What's wrong with Euclid Book V

Not having the real numbers, Euclid defined ratios abstractlyin Book V for use in geometric theorems, but failed to defineratios of ratios, in effect blocking certain further developmentsof Greek mathematics. We introduce a new axiom that can be usedinstead of subtraction, not only to prove the propositions ofBook V but also to define ratios of ratios.     

A treatise on proportion in the tradition of Thomas Bradwardine: The De proportionibus libri duo (1528) of Jean Fernel

The famous French physician Jean Fernel published in 1528 in Paris the De proportionibus libri duo. This treatise belongs to the tradition of texts on proportion that follow Bradwardine?s Tractatus de proportionibus seu de proportionibus velocitatum in motibus (1328). In the first book, Fernel presented a theory of ratios that is traditional but contains some distinctive features, on denominating ratios, on fractions, on irrational ratios. The second book is devoted to a theory of ratio of ratios of which I give an account in this paper.     

General and multiplicative non-parametric corporate performance models with interval ratio data

The increasing intensity of global competition has led organizations to utilize various types of performance measurement tools for improving the quality of their products and services. Data envelopment analysis (DEA) is a methodology for evaluating and measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. All the data in the conventional DEA with input and/or output ratios assumes the form of crisp numbers. However, the observed values of data in real-world problems are sometimes expressed as interval ratios. In this paper, we propose two new models: general and multiplicative non-parametric ratio models for DEA problems with interval data. The contributions of this paper are fourfold: (1) we consider input and output data expressed as interval ratios in DEA; (2) we address the gap in DEA literature for problems not suitable or difficult to model with crisp values; (3) we propose two new DEA models for evaluating the relative efficiencies of DMUs with interval ratios, and (4) we present a case study involving 20 banks with three interval ratios to demonstrate the applicability and efficacy of the proposed models where the traditional indicators are mostly financial ratios.     

Diagonalenverhältnisse im regelmäßigen Vieleck

In a regular polygon there are constant ratios based on the lengths of the diagonals. By folding a regular polygon along the diagonals through one vertex we get a zigzag line which yields special ratios. These ratios are generalizations of the golden ratio.     

The speed of adjustment of financial ratios: A hierarchical Bayesian approach using mixtures

This paper presents a hierarchical Bayesian analysis of the partial adjustment model of financial ratios using mixture models, an approach that allows us to estimate the distribution of the adjustment coefficients. More particularly, it enables us to analyse speed of reaction in the presence of shocks affecting financial ratios objectives as a basis to establish homogenous groups of firms. The proposed methodology is illustrated by examining a set of ratios for a sample of firms operating in the U.S. manufacturing sector. Copyright © 2007 John Wiley & Sons, Ltd.     

An algorithm for generalized fractional programs

An algorithm is suggested that finds the constrained minimum of the maximum of finitely many ratios. The method involves a sequence of linear (convex) subproblems if the ratios are linear (convex-concave). Convergence results as well as rate of convergence results are derived. Special consideration is given to the case of (a) compact feasible regions and (b) linear ratios.The research of S. Schaible was supported by Grant Nos. A4534 and A5408 from NSERC. The authors thank two anonymous referees for their helpful remarks.     

DEA/AR profit ratios and sensitivity of 100 large U.S. banks

Several important DEA/AR concepts were applied here to banking for the first time. This application includes classification, sensitivity, uniqueness, linked cones (LCs), and profit ratios. Notably, large bank behavior seems to be explained better by profit ratios than by relative efficiency. Measures of DEA efficiency, AR efficiency, and LC profit ratios were made for a bank panel of the U.S.'s 100 largest banks in asset size from 1986 to 1991. High levels of inefficiency were found, as in previous studies. Classification of the DEA efficiency measures identified the inefficient DMUs with some positive primal slacks. Sensitivity analysis of the DEA efficiency measures showed that the extreme-efficient classification was generally relatively insensitive to errors in the data. The ARs eliminated (i) 44% to 60% of the DEA-extreme-efficient DMUs and (ii) all of the banks with unprofitable actual profit ratios. Some statistical analyses highlight the superiority of the LC profit ratios, relative to the AR efficiency measures.     

Financial statement analysis: A data envelopment analysis approach

Ratio analysis is a commonly used analytical tool for verifying the performance of a firm. While ratios are easy to compute, which in part explains their wide appeal, their interpretation is problematic, especially when two or more ratios provide conflicting signals. Indeed, ratio analysis is often criticized on the grounds of subjectivity, that is the analyst must pick and choose ratios in order to assess the overall performance of a firm.In this paper we demonstrate that Data Envelopment Analysis (DEA) can augment the traditional ratio analysis. DEA can provide a consistent and reliable measure of managerial or operational efficiency of a firm. We test the null hypothesis that there is no relationship between DEA and traditional accounting ratios as measures of performance of a firm. Our results reject the null hypothesis indicating that DEA can provide information to analysts that is additional to that provided by traditional ratio analysis. We also apply DEA to the oil and gas industry to demonstrate how financial analysts can employ DEA as a complement to ratio analysis.     

期货套期保值最大概率与最小风险分析

研究套期保值的最大概率和最小风险问题 ,导出最大概率的套期比和最小风险的套期比 ,并且说明它们是一致的 .因此 ,所得到的套期比具有最大概率和最小风险这两大优点 .使投资者用这样的套期比进行套期保值 ,就可以最大概率保证其收益 ,并且使其风险最小 .     

