关于线性递推数列通项的求法及其收敛特征

运用方程及化归的思想给出一阶线性递推数列、一阶分式线性递推数列、二阶线性递推数列以及二元一阶线性递推数列的通项,然后在此基础上建立若干关于线性递推数列的收敛特征定理.     

Stolz定理的相关问题及应用探讨

给出并证明了Stolz定理的推广形式,给出了推广形式的Stolz定理在证明L'Hospital法则、求待定型数列的极限、研究具有非线性递推关系数列的渐近性方面的应用.     

四种数列递推关系式的统一及应用

本文指出一类递推数列存在四种形式各异的递推关系式,并举例说明其重要应用.一、定理与推论定理若实数数列{an}有下列四种递推关系式之一:     

压缩映象定理与递推形式极限

文〔1〕对数列极限存在性的讨论主要介绍了单调有界定理与Cauchy准则,本文试图阐述利用压缩映象定理求由递推形式给出的数列极限方面的应用,同时还试图给出由压缩映象定理得到的一些推广.     

不动点原理与递推数列的极限

递推数列的极限问题,常是用单调有界原理来解决．但当递推数列不是单调时,其方法失效．文章利用不动点原理的思想,得到解决递推数列极限的存在性问题的一个定理,使得其解法变得更为有效且简洁．     

递推数列类极限的一种求法

对由递推关系式定义的数列,给出了一个新的求极限定理,其避开了对数列单调性的讨论,首先推测数列极限的可能值,然后直接从数列极限的定义出发,判断推测的正确性,并通过例题说明了这种方法的实际应用.     

压缩映射原理在数列极限中的应用

压缩映射原理是泛函分析中最基本的存在性定理.本文通过对考研中数列极限的典型例题的解析,归纳总结出适合压缩映射原理求极限数列的一般形式,展示压缩映射原理在解决递推数学列极限中的优越性.     

数列的递推式在解题中的一些应用

由数列的递推式求数列的通项公式的方法在各类数学杂志上已发表了许多系统的论述.但是,对于如何利用数列的递推式来解决有关的数学问题却没有引起人们的足够重视,因此,在学生中造成了为求数列的通项而求通项的被动状态.其实,在现实中,许多事物间的关系具有递推的性质,特别是与自然数有关的一些问题,常常可以借助于数列的速推式而得到解决.为此,本文着重介绍数列递推式的一些应用. 首先,介绍下面两个定理.     

广义Lucas数列的求和公式

由二次线性递推公式所定义的Lucas数列{Fn}在数学的理论研究中有重要的作用,不少学者对这个数列的一些特征进行了深入细致的研究.本文在已有的有关广义Lucas数列相关定理的基础上进一步推广,给出了更为广泛的广义Lucas数列的求和公式,采用了递推归纳的方法证明.     

常见递推数列通项公式的求法

递推数列是一类广泛而复杂的问题,具有逻辑推理性强,求解方法开放、灵活等特点.递推数列是数列中的重要内容,通过递推关系,观察、探求数列的规律,进而可求出数列的通项公式.通过对递推关系的学习,培养学生的观察能力、归纳与转化能力、综合运用知识等能力.     

The complexity of child protection recurrence: The case for a systems approach

ObjectiveResearch on child protection recurrence has found consistent child, family, and case characteristics associated with repeated involvement with the child protection system. Despite the considerable body of empirical research, knowledge about why recurrence occurs, and what can be done to reduce it, is limited.MethodThis paper reviews the empirical literature and analyses the approaches of prior recurrence research. Four related conceptual challenges are identified: (1) a tendency to conflate child protection recurrence with repeated child maltreatment; (2) uncertainty about how best to operationalize and measure child protection recurrence in research; (3) inconsistency between prevailing explanations for the most frequently observed patterns of recurrence; and (4) difficulty in developing coherent strategies to address child protection recurrence based on research.ResultsAddressing these challenges requires a greater consideration of the effects of decision-making in the child protection system on recurrence. This paper proposes a methodology based in systems theory and drawing on existing administrative data to examine the characteristics of the child protection system that may also produce recurrence.     

Recurrence of maltreatment: an application of the National Child Abuse and Neglect Data System (NCANDS).

OBJECTIVE: The research describes and compares patterns of maltreatment recurrence across multiple states using large samples, confirms the patterns of recurrence found in the literature, and explores unreported patterns of recurrence. METHOD: A recurrence data set for calendar years 1994 and 1995 was constructed from the multi-state case level data from the National Child Abuse and Neglect Data System. These data were available for 10 states and included a range from 2,419 to 99,288 substantiated or indicated report-child pairs per state. A common set of data constructs lent consistency to data construction and analysis, while preserving differences in policy. Event History Analysis (survival) techniques were used. RESULTS: Single site studies were confirmed across the 10 states. These include the pattern where neglect is most likely to recur, followed by physical abuse and then sexual abuse. Similarly, younger children are more likely to recur. A finding of the analysis is that the likelihood of recurrence increases in a systematic and consistent fashion based upon the sequential ordering of recurrent maltreatment events. Also, the likelihood of recurrence is associated with the provision of postinvestigative services. CONCLUSION: Highly consistent patterns of recurrence were observed across states. Children experiencing multiple recurrences compared to no recurrence or one recurrence may represent a special at risk population requiring additional research. Adequate baselines and an understanding of recurrence is needed when considering recurrence as an outcome indicator or in developing risk assessment tools. Important recurrence patterns may be difficult to detect reliably with relatively small samples.     

