An inequality between the diameter and the inverse dual degree of a tree

Let T be a nontrivial tree with diameter D(T) and radius R(T). Let I(T) be the inverse dual degree of T which is defined to be , where for uV(T). For any longest path P of T, denote by a(P) the number of vertices outside P with degree at least 2, b(P) the number of vertices on P with degree at least 3 and distance at least 2 to each of the end-vertices of P, and c(P) the number of vertices adjacent to one of the end-vertices of P and with degree at least 3. In this note we prove that . As a corollary we then get with equality if and only if T is a path of at least four vertices. The latter inequality strengthens a conjecture made by the program Graffiti.postprin

Analysis and optimization of inventory variance and bullwhip in a manufacturing/remanufacturing system

This is the author accepted manuscript.Quantitative analysis of closed-loop supply chains is often based on a specified cost function; dynamical performance of the system is rarely considered. This paper adopted a control theory approach to build a simple dynamic model of a hybrid manufacturing/remanufacturing system. It highlights the effect of remanufacturing (and return) lead-time and the return rate on the inventory variance and bull whip produced by the ordering policy. The results show that a larger return rate leads to less bullwhip and less inventory variance. Thus returns can be used to improve dynamic performance by absorbing some of the demand fluctuations. Longer remanufacturing (and return) lead-times have less impact on reducing inventory variance and bullwhip than shorter lead-times. It is concluded that within our specified system the inventory variance and bullwhip is always less in supply chain with returns than that without returns

Ozone production in four major cities of China: sensitivity to ozone precursors and heterogeneous processes

Abstract. Despite a large volume of research over a number of years, our understandings of the key precursors that control tropospheric ozone production and the impacts of heterogeneous processes remain incomplete. In this study, we analyze measurements of ozone and its precursors made at rural/suburban sites downwind of four large Chinese cities – Beijing, Shanghai, Guangzhou and Lanzhou. At each site the same measurement techniques were utilized and a photochemical box model based on the Master Chemical Mechanism (v3.2) was applied, to minimize uncertainties in comparison of the results due to differences in methodology. All four cities suffered from severe ozone pollution. At the rural site of Beijing, export of the well-processed urban plumes contributed to the extremely high ozone levels (up to an hourly value of 286 ppbv), while the pollution observed at the suburban sites of Shanghai, Guangzhou and Lanzhou was characterized by intense in-situ ozone production. The major anthropogenic hydrocarbons were alkenes and aromatics in Beijing and Shanghai, aromatics in Guangzhou, and alkenes in Lanzhou. The ozone production was found to be in a VOCs-limited regime in both Shanghai and Guangzhou, and a mixed regime in Lanzhou. In Shanghai, the ozone formation was most sensitive to aromatics and alkenes, while in Guangzhou aromatics were the predominant ozone precursors. In Lanzhou, either controlling NOx or reducing emissions of olefins from the petrochemical industry would mitigate the local ozone production. The potential impacts of several heterogeneous processes on the ozone formation were assessed. The hydrolysis of dinitrogen pentoxide (N2O5), uptake of the hydroperoxyl radical (HO2) on particles, and surface reactions of NO2 forming nitrous acid (HONO) present considerable sources of uncertainty in the current studies of ozone chemistry. Further efforts are urgently required to better understand these processes and refine atmospheric models

Luminescent Cyclometalated Gold(III) Alkyl Complexes: Photophysical and Photochemical Properties

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Low-level interpretability and high-level interpretability: a unified view of data-driven interpretable fuzzy system modelling

This paper aims at providing an in-depth overview of designing interpretable fuzzy inference models from data within a unified framework. The objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real-world systems whose first-principle models are unknown. Because system behaviour can be described naturally as a series of linguistic rules, data-driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose. However, fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability. Model accuracy and interpretability are two conflicting objectives, so interpretation preservation during adaptation in data-driven fuzzy system modelling is a challenging task, which has received much attention in fuzzy system modelling community. In order to clearly discriminate the different roles of fuzzy sets, input variables, and other components in achieving an interpretable fuzzy model, a taxonomy of fuzzy model interpretability is first proposed in terms of low-level interpretability and high-level interpretability in this paper. The low-level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level, while the high-level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage, completeness, and consistency of the rules in terms of the criteria on fuzzy rule level. Some criteria for low-level interpretability and high-level interpretability are identified, respectively. Different data-driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low-level interpretability and high-level interpretability. Furthermore, some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested. Crown Copyright © 2008

