The predicative Frege hierarchy

In this paper, we characterize the strength of the predicative Frege hierarchy, , introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that and are mutually interpretable. It follows that is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess’ PV is Robinson’s Q, The Journal of Symbolic Logic 72 (2) (2007) 619–624] using a different proof. Another consequence of the our main result is that is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, , Q3). The fact that interprets EA was proved earlier by Burgess. We provide a different proof.Each of the theories is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, , is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with is finitely axiomatizable.     

综合评价中权数的函数性质及其生成研究

通过对权数进行系统的研究,本文认为从"权数的性质"这个角度出发,权数具有"重要性"、"模糊性"、"主观性"和"时序性".将区域因素考虑进去,提出了权数具有"空间性","时空性"的观点.进一步,随着时间或区域的连续变化时,若权数也连续取值,则此时权数可以看成函数的形式,并给出两种具体的方法求权函数.特别针对权函数取值范围在0和1之间,将其进行函数变换后,通过新函数(权函数的Logit函数)的函数生成方法,进而得出权函数的生成方法.最后通过例子说明该方法比较符合权数的实际取值.     

The effect of quantitative reasoning on prospective mathematics teachers’ proof comprehension: The case of real numbers

We report a mixed-methods research study investigating the effect of quantitative reasoning on prospective mathematics teachers’ comprehension of a proof on real numbers. Nineteen prospective mathematics teachers engaged in quantitative reasoning while developing real numbers as rational number sequences in a series of instructional activities. All participants completed a proof comprehension assessment prior to and upon completion of the instruction. Six of the prospective mathematics teachers also participated in semi-structured interviews after the post-test. Results showed a significant difference in proof comprehension performance between the pre- and post-tests. Moreover, results from the interviews showed that prospective teachers reasoned quantitatively on the proof comprehension dimensions. Results suggest that engaging in quantitative reasoning during instruction may help to develop proof comprehension, particularly in situations involving the analysis of proofs entailing properties of the real number system. We recommend embedding quantitative reasoning in teacher education and professional development programs to facilitate mathematics teachers’ proof comprehension and proving activities.     

Programming—an integral part of an elementary mathematics methods course

This paper argues that programming is not only feasible but a necessary component in elementary mathematics teacher training. The paper also presents a model for a teacher training course which emphasizes the meta‐morphic stages of readiness that programming must go through to be of benefit to pre‐service teachers.     

Classroom sociomathematical norms for proof presentation in undergraduate in abstract algebra

This paper is a case study of the teaching of an undergraduate abstract algebra course with a particular focus on the manner in which the students presented proofs and the class engaged in a subsequent discussion of those proofs that included validating the work. This study describes norms for classroom work that include a set of norms that the presenter of a proof was responsible for enacting, including only using previously agreed upon results, as well as a separate set that the audience was to enact related to developing their understanding of the presented proof and validating the work. The study suggests that the students developed a sense of communal and individual responsibility for contributing to growing the body of mathematical knowledge known by the class, with an implied responsibility for knowing the already developed mathematics. Moreover, the behaviors that norms prompted the students to engage were those that literature suggests leads to increased comprehension of proofs.     

Language and Mathematics - Key Factors influencing the Comprehension Process in reality-based Tasks

Solving reality-based tasks is an important goal in mathematics instruction and is anchored in education standards determined by mathematical modeling skills. These tasks demand a serious examination of the real-world as well as text comprehension to successfully solve them. Therefore, this study empirically reconstructed the comprehension process during the solution of reality-based tasks and examined how it correlates with process-, person-, and task-related attributes. Fifty-five seventh graders using the Think Aloud Method solved reality-based tasks that were varied in their linguistic and situational complexity level. Their mathematical performance as well as their reading ability were measured. Based on detailed analyses of solution processes combined with the performance data, we point out the relevance of comprehension activities and empirically identify factors that influence the comprehension process.     

Students’ ways of understanding a proof

We present a model for describing the growth of students’ understandings when reading a proof. The model is composed of two main paths. One is focused on becoming aware of the deductive structure of the proof, in other words, understanding the proof at a semantic level. Generalization, abstraction, and formalization are the most important transitions in this path. The other path focuses on the surface-level form of the proof, and the use of symbolic representations. At the end of this path, students understand how and why symbolic computations formally establish a claim, at a syntactic level. We make distinctions between states in the model and illustrate them with examples from early secondary students’ mathematical activity. We then apply the model to one student’s developing understanding in order to show how the model works in practice. We close with some suggestions for further research.     

Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.     

Weight of the comprehension axiom in a theory based on logic without contractions

A set-theoretic LST system based on a logic without rules of contraction of identical formulas in the antecedent or succedent of sequents is introduced. The set-theoretic axioms used are the comprehension principle,

绝对连续测度的Loeb代数

It is proved that if μ and v be finite measures on a measurable space (X,S) and v is absolutely continuous with respect to v, then it is holds that L(^*S,^*v) C L(^*S,^*v), while L(^*S,^*v) and L(^*S,^*v) are the Loeb algebras with respect to measure spaces (X,S,μ) and(X,S,μ).     

数字图像处理技术在客观题阅卷中的应用

针对光电式阅卷机价格昂贵,使用成本高,设备利用率低、维护困难的缺点,介绍了一种采用数字图像处理技术,分析、理解答题卡图像,正确提取考试信息,结合数据库技术实现智能阅卷和成绩统计的新方法.首先对答题卡图像进行纠斜、定位、分割等预处理,然后通过比较答题卡图像中各选项的平均灰度值大小,不仅能识别出正确选项,而且能识别出错选、重选、漏选等选项.实验表明用此法实现客观题阅卷具有硬件价格低,使用成本低、设备利用率高、无需维护、阅卷速度快的优点,具有极大的推广应用价值.     

基于局部评价环境的群组评价状态测评研究

群组评价状态测评是提高群组评价方法适用性和有效性的途径之一,以评价参与者为节点,构建具有评价群体之间链接关系的群体网络结构,即评价局部环境,将评价群体网络结构信息与属性信息进行整合构造局部评价环境的群组评价状态测度,实现协商过程有效跟随,实时对评价参与者和群组评价状态进行测评。根据测评结果给出具有评价参与者核心性测度、群体协商充分性和有效性测度的建议,对评价参与者的评价行为进行引导,给出具有导向性的评价信息集结方法建议。给出示例对群组评价状态测评进行示范解释。     

Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic

In , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of , and we prove an analogy of Hájek’s theorem with a very simple procedure.      

LEVELS OF IMPLICATION AND TYPE FREE THEORIES OF CLASSIFICATIONS WITH APPROXIMATION OPERATOR

We investigate a theory of Frege structures extended by the Myhill-Flagg hierarchy of implications. We study its relation to a property theory with an approximation operator and we give a proof theoretical analysis of the basic system involved. MSC: 03F35, 03D60.     

On a positive set theory with inequality

We introduce a quite natural Frege‐style set theory, which we call Strong‐Frege‐2 $(\mathsf {SF}_2)$, a sort of simplification of the theory considered in 13 (under the name Strong‐Frege‐3) and 1 (under the name F2). We give a model of a weaker variant of $\mathsf {SF}_2$, called $\mathsf {SF}_2\mathsf {AC}$, where atoms and coatoms are allowed. To construct the model we use an enumeration “almost without repetitions” of the Π¹₁ sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of 5 and 15 . © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

CONTENTS OF VOLUME 30

正~~     

Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with