ABUNDANCE OF UNIMODAL MAPS WITH DENSE CRITICAL ORBIT AND PREFIXED CRITICAL ORBIT

For the so-called quadratic family, it is proved that, there exists a positive Lebesgue measure set E in the parameter space such that the corresponding map has a dense critical orbit in the support set of the invariant measure for almost all parameters in E; it is also proved that there is a dense set in E such that the critical orbit of the corresponding map enter the reversed fixed point.

Managing Populations with Unimodal Dynamics

In this work, we analyzed the impact of interventions on populations which exhibit unimodal dynamics. The six landmarks that characterize the “shape” of the unimodal reproduction curve f ( x ) of the difference equation, X n+1 = f ( X n )?, are defined and used in order to examine and determine the behavior of dynamics of populations. By using the Li-Yorke criterion for determination of chaos we propose a qualitative intervention rule that can be applied without any explicit population equation. This proposed strategy for intervention brings out many interesting behaviors in population dynamics. A qualitative decision rule can be applied with a straight edge without any population equation and therefore offers a robust strategy for the management of populations.

Parallel Minimax Searching Algorithm for Extremum of Unimodal Unbounded Function

In this paper we consider a parallel algorithm that detects the maximizer of unimodal function f(x) computable at every point on unbounded interval (0, ∞). The algorithm consists of two modes: scanning and detecting. Search diagrams are introduced as a way to describe parallel searching algorithms on unbounded intervals. Dynamic programming equations, combined with a series of liner programming problems, describe relations between results for every pair of successive evaluations of function f in parallel. Properties of optimal search strategies are derived from these equations. The worst-case complexity analysis shows that, if the maximizer is located on a priori unknown interval (n-1], then it can be detected after cp(n)=「2log「p/2」+1(n+1)」-1 parallel evaluations of f(x), where p is the number of processors.

Some Dynamic and Combinatorial Properties of One Parameter Families of Unimodal Maps with Monotonicity

It is known that certain one parameter families of unimodal maps of the interval have a topological universality with regard to their dynamic behavior [ 1, 2]. As a parameter is smoothly increased, a fascinating variety of dynamic behaviors are produced. For some families the behaviors are monotonic in the parameter, while in others they are not [3]. The question is what sort of conditions on a one parameter family will ensure this monotonicity of the behavior with the parameter？ The answer is unknown and will not be given here. What we do instead is to investigate certain geometric-dynamic-combinatorial consequences of assuming that the family has this monotonicity. Specifically, using tools of symbolic dynamics, state space is ＂course grained＂ with a finite alphabet. We decompose a non-invertible map into nonlinear but invertible pieces. From these invertible pieces, we form inverse maps via composition along words. Equations of motion are developed for both forward and inverse orbits （in both the variables of state space and the parameter）, and an equation relating forward and inverse motions at fix-points is exhibited. Finally, we deduce a list of conditions, each of which is equivalent to monotone behavior. One of these conditions states that simple parity characteristics of words correspond to definite dynamics near fixed-points and vice versa.

Decay of Correlations for Fibonacci Unimodal Interval Maps

We consider a class of generalized Fibonacci unimodal maps for which the central return times {Sn} satisfy that sn= sn-1 ＋ ksh-2 for some k≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in （1, k ＋ 1）.

UNIMODAL ORBITS IN CONTINUOUS FUNCTIONS AND GENERALIZATION OF SARKOVSKII'S THEOREM

Replete unimodal function and other conceptions are defined. It is proved that for any continuous function φ, if φ has a unimodal orbit , then φ has all unimodal orbits whose types precede the orbit type of. They are altogether distributed on a compact subset X of I. The restriction φ|X of φ to X is a replete unimodal function. Moreover, an ordered classification φ of the function space C°(I,R) is given. It is a refinement of the famous Sarkovskii’s ordered classification F, and also a generalization of the conclusion obtained by Bhatia and Egerland not long ago.