Constructing accurate and parsimonious fuzzy models with distinguishable fuzzy sets based on an entropy measure

Parsimony is very important in system modeling as it is closely related to model interpretability. In this paper, a scheme for constructing accurate and parsimonious fuzzy models by generating distinguishable fuzzy sets is proposed, in which the distinguishability of input space partitioning is measured by a so-called "local" entropy. By maximizing this entropy measure the optimal number of merged fuzzy sets with good distinguishability can be obtained, which leads to a parsimonious input space partitioning while preserving the information of the original fuzzy sets as much as possible. Different from the existing merging algorithms, the proposed scheme takes into account the information provided by input-output samples to optimize input space partitioning. Furthermore, this scheme possesses the ability to seek a balance between the global approximation ability and distinguishability of input space partitioning in constructing Takagi-Sugeno (TS) fuzzy models. Experimental results have shown that this scheme is able to produce accurate and parsimonious fuzzy models with distinguishable fuzzy sets. © 2005 Elsevier B.V. All rights reserved

Constructing L2-SVM-based fuzzy classifiers in high-dimensional space with automatic model selection and fuzzy rule ranking

In this paper, a new scheme for constructing parsimonious fuzzy classifiers is proposed based on the L2-support vector machine (L2-SVM) technique with model selection and feature ranking performed simultaneously in an integrated manner, in which fuzzy rules are optimally generated from data by L2-SVM learning. In order to identify the most influential fuzzy rules induced from the SVM learning, two novel indexes for fuzzy rule ranking are proposed and named as α-values and ω-values of fuzzy rules in this paper. The α-values are defined as the Lagrangian multipliers of the L2-SVM and adopted to evaluate the output contribution of fuzzy rules, while the ω-values are developed by considering both the rule base structure and the output contribution of fuzzy rules. As a prototype-based classifier, the L2-SVM-based fuzzy classifier evades the curse of dimensionality in high-dimensional space in the sense that the number of support vectors, which equals the number of induced fuzzy rules, is not related to the dimensionality. Experimental results on high-dimensional benchmark problems have shown that by using the proposed scheme the most influential fuzzy rules can be effectively induced and selected, and at the same time feature ranking results can also be obtained to construct parsimonious fuzzy classifiers with better generalization performance than the well-known algorithms in literature. © 2007 IEEE

Extracting takagi-sugeno fuzzy rules with interpretable submodels via regularization of linguistic modifiers

In this paper, a method for constructing Takagi-Sugeno (TS) fuzzy system from data is proposed with the objective of preserving TS submodel comprehensibility, in which linguistic modifiers are suggested to characterize the fuzzy sets. A good property held by the proposed linguistic modifiers is that they can broaden the cores of fuzzy sets while contracting the overlaps of adjoining membership functions (MFs) during identification of fuzzy systems from data. As a result, the TS submodels identified tend to dominate the system behaviors by automatically matching the global model (GM) in corresponding subareas, which leads to good TS model interpretability while producing distinguishable input space partitioning. However, the GM accuracy and model interpretability are two conflicting modeling objectives, improving interpretability of fuzzy models generally degrades the GM performance of fuzzy models, and vice versa. Hence, one challenging problem is how to construct a TS fuzzy model with not only good global performance but also good submodel interpretability. In order to achieve a good tradeoff between GM performance and submodel interpretability, a regularization learning algorithm is presented in which the GM objective function is combined with a local model objective function defined in terms of an extended index of fuzziness of identified MFs. Moreover, a parsimonious rule base is obtained by adopting a QR decomposition method to select the important fuzzy rules and reduce the redundant ones. Experimental studies have shown that the TS models identified by the suggested method possess good submodel interpretability and satisfactory GM performance with parsimonious rule bases. © 2006 IEEE

Development and evaluation of a train-the-trainer workshop for Hong Kong community social service agency staff

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Exploring community stakeholders? Perceptions of the enhancing family well-being project in Hong Kong: A qualitative study

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