VC‐dimension on manifolds: a first approach

The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd.     

Algorithms for finding logical regularities in pattern recognition

Three algorithms for finding logical regularities of classes in the precedent-based recognition problem are proposed. Logical regularities of classes are defined as conjunctions of special oneplace predicates that determine the membership of a value of a feature in a certain interval of the real axis. The conjunctions are true on as large subsets of reference objects of a certain class as possible. The problem of finding logical regularities is formulated as a special integer programming problem. Relaxation, genetic, and combinatorial algorithms are proposed for solving this problem. Comparison results for these algorithms using model and real-time problems are presented. Comparison results for various estimate evaluation recognition algorithms that use logical regularities of classes in voting procedures are also presented.     

A memetic-based fuzzy support vector machine model and its application to license plate recognition

In this paper, a novel fuzzy support vector machine (FSVM) coupled with a memetic particle swarm optimization (MPSO) algorithm is introduced. Its application to a license plate recognition problem is studied comprehensively. The proposed recognition model comprises linear FSVM classifiers which are used to locate a two-character window of the license plate. A new MPSO algorithm which consists of three layers i.e. a global optimization layer, a component optimization layer, and a local optimization layer is constructed. During the construction process, MPSO performs FSVM parameters tuning, feature selection, and training instance selection simultaneously. A total of 220 real Malaysian car plate images are used for evaluation. The experimental results indicate the effectiveness of the proposed model for undertaking license plate recognition problems.     

Categorical set theory: A characterization of the category of sets

We consider Zermelo-Fraenkel set theory ZF and the theory ETS(ZF) of the elementary topos of ZF sets, which is an extension of Lawvere-Tierney's theory of elementary topoi, and prove that the theory ETS(ZF) characterizes the category (topos) of ZF-sets in the following sense. The category (topos) of ZF sets satisfies the axioms of ETS(ZF), and conversely we can define within topos theory ETS(ZF) the model of set-objects in which the ZF axiloms hold, and, furthermore, the model of set-objects in the topos of ZF sets is “equivalent” to set theory ZF and the topos of set-objects in ETS(ZF) is “logical equivalent” to topos theory ETS(ZF). Actually, the corresponding result for a weak set theory Z and the theory ETS(Z) of the elementary topos of Z sets is proved. Adding further axioms (axiom of choice, continuum hypothesis, etc.), various results of the same character are obtained.The construction of the model of set-objects uses a universal mapping property of transitive sets, which enables us to introduce transitive-set-objects in any elementary topos and to prove their basic properties.     

Low-order A0-stable Adams-type correctors

This paper presents two low-order, A₀-stable Adams-type correctors. The correctors have better absolute stability properties than the corresponding Adams-Moulton correctors but at the same time preserve other desirable characteristics of the Adams correctors. In particular, the new correctors may be implemented with trivial modifications in the highly regarded GEAR mathematical software packages. Empirical evidence will be given suggesting that the resulting methods enhance the capabilities of the packages.     

Classification Modulo Invariance,With Application to Face Recognition

This article presents techniques for constructing classifiers that combine statistical information from training data with tangent approximations to known transformations; it demonstrates the techniques by applying them to a face recognition task. Our approach is to build Bayes classifiers with approximate class-conditional probability densities for measured data. The high dimension of the measurements in modern classification problems such as speech or image recognition makes inferring probability densities from feasibly sized training datasets difficult. We address the difficulty by imposing severely simplifying assumptions and exploiting a priori information about transformations to which classification should be invariant. For the face recognition task, we used a five-parameter group of such transformations consisting of rotation, shifts, and scalings. On the face recognition task, a classifier based on our techniques has an error rate that is 20% lower than that of the best algorithm in a reference software distribution.     

A belief function classifier based on information provided by noisy and dependent features

A model and method are proposed for dealing with noisy and dependent features in classification problems. The knowledge base consists of uncertain logical rules forming a probabilistic argumentation system. Assumption-based reasoning is the inference mechanism that is used to derive information about the correct class of the object. Given a hypothesis regarding the correct class, the system provides a symbolic expression of the arguments for that hypothesis as a logical disjunctive normal form. These arguments turn into degrees of support for the hypothesis when numerical weights are assigned to them, thereby creating a support function on the set of possible classes. Since a support function is a belief function, the pignistic transformation is then applied to the support function and the object is placed into the class with maximal pignistic probability.     

