Heights of representative systems

Representative systems are hierarchical aggregation schemes that are applicable in social choice theory, multiattribute decision making, and in the study of three-valued logics. For example, many procedures for voting on issues—including simple majority voting and weighted voting—can be characterized as representative system. Such systems also include procedures in which vote outcomes of constituencies are treated as votes in a higher level of an election system. The general form of a representative system consists of a “supreme council” which aggregates vote outcomes of secondary councils, which in turn aggregate vote outcomes of tertiary councils, and so forth.An n-variable representative system maps n-tuples of 1's, 0's and ?1's into {1,0,?1} through a nested hierarchy of sign functions. The height of a representative system is the fewest number of hierarchical levels that are needed to characterize the system. The height μ(n) of all n-variable representative systems is the largest height of such systems. It was shown previously that μ(n) ? n ? 1 for all positive integers n and that μ(n) = n ? 1 for n from 1 to 4 inclusive. The present paper proves that μ(5) = μ(6) = 4 and that μ(n) ? ?2 for all n ? 6. The height function μ is known to be unbounded.     

Content and pedagogical content knowledge in argumentation and proof of future teachers: a comparative case study in Germany and Hong Kong

The results of a comparative case study on mathematical and pedagogical content knowledge in the area of argumentation and proof of future teachers in Germany and Hong Kong are reported in this article. The study forms part of a qualitatively oriented comparative study on future teachers in Australia, Germany, and Hong Kong. Six case studies based on interviews and written questionnaires are described. These case studies show the strengths of the Hong Kong future teachers in mathematical knowledge in the area of argumentation and proof, whereas the three German future teachers perform stronger in the related pedagogical content domain. Furthermore, regarding the German future teachers, it seems that the two domains of knowledge are more strongly connected to each other. The results are interpreted in the light of related research, such as the MT21 study.     

Igusa–Todorov functions for Artin algebras

In this paper we study the behavior of the Igusa–Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the ?-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite ?-dimension and finite ψ-dimension.     

Voting power in the European Union enlargement

The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in O(n2ⁿ). We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.     

Comparing influence theories in voting games under locally generated measures of dissatisfaction

The Isbell desirability relation (I), the Shapley?CShubik index (SS) and the Banzhaf?CColeman index (BC) are power theories that grasp the notion of individual influence in a yes?Cno voting rule. Also, a yes?Cno voting rule is often used as a tool for aggregating individual preferences over any given finite set of alternatives into a collective preference. In this second context, Diffo Lambo and Moulen (DM) have introduced a power relation which ranks the voters with respect to how ably they influence the collective preference. However, DM relies on the metric d that measures closeness between preference relations. Our concern in this work is: do I, SS, BC and DM agree when the same yes?Cno voting rule is the basis for collective decision making? We provide a concrete and intuitive class of metrics called locally generated (LG). We give a characterization of the LG metrics d for which I, SS, BC and DM agree on ranking the voters.     

Asymptotics for Kernel Estimation of Slicing Average Third-Moment Estimation

To estimate central dimension-reduction space in multivariate nonparametric rcgression, Sliced Inverse Regression （SIR）, Sliced Average Variance Estimation （SAVE） and Slicing Average Third-moment Estimation （SAT） have been developed, Since slicing estimation has very different asymptotic behavior for SIR, and SAVE, the relevant study has been madc case by case, when the kernel estimators of SIH and SAVE share similar asymptotic properties. In this paper, we also investigate kernel estimation of SAT. We. prove the asymptotic normality, and show that, compared with tile existing results, the kernel Slnoothing for SIR, SAVE and SAT has very similar asymptotic behavior,     

Profinite groups of finite cohomological dimension

Let 1→N→G→G/N→1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n:=cd_p(G)<∞ and if the order of $\text{H}{}^{\mathrm{k}}\text{(N,}\text{F}{}_{\mathrm{p}}\text{)}$ is finite for k:=cd_p(N), the virtual cohomological p-dimension of G/N equals n?k. To cite this article: T. Weigel, P. Zalesskii, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Auxiliary space preconditioning in H 0(curl; Ω)

We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215–235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space method. This author was fully supported by Hong Kong RGC grant (Project No. 403403) This author acknowledges the support from a Direct Grant of CUHK during his visit at The Chinese University of Hong Kong.     