标准存货质押融资业务贷款价值比率研究

确定合适的质押存货贷款价值比率能够使银行有效地缓释存货质押融资业务的信用风险。沿着简化式的思路，本文综合考虑了外生的企业违约概率，质押存货的价格波动率，贷款的周期和盯市频率等因素的影响，为银行在保持风险容忍水平一致的情况下确定特定存货质押融资业务的相应贷款价值比率提供了一个基本模型。此外，针对存货质押融资的现状，本文还将清算延迟、流动性风险和非零的触发水平等情况引入基本模型中进行了拓展研究。     

Note on incremental benefit/cost ratios in analytic hierarchy process

Scholars have frequently applied benefit-cost analysis to policy issue decisions. They hope to measure the relative importance among alternatives and provide systematical decision-making information using the benefit and cost factors. In this paper, the work of Bernhard and Canada [1] is studied based on incremental benefit/cost ratios with a cutoff ratio to analyze and extend the benefit/cost problem. This note has three purposes. First, we derive some theoretical properties for the incremental benefit/cost ratios with the cutoff ratio and a corrected method to show that the previous results are incomplete. Second, we provide an efficient method for solving the incremental benefit/cost ratios with cutoff ratio problems. Third, Bernhard and Canada [1] assumed that the desired incremental benefit/cost ratios could be derived using a comparison matrix under the perfectly consistent condition. We offer a counter-example to highlight the difference between the normalized benefit from the benefit and the normalized benefit vector from the AHP. We hope these findings will be useful for decision makers in applying benefit/cost analysis.     

Mathematics education graduate students’ understanding of trigonometric ratios

This study describes mathematics education graduate students’ understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students’ understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students’ responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students’ concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students’ understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.     

Using concave envelopes to globally solve the nonlinear sum of ratios problem

This article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm works by globally solving a sum of ratios problem that is equivalent to problem (P). In the algorithm, upper bounds are computed by maximizing concave envelopes of a sum of ratios function over intersections of the feasible region of the equivalent problem with rectangular sets. The rectangular sets are systematically subdivided as the branch and bound search proceeds. Two versions of the algorithm, with convergence results, are presented. Computational advantages of these algorithms are indicated, and some computational results are given that were obtained by globally solving some sample problems with one of these algorithms.     

基于动态条件相关系数模型的石油市场套期保值比估计

以WTI和Brent两地的原油现货市场和期货市场为研究对象,选择对角化的动态条件相关(DCC)模型估计了市场间的动态条件相关系数,求解了WTI市场、Brent市场及跨市的动态套期保值比,评价了各种市场组合的套期保值效果.得到如下几点结论:第一,WTI市场的一体化程度高于Brent市场;第二,两个月期货的套期保值比高于1个月期货的套期保值比,WTI相应市场组合的套期保值比要高于Brent市场;第三,采取Brent期货对WTI现货进行对冲时,其套期保值比要高于用WTI期货对Brent现货对冲时的情形,也高于Brent市场的套期保值比;第四,套期保值比越高,套期保值效果越好.     

财务危机预警中财务比率的选择研究

本文采用定性与定量相结合的方法对财务危机预警中财务比率的选择进行了研究 .首先根据初选原则对财务比率进行了初步选择 ,然后提出了一种基于神经网络的变量选择方法 ,对初选的财务比率进行定量筛选并进行了实证研究 ,结果表明 ,这是一种科学、通用的变量选择方法     

Study on Discharge Coefficients of Drain Orifices in Closed Cavities北大核心CSCD

该文利用数值方法模拟了封闭腔体的排液流动,获取了排液孔的流量系数.通过量纲分析法研究了小孔流量系数的主要影响因素,拟合了计算小孔流量系数的经验公式.结果表明：当水头高度小于200 mm时,小孔流量系数随水头高度的增加而减小;当水头高度大于200 mm时,小孔流量系数稳定在0.61附近.不同厚径比的小孔流量系数表现为两种不同的形式：小厚径比的小孔呈现薄孔流动特性,流量系数为0.6左右;大厚径比的小孔呈现厚孔流动特性,流量系数为0.8左右.     

A universality theorem for ratios of random characteristic polynomials

We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a universality limit for these ratios.     

Estimating criterion weights using eigenvectors: A comparative study

In addition to Saaty's eigenvector, there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios. As the judgment of the ratios are dependent on personal experience, learning, situations and state of mind, inconsistencies and degree of easiness or judgment on these individual ratios can be different, we study the properties of the different eigen weight vectors, including that of Saaty and that recently proposed by Cogger and Yu. A general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed.     

On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix

In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.     

Distributed Online Frequency Assignment in Cellular Networks

A cellular network is generally modeled as a subgraph of the triangular lattice. The distributed online frequency assignment problem can be abstracted as a multicoloring problem on a weighted graph, where the weight vector associated with the vertices models the number of calls to be served at the vertices and is assumed to change over time. In this paper, we develop a framework for studying distributed online frequency assignment in cellular networks. We present the first distributed online algorithms for this problem with proven bounds on their competitive ratios. We show a series of algorithms that use at each vertex information about increasingly larger neighborhoods of the vertex, and that achieve better competitive ratios. In contrast, we show lower bounds on the competitive ratios of some natural classes of online algorithms.