Implicit Sampling, with Application to Data Assimilation

There are many computational tasks, in which it is necessary to sample a given probability density function (or pdf for short), i.e., to use a computer to construct a sequence of independent random vectors x i (i = 1, 2, ··· ), whose histogram converges to the given pdf. This can be difficult because the sample space can be huge, and more importantly, because the portion of the space, where the density is significant, can be very small, so that one may miss it by an ill-designed sampling scheme. Indeed, Markovchain Monte Carlo, the most widely used sampling scheme, can be thought of as a search algorithm, where one starts at an arbitrary point and one advances step-by-step towards the high probability region of the space. This can be expensive, in particular because one is typically interested in independent samples, while the chain has a memory. The authors present an alternative, in which samples are found by solving an algebraic equation with a random right-hand side rather than by following a chain; each sample is independent of the previous samples. The construction in the context of numerical integration is explained, and then it is applied to data assimilation.     

The Potential for Endogeneity Bias in Data Envelopment Analysis

Data envelopment analysis has become an important technique for modelling the relationship between inputs and outputs in the production process, particularly in the public sector. However, whenever measures of the output of public sector activity receive public attention, there is a strong possibility that there will be a feedback from the achieved output to the resources devoted to the activity. In other words, the level of resources is endogenous. The implications of such endogeneity for standard econometric estimation techniques are well known, and methods exist to deal with the problem. Most commentators have assumed that endogeneity poses no analogous problems for DEA because the technique merely places an envelope around feasible production possibilities. Using Monte Carlo simulation techniques, however, this paper shows that the efficiency estimates generated by DEA in the presence of endogeneity can be subject to bias, in the sense that inefficient units using low levels of the endogenous resource may be set tougher efficiency targets than equally inefficient units using more of the resource, particularly when sample sizes are small. The paper concludes that, in such circumstances, great caution should be exercised when comparing efficiency measures for units using different levels of the endogenous input.     

Altering the Bias in Differential School Effectiveness Using Data Envelopment Analysis

There is evidence that many schools are differentially effective in that they promote with different effectiveness the academic attainments of different groups of pupils, divided by such characteristics as prior academic attainments, socio-economic background and so on. Awareness of the extent and direction of a school's differential effectiveness is important. Measures of overall performance of a differentially effective school can be misleading and targets seeking to improve the overall performance of such a school may not address its underlying inadequacies in raising the attainments of specific groups of pupils. This paper develops a data envelopment analysis based method for setting targets for diffentially effective schools to enable them to alter the bias of their differential effectiveness. The method also identifies role model schools which a differentially effective school can emulate in seeking to redress the imbalances in its effectiveness.     

Implicit functionals and eigenvalue problems

An approach to nonlinear, positively homogeneous eigenvalue problems based on the use of the spectral parameter as a functional of Euler type is suggested. It allows one to present the spectral-parameter domain as a bifurcation diagram of the problem. Fučik spectrum problems (classical and for p-Laplacians) and the problem with a nonlinear dependence of the weight function on the spectral parameter are considered. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 169–186, 2005.     

Implicit Assumptions and Communication in Statistics

Statistics is becoming an increasingly important component of the school curriculum. The subject provides students with opportunities to explore situations that they read about or face in their everyday lives. This article describes a statistics project that focuses on communicating conclusions and on understanding implicit assumptions that may affect conclusions based on the information. The ability to make a table and draw graphs and charts is an important component of statistical education. However, the ability to communicate ideas to others and understand implicit assumptions associated with data are key objectives of educating students for a world that increasingly relies on statistical information as a means of mass communication.     

Scientists and Science Educators Mentoring Secondary Science Teachers

This paper examines secondary science teachers' perspectives of the role that mentoring by a scientist and science educator pair played in their professional development. Multiple data sources from three years of a professional development project, including interviews, participant reflections, and a focus group, were used to examine the benefits, supporting characteristics, and challenges of the mentoring relationship. Results indicated that primary benefits of the mentoring included assistance in translating science content and inquiry‐based pedagogy from the professional development into practice and breaking the isolation felt by secondary science teachers. Specific characteristics that were found to support the teachers in the mentoring relationship included: (1) mentors who were seen as objective, outside observers; (2) a sustained relationship with the mentors; and (3) accountability. Challenges included matching scientists' and science educators' content expertise with teachers' curriculum and the negotiation of roles and expectations between the teachers and mentors.     

Implicit function theorems and lagrange multipliers

Several implicit function theorems are proved, for systems of inequalities as well as equalities, assuming weaker differentiability than continuous Fréchet. These results give sufficient hypotheses for the Kuhn-Tucker conditions to hold, in a constrained minimization problem, with cone constraints and in any dimension. A condition is given for an inequality constraint system to be equivalently replaced by another, for which the function involved has surjective derivative. This allows some known implicit function theorems to be applied to inequality systems.     

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.     

Gwinner变分不等式和隐变分不等式

本文在研究Gwinner变分不等式的基础上，利用新的集值映象不动点定理，探讨一类具广泛意义的隐变分不等式问题，改进了迄今相关结果中对紧性条件的要求，在非紧设置下获得了解的存在性定理．     

A-Stable and L-Stable Block Implicit One-Step Method

A class of methods for solving the initial problem for ordinary differential equations are studied. We develop k-block implicit one step methods whose nodes in a block are nonequidistant. When the components of the node vector are related to the zeros of Jacobi's orthogonal polynomials, we can derive a subclass of formulas which are A or L-stable. The order can be arbitrarily high with A- or L-stability. We suggest a modified algorithm which avoids the inversion of a $km×km$ matrix during Newton-Raphson iterations, where $m$ is the number of differential equations. When k=4, for example, only a couple of $m×m$ matrices have to be inversed, but four values can be obtained at one time.     

Implicit functions and sensitivity of stationary points

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ⁿ ,D being an open bounded subset of ⁿ . LetF belong toL(D) and suppose that solves the equationF(x) = 0. In case that the generalized Jacobian ofF at is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.     

关于“创新环境”内隐观的调查研究:基于新疆高校师生大样本调查数据

采用内隐研究方法,以新疆高校师生为调查样本,探究普通人对创新环境的认知,研究结果表明:1)人们对创新环境的认知以某种特征结构形式存在于头脑中,并呈现出丰富的内涵;2)经过因子分析,发现人们心目中理想的创新环境由创新网络、社会保障、组织支持、物质基础、人文氛围、创新资源六个维度35个指标构成;3)不同类型学校、教师与学生对创新环境特征的认知上存在显著性差异.