance Schemes for Ship Collision Avoidance

The best strategy for collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between the host ship and the intruder ship. In a restricted waterway area, two main constraints must be satisfied: the lateral deviation of the host ship from the original course is to be contained within certain limits; the longitudinal distance covered by the host ship is to be subject to a prescribed bound. At the maximin point of the encounter, the time derivative of the relative distance vanishes; this yields an inner boundary condition (orthogonality between the relative position vector and the relative velocity vector) separating the main phases of the maneuver: the avoidance and recovery phases. In this way, the optimal trajectory problem (a Chebyshev problem) can be converted into a Bolza problem with an inner boundary condition. Numerical solutions are obtained via the multiple-subarc sequential gradient-restoration algorithm (SGRA). Because the optimal trajectory is not suitable for real-time implementation, a guidance scheme approximating the optimal trajectory in real time is to be developed. For ship collision avoidance, the optimal trajectory results show that the rudder angle time history has a bang-bang form characterized by the alternation of saturated control subarcs of opposite signs joined by rapid transitions. Just as the optimal trajectory can be partitioned into three phases (avoidance phase, recovery phase, steady phase), a guidance trajectory can be constructed in the same way. For the avoidance and recovery phases, use of decomposition techniques leads to an algorithm computing the time lengths of these phases in real time. For the steady phase, a feedback control scheme is used to maneuver the ship steadily. Numerical results are presented. Portions of this paper were presented by the senior author at the 13th International Workshop on Dynamics and Control, Wiesensteig, Germany, 22-26 May 2005, in honor of George Leitmann. This research was supported by NSF Grant CMS-02-18878.     

Multi-color display and its model for a white OLED combined with optical color filters

A bright white organic light-emitting diode (OLED) has been successfully fabricated in which a greenish-white emitting host Bis-(2-(2-hydroxyphenyl) benzothiazole)zinc (Zn(BTZ)₂) is doped with an orange-red fluorescent dye 5,6,11,12-tetraphenylnaphthacene (rubrene). The pure white light emission with CIE coordinates of (0.341, 0.334) is achieved at the driving voltage of 20 V, and the corresponding luminance and external quantum efficiency are 4000 Cd/m² and 0.63%. On the other hand, the pure red, green, and blue (RGB) emissions are obtained from the white OLED combined well with RGB custom-built optical color filters (CFs) with peaks at 616, 556, and 456 nm, respectively. Moreover, based upon the color mixture principles, we build a model of multi-color display mixed by the pure RGB emissions under the controllable bias.     

Global approximations to the principal real-valued branch of the Lambert W-function

W(z) is defined implicitly as the root of W exp(W) = z. It is shown that a simple analytic approximation has a relative error of less than 5% over the whole domain z ε [−exp(−1), ∞] of the principle branch—sufficiently accurate so that four Newton iterations refine this approximation to a relative error smaller than 1.E-12. As a second form of global approximation, the W-function is expanded as a series of rational Chebyshev functions TB_j in a shifted, logarithmic coordinate with an error that decreases exponentially fast with the series truncation.     

利用非线性提升的信号去噪

本文提出了一种基于非线性滤波和提升格式进行信号去噪的算法。利用提升格式设计的灵活性取非线性滤波为预测算子 ,数值算例表明该方法的去噪效果优于仅使用非线性滤波去噪。     

Chebyshev Centers in Hyperplanes of c 0

We give a full characterization of the closed one-codimensional subspaces of c ₀, in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.