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.

A Simple Cement Hydration Model Considering the Influences ofWater-to-Cement Ratio and Mineral Composition

A simple hydration model is used here by taking the composition of the cement and the initial water: cementratio (w/c) into account explicitly. Its conceptual basis is a combination of the Avrami equation and Bentz’s modelbased on simple spatial considerations. In this model, the Avrami equation determines the initial reaction, andBentz’s model describes the following hydration stage. The model favors engineers for it relies on one experimentalparameter and has a reliable approximation in the practice.

Estimation of Aggregate Losses of Secondary Cancer Using PH-OPPL and PH-TPPL Distributions

Kenyan insurance firms have introduced insurance policies of chronic illnesses like cancer; however, they have faced a huge challenge in the pricing of these policies as cancer can transit into different stages, which consequently leads to variation in the cost of treatment. This has made the estimation of aggregate losses of diseases which have multiple stages of transitions such as cancer, an area of interest of many insurance firms. Mixture phase type distributions can be used to solve this setback as they can in-cooperate the transition in the estimation of claim frequency while also in-cooperating the heterogeneity aspect of claim data. In this paper, we estimate the aggregate losses of secondary cancer cases in Kenya using mixture phase type Poisson Lindley distributions. Phase type (PH) distributions for one and two parameter Poisson Lindley are developed as well their compound distributions. The matrix parameters of the PH distributions are estimated using continuous Chapman Kolmogorov equations as the disease process of cancer is continuous while severity is modeled using Pareto, Generalized Pareto and Weibull distributions. This study shows that aggregate losses for Kenyan data are best estimated using PH-OPPL-Weibull model in the case of PH-OPPL distribution models and PH-TPPL-Generalized Pareto model in the case of PH-TPPL distribution models. Comparing the two best models, PH-OPPL-Weibull model provided the best fit for secondary cancer cases in Kenya. This model is also recommended for different diseases which are dynamic in nature like cancer.