Polynomials with Palindromic and Unimodal Coefficients

Let f（q） = arq^r ＋…＋asqs, with ar ≠ 0 and as ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r＋s= n and ar＋i =as-i for all i. Polynomials of darga n form a linear subspace Pn（q） of R（q）n＋l of dimension [n/2] ＋ 1. We give transition matrices between two bases {q^j（1 ＋ q ＋… q^n-2j）}, {q^j（1 ＋ q）^n-2j } and the standard basis {q^j（1 ＋ q^n-2j）} of Pn （q）. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.

The local Hlder exponent for the entropy of real unimodal maps In Memory of Lei Tan

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ(equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Hlder continuous where h(θ) > 0, and more precisely for any θ which does not lie in a plateau the local Hlder exponent equals exactly, up to a factor log 2, the value of the function at that point.This confirms a conjecture of Isola and Politi(1990), and extends a similar result for the dimension of invariant subsets of the circle.

Unimodality of Independence Polynomials of the Cycle Cover Product of Graphs

An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.

The Unimodality of Initial B-Spline Approximations in Spline Fitting

Finding optimal knots is a challenging problem in spline fitting due to a lack of prior knowledge regarding optimal knots.The unimodality of initial B-spline approximations associated with given data is a promising characteristic of locating optimal knots and has been applied successfully.The initial B-spline approximations herein are required to approximate given data well enough and characterized by the unimodality if jumps from the highest-order derivatives of the approximations at some interior knots are local maxima.In this paper,we prove the unimodality of the initial B-spline approximations that are constructed under two assumptions:Data points are sampled uniformly and sufficiently from B-spline functions,and initial knots are chosen as the parameters of sampling points.Our work establishes the theoretical basis of the unimodality of initial B-spline approximations and pioneers the theoretical study of locating optimal knots.

TOPOLOGICAL AND METRICAL CONDITIONS FOR COLLET-ECKMANN UNIMODAL MAPS

In this paper we prove that Sands' topological condition for Collet-Eckmann maps implies Tsujii's metrical condition; on the other hand, if a Collet-Eckmann map satisfies Tsujii's metrical condition, then it satisfies Sands' topological condition. Thus we obtain three different versions of Benedicks-Carleson Theorem by using topological conditions.

Distribution Patterns of Medicinal Plants along an Elevational Gradient in Central Himalaya,Nepal

This study aimed to compare the distribution patterns and trends of plant parts used among different groups of medicinal plants, geographical regions,and between medicinal plants and all vascular plants.We used the published sources for elevation records of 2,331 medicinal plant species to interpolate presence between minimum and maximum elevations and estimated medicinal plant richness for each 100-m elevational band. Monte Carlo simulations were used to test whether differences in elevational distribution between different groups of medicinal plants were significant. Total number of medicinal plants as well as different groups showed unimodal relationship with elevation. The elevational distributions of medicinal plants significantly differ between regions and between medicinal plant groups.When comparing the richness of all medicinal plants to all vascular plants,Monte Carlo simulations indicated that the numbers of medicinal plants are higher than expected at low elevations.The highest richness of medicinal plants at low elevation could be possibly due to favorable environmental factors such as high temperature, rainfall,sunlight or due to higher density of human population and thus higher pressure on use of any plants in lower elevations.

Changes in the relationship between species richness and belowground biomass among grassland types and along environmental gradients in Xinjiang, Northwest China