基于递归图的电价信号复杂度描述

为有效描述电价的波动特性,规避电力市场风险,引入递归图及其定量递归分析法对美国加州电力市场电价信号的进行研究.研究表明电灾期与非电灾期的递归图有明显不同的结构特征,电价信号的定量递归分析中的特征量变化进一步反映了这种差异.因此,采用递归图作为电价序列的定性分析工具是有效的,用递归分析法描述电价信号复杂度是可行的.     

Factors associated with child protection recurrence in Australia

The aim of the current research was to advance understanding of child protection in Australia by examining the factors associated with recurrence of child protection notifications to the formal child protection system. Extant research has been primarily undertaken in the USA and it is important to understand whether similar factors associated with recurrence actually hold in the Australian context. Administrative data were obtained for a sample of 9608 children first subject to a screened-in report in 2011–12. Children were followed for 12 months. Cox Proportional Hazard models were used to measure associations between 26 independent variables and four types of recurrence: subsequent reports, subsequent investigations, subsequent substantiations, and subsequent intervention. Factors associated with recurrence in Australia were broadly similar to those identified in other jurisdictions, including reports and substantiation for neglect, younger age, prior child protection involvement in the household, and parental characteristics including drug use, mental health problems, and history of maltreatment as a child. As in previous studies, post-investigative service provision was positively associated with recurrence. In prior US research, race did not predict recurrence. However, in the present study, Indigenous Australian children were significantly more likely to be subject to all types of recurrence measured. Future research on recurrence should aim to disentangle the complex relationships between child protection recurrence, child maltreatment, and service delivery. Recurrence is not a good proxy indicator of child safety. The findings have implications for the equity of recurrence-based risk assessment tools as they are applied to indigenous populations.     

A developmental approach to the risk of a first recurrence in child protective services

The objective of this study is to estimate the risk of a first recurrence over a five-year period following initial child protective services (CPS) intervention and identify the characteristics associated with the risk of recurrence for three different age groups. Recurrence is defined as the first substantiated report within the observation period after initial services have ended. The study involved a cohort of 25,897 Quebec children who received postinvestigation services for the first time and whose cases were closed between 2005 and 2009. Survival analysis was used to estimate the five-year risk of recurrence and Cox regression to model the risk of recurrence for three age groups. The covariates introduced into the regression analyses were characteristics of the child and initial services. The risk of recurrence in the five years following termination of initial CPS services was 36% for the entire cohort and varied depending on the child's age at the time of case closure. Children aged 6–11 when their cases were closed had the highest risk of recurrence. Although Aboriginal descent and prior CPS investigations have a consistent effect on the risk of recurrence in all three age groups, the effects of other covariates, such as out-of-home placement and court involvement, vary or are even reversed, depending on the child's age. These findings highlight the need to adopt a differential approach that takes into account the child's age, both in the provision of protective services and in research involving the population receiving such services.     

一类递推数列极限问题的推广与应用

求递推数列的极限是数学分析教材和一些高校硕士研究生入学考试中经常出现的问题．通过对一类递推数列的极限问题作推广,对推广的结论给出了具体应用．     

The effect of services on the recurrence of child maltreatment

OBJECTIVE: Building on a previous model of recurrence, this study examined the relationship of services on the hazard of child maltreatment recurrence during CPS intervention for families who were provided continuing intervention following a confirmed index report of physical abuse or neglect. METHOD: This nonconcurrent prospective study selected 434 subject families who met study eligibility requirements from 1181 families randomly selected from the 2902 families who had experienced a substantiated report of child abuse or neglect during the sampling year. Data were collected and coded from archival sources for 5 years following the index report. Each record was coded by two research analysts to increase inter-rater reliability. Data were analyzed with the Cox Proportional Regression Model. RESULTS: Case characteristics that predicted recurrence were: child vulnerability, family stress, partner abuse, and social support deficits. After examining the potential effect of nine service-related variables only attendance at services predicted recurrence while controlling for other variables in the model. Families who were noted to attend the services identified in their service plans were 33% less likely to experience a recurrence of child maltreatment while their case was active with CPS. CONCLUSIONS: Implications of this and other research suggests that actively engaging families in a helping alliance and helping them accept and receive services may reduce the likelihood of future maltreatment.     

也谈二阶线性递推数列的周期性

对二阶线性递推数列的周期性进行探究,推导出其为周期数列的一个充分必要条件。     

二阶线性递推数列的通项公式

在数学建模中常常用数列的递推公式求数列通项,由递推公式求数列通项既可考查等价与化归数学思想,又能加深考生对等差与等比数列的理解,因而这类题目在高考和数学竞赛中经常出现.故以一阶线性递推数列的通项公式为基础,推导出二阶线性递推数列的通项公式.