The Place of Logic in Reasoning

Reasoning is a goal-oriented activity. The logical steps are at best the median part of a full reasoning: before them, a language has to be defined, and a model of the goal in this language has to be developed; after them, their result has to be checked in the real world with respect to the goal. Both the prior and the subsequent steps can be conducted rationally; none of them has a logical counterpart. Furthermore, Logic aims at prescribing what a correct reasoning is. But correct with respect to what? If the answer is: with respect to truth, the next question is whether the truth in everyday life, physics, economy, is the same as the truth that logicians have in mind. Resorting to Logic is justified only if an idealization in terms of true propositions in the logical sense is compatible with the goal. If such an idealization is legitimate, so is the use of classical Logic. If not, there is no authority forbidding to skew Logic in order to better reflect the nature of the reasoning required for the task.     

Logical regularities in pattern recognition (Parametric Approach)

The search for logical regularities of classes in the recognition by precedents problems and the use of logical regularities for solving recognition and prediction problems are considered. Logical regularities of classes are defined as conjunctions of one-place predicates that determine the membership of a value of a feature in a certain interval of the real axis. The conjunctions are true on the subsets of reference objects of a certain class and are optimal. Various optimality criteria are considered and the problem of finding logical regularities is formulated as an integer programming problem. A qualitative analysis of these problems is performed. Models for evaluating estimates on the basis of systems of logical regularities are considered. Modifications of linear decision rules for finding estimates of how close the reference objects are to classes are proposed that are based on the search for the maximum gap. Approximations of logical regularities of classes by smooth functions is proposed. The concept of the dynamic logical regularity of classes is introduced, an algorithm for finding dynamic logical regularities is proposed, and a prediction method is developed.     

基于遗传算法的固定起讫点危险品配送路线优化

为科学选择危险品配送路线,保障运输安全,将传统TSP(Travelling SalesmanProblem)问题加以推广和延伸,建立以路段交通事故率、路侧人口密度、环境影响因子和路段运输费用为指标的固定起讫点危险品配送路线优化模型.以遗传算法基本框架为基础,引入新的遗传算子,构建了可用于实现模型的多目标遗传算法.实例仿真表明,所建模型和算法在求解固定起讫点危险品配送路线优化问题中有较好的实用性.     

Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case

We perform a qualitative investigation of critical Hamilton–Jacobi equations, with stationary ergodic Hamiltonian, in dimension 1. We show the existence of approximate correctors, give characterizing conditions for the existence of correctors, provide Lax-type representation formulae and establish comparison principles. The results are applied to look into the corresponding effective Hamiltonian and to study a homogenization problem. In the analysis a crucial role is played by tools from stochastic geometry such as, for instance, closed random stationary sets.     

A characterisation of elementary fibrations

In the categorical approach to logic proposed by Lawvere, which systematically uses adjoints to describe the logical operations, equality is presented in the form of a left adjoint to reindexing along diagonal arrows in the base. Taking advantage of the modular perspective provided by category theory, one can look at those Grothendieck fibrations which sustain just the structure of equality, the so-called elementary fibrations, aka fibrations with equality.The present paper provides a characterisation of elementary fibrations which is a substantial generalisation of the one already available for faithful fibrations. The characterisation is based on a particular structure in the fibres which may be understood as proof-relevant equality predicates equipped with a principle of indiscernibility of identicals à la Leibniz. We exemplify this structure for several classes of fibrations, in particular, for fibrations used in the semantics of the identity type of Martin-Löf type theory. We close the paper discussing some fibrations related to Hofmann and Streicher's groupoid model of the identity type and showing that one of them is elementary.     

Interpretability assessment of fuzzy knowledge bases: A cointension based approach