香港艾滋病近期传播特征与动态预测

首次研究出香港艾滋病目前传播势态为一元线性回归方程,并由方程的预测值和斜率的变化动态监测香港艾滋病近期发病趋势,艾滋病病毒感染者年平均发病率为255至271个,艾滋病病患者年平均发病率为56个左右.本文研究方法简单易行,对香港政府近期预防艾滋病有重要的理论和实用价值.     

Optimization of decisional power in representational systems with vote binding

The theory of decisional power of Hoede and Bakker is applied to representational systems in which all representatives of a certain group vote alike. An equivalent system is one in which a group is represented by one representative with a voting weight. The main theorem of this paper gives the optimal distribution of weights for such a system. As an example the distribution of seats in the european parliament of the european economic community is reconsidered. The effect of vetoes is discussed as well as other ways of voting towards inaction used for minority protection. The decrease in overall power is accompanied by a tendency towards greater equality of power.     

Bargaining set theory and stability in coalition governments

Various bargaining set theories are compared as predictors of coalition government portfolio distribution. While the kernel and B₁-bargaining set are known to exist in voting games with side payments, it is argued that the kernel, and thus B₁, are poor predictors. The B₂-bargaining set, a subset of B₁, when it exists is shown to be a good payoff predictor in a fractionalized and depolarized parliamentary situation (Finland: 1945ndash;1971). Moreover this predictor provides some explanation for the formation of surplus (winning but not minimal) coalitions.     

Evaluating correct classification probability for weighted voting classifiers with plurality voting

Weighted voting classifiers (WVCs) consist of N units that each provide individual classification decisions. The entire system output is based on tallying the weighted votes for each decision and choosing the winning one (plurality voting) or one which has the total weight of supporting votes greater than some specified threshold (threshold voting). Each individual unit may abstain from voting. The entire system may also abstain from voting if no decision is ultimately winning. Existing methods of evaluating the correct classification probability (CCP) of WVCs can be applied to limited special cases of these systems (threshold voting) and impose some restrictions on their parameters. In this paper a method is suggested which allows the CCP of WVCs with both plurality and threshold voting to be exactly evaluated without imposing constraints on unit weights. The method is based on using the modified universal generating function technique.     

Voting Power in the Governance of the International Monetary Fund

In general in an organisation whose system of governance involves weighted voting, a member's weight in terms of the number of votes and the formal power it represents differ. Power indices provide a means of analysing this difference. The paper uses new algorithms for computing power indices for large games. Three analyses are carried out: (1) the distribution of Banzhaf voting power among members in 1999; the results show that the United States has considerably more power over ordinary decisions than its weight of 17% but that the use of special supermajorities limits its power; (2) the effect of varying the majority requirement on the power of the IMF to act and the powers of members to prevent and initiate action (Coleman indices); the results show the effect of supermajorities severely limits the power to act and therefore renders the voting system ineffective in democratic terms, also the sovereignty of the United States within the IMF is effectively limited to just the power of veto; (3) the paper proposes the determination of the weights instrumentally by means of an iterative algorithm to give the required power distribution; this would be a useful procedure for determining appropriate changes in weights consequent on changes to individual countries' quotas; this is applied to the 1999 data. Policy recommendations are, first, that the IMF use only simple majority voting, and discontinue using special supermajorities, and, second, allocate voting weight instrumentally using power indices.     