The association between biodiversity and belowground biomass(BGB) remains a central debate in ecology.In this study, we compared the variations in species richness(SR) and BGB as well as their interaction in the top(0–20 cm), middle(20–50 cm) and deep(50–100 cm) soil depths among 8 grassland types(lowland meadow, temperate desert, temperate desert steppe, temperate steppe desert, temperate steppe, temperate meadow steppe, mountain meadow and alpine steppe) and along environmental gradients(elevation, energy condition(annual mean temperature(AMT) and potential evapotranspiration(PET)), and mean annual precipitation(MAP)) based on a 2011–2013 survey of 379 sites in Xinjiang, Northwest China.The SR and BGB varied among the grassland types.The alpine steppe had a medium level of SR but the highest BGB in the top soil depth, whereas the lowland meadow had the lowest SR but the highest BGB in the middle and deep soil depths.The SR and BGB in the different soil depths were tightly associated with elevation, MAP and energy condition;however, the particular forms of trends in SR and BGB depended on environmental factors and soil depths.The relationship between SR and BGB was unimodal in the top soil depth, but SR was positively related with BGB in the middle soil depth.Although elevation, MAP, energy condition and SR had significant effects on BGB, the variations in BGB in the top soil depth were mostly determined by elevation, and those in the middle and deep soil depths were mainly affected by energy condition.These findings highlight the importance of environmental factors in the regulations of SR and BGB as well as their interaction in the grasslands in Xinjiang.

Simulation of the pore size distribution function for a deformable soil

In order to obtain an indirect estimation method of the pore size distribution function(PSDF)for a deformable soil,both the soil-water characteristic curve in the form of gravimetric water content(w-SWCC)and the shrinkage curve(SC)are used as the input parameters.The w-SWCC defines the relationship between the gravimetric water content and soil suction.The SC illustrates the variation of the void ratio with respect to different water contents.10 points in the w-SWCC were selected as initial conditions.By adopting different void ratios,a group of soil-water characteristic curve in the form of the degree of saturation(S-SWCC)can be obtained.Based on Kelvin's capillary law,the S-SWCCs can be converted into a group of PSDFs.In the group of PSDFs,each PSDF represents the geometric pore space in soil corresponding to a given void ratio.From the proposed methodology,it is observed that a bimodal PSDF can be gradually changed into a unimodal PSDF when the soil is compressed.The Chataignier clay is selected as the verification and it shows that the simulation results agree well with the measured results from the mercury intrusion porosimetry(MIP)test.In addition,the discrepancies between both direct measurement data using the MIP test and the indirect estimated results from the proposed method are also discussed.

On Real-rootedness of Independence Polynomials of Rooted Products of Graphs

An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwenk and Erd os conjectured that the independence polynomial of any tree or forest is unimodal.Although this unimodality conjecture has attracted many researchers’ attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal.In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.

ON THE FEIGENBAUM'S FUNCTIONAL EQUATION_f^P (λx )=λf (x)

The author considers the Feigenbaum's functional equation fP(λx) =λf(x) for each p ≥ 2.The existence of even unimodal C1 solutions to this equation is discussed and a feasible methodto construct such solutions is given.

Highly linear polyethylenes using the2-(1-(2,4-dibenzhydrylnaphthylimino)ethyl)-6-(1-(arylimino)ethyl)-pyridylcobalt chlorides: synthesis, characterization and ethylene polymerization

The series of 2-（1-（2,4-dibenzhydrylnaphthylimino）ethyl）-6-（1-（arylimino）ethyl）pyridine derivatives and their cobalt chlorides （Col-Co5） were synthesized and fully characterized. The representative complexes Co2 and Co3 were confirmed by single crystal X-ray diffraction analyses, indicating pseudo-square-pyramidal geometry around the cobalt centre. All cobalt complexes, activated with either methylaluminoxane （MAO） or modified methylaluminoxane （MMAO）, exhibited high activities towards ethylene polymerization, and produced polyethylenes with high molecular weight and highly linear feature as well as unimodal distribution.

3-PERIODIC ORBIT IMPLYING 683172687698650885-PERIODIC ORBITS——INFIMUMS OF NUMBERS OF PERIODIC ORBITS IN CONTINUOUS FUNCTIONS

For any continuous function f on the interval I=[0, 1] and any m, n≥1, let N(n, f)denote the number of n-periodic orbits in f. Put N(n, m)=min{N(n, f):f is a continuousfunction on I, and N(m, f)≥1}. The famous Sarkovskii’s theorem can be stated as follows:If n?m, then N(n,m)≥1. In this paper, we further obtain analytic expressions of the precisevalue of N(n, m) for all positive integers m and n, which are convenient for computing.