Computing with words (CWW) relies on linguistic representation of knowledge that is processed by operating at the semantical level defined through fuzzy sets. Linguistic representation of knowledge is a major issue when fuzzy rule based models are acquired from data by some form of empirical learning. Indeed, these models are often requested to exhibit interpretability, which is normally evaluated in terms of structural features, such as rule complexity, properties on fuzzy sets and partitions. In this paper we propose a different approach for evaluating interpretability that is based on the notion of cointension. The interpretability of a fuzzy rule-based model is measured in terms of cointension degree between the explicit semantics, defined by the formal parameter settings of the model, and the implicit semantics conveyed to the reader by the linguistic representation of knowledge. Implicit semantics calls for a representation of user’s knowledge which is difficult to externalise. Nevertheless, we identify a set of properties - which we call “logical view” - that is expected to hold in the implicit semantics and is used in our approach to evaluate the cointension between explicit and implicit semantics. In practice, a new fuzzy rule base is obtained by minimising the fuzzy rule base through logical properties. Semantic comparison is made by evaluating the performances of the two rule bases, which are supposed to be similar when the two semantics are almost equivalent. If this is the case, we deduce that the logical view is applicable to the model, which can be tagged as interpretable from the cointension viewpoint. These ideas are then used to define a strategy for assessing interpretability of fuzzy rule-based classifiers (FRBCs). The strategy has been evaluated on a set of pre-existent FRBCs, acquired by different learning processes from a well-known benchmark dataset. Our analysis highlighted that some of them are not cointensive with user’s knowledge, hence their linguistic representation is not appropriate, even though they can be tagged as interpretable from a structural point of view.     

Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

Further development of multiple centrality correctors for interior point methods

This paper addresses the role of centrality in the implementation of interior point methods. We provide theoretical arguments to justify the use of a symmetric neighbourhood, and translate them into computational practice leading to a new insight into the role of re-centering in the implementation of interior point methods. Second-order correctors, such as Mehrotra’s predictor–corrector, can occasionally fail: we derive a remedy to such difficulties from a new interpretation of multiple centrality correctors. Through extensive numerical experience we show that the proposed centrality correcting scheme leads to noteworthy savings over second-order predictor–corrector technique and previous implementations of multiple centrality correctors.     

运载火箭姿态运动并行实时数字仿真

1.前言 大型运载火箭的姿态运动是指火箭绕其质心的运动,它是火箭姿态稳定控制系统的控制对象.火箭的姿态运动是多种运动的复合,诸如火箭壳体的弹性弯曲振动、液体推进剂在贮箱内的晃动,都会使其发生弱阻尼或不衰减的振荡.另外,火箭的参数,如转动惯量、重心位置、谐振频率和气动特性等都是随时间和飞行状态变化的,从而使运动特性变得非常复杂.     

Using a Functional Model to Develop a Mathematical Formula

The unifying theme of models was incorporated into a required Science Capstone course for pre‐service elementary teachers based on national standards in science and mathematics. A model of a teeter‐totter was selectedfor use as an example of a functional model for gathering data as well as a visual model of a mathematical equation for developing the mathematical relationship for a Class 1 lever, M1D1=M1D1. In this study, 20 student groups (n=72) collected data using the model in an inquiry‐based activity. All groups developed the qualitative relationship, 13 groups developed a correct mathematical formula, 6 groups developed one‐half of the relationship (X = mass × distance), and 1 group attempted to develop a procedural relationship. The pre‐service elementary teachers used a variety of model types in the activity including visual/pictorial, functional/physical and mathematical‐both graphs and formulas. The use of the teeter‐totter model as a visual and functional model of a mathematical formula was a factor in developing the mathematical relationship.     

On classes of structures axiomatizable by universal d-Horn sentences and universal positive disjunctions

We provide universal algebraic characterizations (in the sense of not involving any “logical notion”) of some elementary classes of structures whose definitions involve universal d-Horn sentences and universally closed disjunctions of atomic formulas. These include, in particular, the classes of fields, of non-trivial rings, and of directed graphs without loops where every two elements are adjacent. The classical example of this kind of characterization result is the HSP theorem, but there are myriad other examples (e.g., the characterization of elementary classes using isomorphic images, ultraproducts and ultrapowers due to Keisler and Shelah).     

基于模糊聚类的语音识别

指出模糊聚类方法影响语间识别的正确率 ,提出用基于分解构造的模糊聚类方法设立语音模式的参考向量 ,说明该方法能提高语章识别的正确率。     

Evidential calibration of binary SVM classifiers

In machine learning problems, the availability of several classifiers trained on different data or features makes the combination of pattern classifiers of great interest. To combine distinct sources of information, it is necessary to represent the outputs of classifiers in a common space via a transformation called calibration. The most classical way is to use class membership probabilities. However, using a single probability measure may be insufficient to model the uncertainty induced by the calibration step, especially in the case of few training data. In this paper, we extend classical probabilistic calibration methods to the evidential framework. Experimental results from the calibration of SVM classifiers show the interest of using belief functions in classification problems.