Multi-parametric optimizations for power dissipation characteristics of Stockbridge dampers based on probability distribution of wind speed

Power dissipation characteristics of Stockbridge dampers (SD) is one of the important indexes in anti-vibration work of transmission line. The study focuses on the optimization of the SD's power dissipation characteristics under the effect of multi-structure parameter coupling. The aeolian vibration of overhead transmission lines is uncertain and random in stochastic dynamics. According to Strouhal formula, the relationship between the vibration frequency of transmission lines and wind speed can be found out. Based on the Weibull wind speed probability distribution, the probability density function of the transmission line conductor and damper coupling system vibration frequency is derived. The SD is considered as a typical 2-dimension of stochastic dynamical system. Based on the random process generated by the power dissipation of the SD, the characteristics of power dissipation and SD's resonant frequencies are analyzed when the multi-structure parameters of the SD are coupled. And the diagrams of the power dissipation at various frequencies are obtained.Based on the probability density function of the vibration frequency of the overhead conductor and damper, the objective function, namely the mathematical expectation of power dissipation (E(PD)), of the optimizations for the SD's power dissipation under the coupling of multiple structural parameters is proposed for the first time according to the author's knowledge. Constraint conditions of the optimizations are built by the quantization processing. The energy dissipation characteristics of the dampers can be evaluated by E(PD), and the power dissipation of SD with different coupled dual structure parameters is optimized based on the proposed method. The optimal values or the optimal value intervals of different coupled dual structure parameters are found, which may provide practical data.     

The Lagrangian Globalization Method for Nonsmooth Constrained Equations

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given. This author's work was done when she was visiting The Hong Kong Polytechnic University. His work is also supported by the Research Grant Council of Hong Kong.     

On maximum likelihood estimation of the drift matrix of a degenerated O–U process

In this work, we consider a 2n-dimension Ornstein–Uhlenbeck (O–U) process with a singular diffusion matrix. This process represents a currently used model for mechanical systems subject to random vibrations. We study the problem of estimating the drift parameters of the stochastic differential equation that governs the O–U process. The maximum likelihood estimator proposed and explored in Koncz (J Anal Math 13(1):75–91, 1987) is revisited and applied to our model. We prove the local asymptotic normality property and the convergence of moments of the estimator. Simulation studies based on representative examples taken from the literature illustrate the obtained theoretical results.     

An appointment-based service center with guaranteed service

Customers are scheduled to arrive at periodic scheduling intervals to receive service from a single server system. A customer must start receiving service within a given departure interval; if this is not the case, the system will pay a penalty and/or transfer the customer to another facility at the system's cost. A complete transient and steady-state analysis of the system is given, the optimal scheduling and departure intervals are determined, and the virtual work in the system is analyzed.     

The General Pay-Word: A Micro-payment Scheme Based on <Emphasis Type="BoldItalic">n</Emphasis>-dimension One-way Hash Chain

In this paper, we extend the Pay-Word micro-payment scheme using the 1-dimension one-way hash chain to generate an n-dimension one-way hash chain. According to the system requirements, a user can flexibly choose the number of dimensions to gain the best time/space trade off. The proposed scheme is based on an n-dimension one-way hash chain, which can improve the efficiency of deriving the pay-words but also increase the temporary storage space. In addition, this scheme is fit for real-time payment.AMS Classification: 14G50, 11T71     

QR-Submanifolds of (p - 1) QR-Dimension in a Quaternionic Projective Space QP (n+p)/4

The purpose of this paper is to study n-dimensional QR-submanifolds of (p - 1) QR-dimension isometrically immersed in a quaternionic projective space QP ^(n+p)/4 and to give sufficient conditions in order for such a submanifold to be a tube over a quaternionic invariant submanifold.     

Minimax portfolio optimization: empirical numerical study

In this paper, we carry out the empirical numerical study of the l_∞ portfolio selection model where the objective is to minimize the maximum individual risk. We compare the numerical performance of this model with that of the Markowitz's quadratic programming model by using real data from the Stock Exchange of Hong Kong. Our computational results show that the l_∞ model has a similar performance to the Markowitz's model and that the l_∞ model is not sensitive to the data. For the situation with only two assets, we establish that the expected return of the minimum variance model is less than that of the minimum l_∞ model when both variance and the return rate of one asset is less than the corresponding values of another asset.