An improved algorithm for computing topological entropy

A new algorithm is presented for computing the topological entropy of a unimodal map of the interval. The accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed.     

约束最小二乘估计的优良性

对于线性模型y=Xθ ε,ε服从椭球等高单峰分布,未知参数θ满足不等式约束a′θ≥0,证明了在参数估计优良性的集中概率准则下,θ的约束最小二乘估计θ~*优于最小二乘估计θ.     

q-Binomials and related symmetric unimodal polynomials

The q-binomial coefficients were conjectured to be unimodal as early as the 1850's but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an ‘elementary’ proof using linear algebra. Finally, in 1989, Kathy O'Hara provided a combinatorial proof of the unimodality of the q-binomial coefficients. Very soon thereafter, Doron Zeilberger translated the argument into an elegant recurrence. We introduce several perturbations to the recurrence to create a larger family of unimodal polynomials. We analyse how these perturbations affect the final polynomial and analyse some specific cases.     

Global stability and the Hopf bifurcation for some class of delay differential equation

In this paper, we present an analysis for the class of delay differential equations with one discrete delay and the right‐hand side depending only on the past. We extend the results from paper by U. Fory? (Appl. Math. Lett. 2004; 17 (5):581–584), where the right‐hand side is a unimodal function. In the performed analysis, we state more general conditions for global stability of the positive steady state and propose some conditions for the stable Hopf bifurcation occurring when this steady state looses stability. We illustrate the analysis by biological examples coming from the population dynamics. Copyright © 2007 John Wiley & Sons, Ltd.     

Symmetry and Unimodality in Linear Inference

Distribution-free results beyond Gauss-Markov theory are found under weak assumptions regarding the errors. Symmetry, unimodality, and location-scale families are studied in estimation; nonstandard versions of Gauss–Markov results are given; and distribution-free confidence sets are tightened under symmetry and unimodality of errors. Normal-theory approximate tests are seen to exhibit monotone power in certain classes of symmetric unimodal errors.     

区间上平顶单峰扩张自映射的周期轨道

设ｔ（０＜ｔ＜１）是一个常数，ｎ≥３是奇数，ｍ≥０及ｋ≥１是整数，Ｐ０（ｘ）＝ｘ－１，Ｐｉ（ｘ）＝（ｘ２ｉ－１－１）Ｐｉ－１（ｘ）（ｉ≥１），ｒｍｎ（ｔ）及ｒｋ（ｔ）分别是方程Ｐｍ（ｘ）（ｘ２ｍｎ－２ｘ２ｍ（ｎ－２）－１）－ｔ（ｘ２ｍｎ－１）（ｘ２ｍ＋１）＝０及Ｐｋ－１（ｘ）－ｔ（ｘ２ｋ－１＋１）＝０在（１，＋∞）上的唯一实根，ｆ是闭区间Ｉ＝［０，１］上的峰顶区间长度为ｔ的平顶单峰扩张自映射．本文证明了，若ｆ的扩张常数λ≥ｒｍｎ（ｔ）（或＞ｒｋ（ｔ）），，则ｆ有２ｍｎ（或２ｋ）周期点．此外，本文还指出，当１＜λ＜ｒｍｎ（ｔ）（或≤ｒｋ（ｔ）时，在Ｉ上存在着具有扩张常数λ及峰顶区间长度ｔ却无２ｍｎ（或２ｋ）周期点的平顶单峰扩张自映射     

A counterexample concerning monotone unimodality

In answer to a question of Dharmadhikari and Jogdeo (1976) we show that there exist multivariate distributions which are symmetric, star unimodal, and linear unimodal, but not monotone unimodal.     

C*-algebras associated with interval maps

For each piecewise monotonic map of , we associate a pair of C*-algebras and and calculate their K-groups. The algebra is an AI-algebra. We characterize when and are simple. In those cases, has a unique trace, and is purely infinite with a unique KMS state. In the case that is Markov, these algebras include the Cuntz-Krieger algebras , and the associated AF-algebras . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and -transformations. For the case of interval exchange maps and of -transformations, the C*-algebra coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

     

Non-autonomous periodic systems with Allee effects

A new class of maps called unimodal Allee maps are introduced. Such maps arise in the study of population dynamics in which the population goes extinct if its size falls below a threshold value. A unimodal Allee map is thus a unimodal map with three fixed points, a zero fixed point, a small positive fixed point, called threshold point, and a bigger positive fixed point, called the carrying capacity. In this paper, the properties and stability of the three fixed points are studied in the setting of non-autonomous periodic dynamical systems or difference equations. Finally, we investigate the bifurcation of periodic systems/difference equations when the system consists of two unimodal Allee maps.     

Projection Method for Moment Bounds on Order Statistics from Restricted Families: II. Independent Case

The method of projection, proposed in Part I, is applied to derive sharp moment bounds for the expectations of order statistics based onindependentsamples from restricted families of distributions. Three families are considered: life distributions with decreasing failure density, decreasing failure rate, and symmetric unimodal ones. The respective bounds are also numerically compared with those for general populations in both the dependent and independent cases.