The L_(p) Shephard problem on entropy of log-concave functions

this paper,we introduce the L_(p) Shephard problem on entropy of log-concave functions,a comparison problem:whether ∏_(p)f≤∏_(p)g implies that Ent(f)≥Ent(g),for 1≤p<n,and Ent(f)≤Ent(g),for n<p,where ∏_(p)f is the L_(p) projection body of a log-concave function f.Our results give a partial answer to this problem.

The Shephard Type Problems for General L_p-Centroid Bodies

In this paper, combining with the L_p-dual geominimal surface area and the general L_p-centroid bodies, we research the Shephard type problems for general L_p-centroid bodies.

On the Analog of Shephard Problem for L_p-polar Projection Bodies

For p > 0, Lutwak, Yang and Zhang introduced the concept of L_p-polar projection body Γ_(-p)K of a convex body K in Rn. Let p ≥ 1 and K, L ? Rnbe two origin-symmetric convex bodies, we consider the question of whether Γ_(-p) K ? Γ_(-p) L implies ?_p(L) ≤ ?_p(K),where ?_p(K) denotes the L_p-affine surface area of K and K = Voln(K)^(-1/p) K. We prove a necessary and sufficient condition of an analog of the Shephard problem for the L_p-polar projection bodies.

平面Shephard问题的一类反例

利用常宽等腰梯形和椭圆举出平面上Shephard问题的一类精确反例,利用圆与Reuleaux三角形比较的经典反例为所得结论的特殊反例.

Diagrams Produced by Secondary Students in Multiplicative Comparison Word Problems

We investigated the use of diagrams in multiplicative comparison word problems. The diagrams have been considered as one of the effective heuristic strategies or solving math problems. However, how students use during their school and the degree development that shows in their performance when applied to specific fields of knowledge is a task to be elucidated. We place our study in the school stage in which it makes the transition from arithmetic to algebra and arithmetic problems we focus on in the underlying multiplicative comparison scheme. In this paper, we analyzed the responses of high school students to the translation of multiplicative comparison word problems to representation graphs. We have used the responses of 12 -14 year old students （freshman year of secondary school） to represent multiplicative comparison word problems to identify and categorize the students responses, which allowed us identify categories for each type of representation and hypothesize priority order and subordination between the categories. Results show that students are not familiar with building diagrams that integrate existing relations in word problems. Most of the students do not use all the quantitative information contained in the word problem, therefore draw diagrams referring to the subject or context of the problem without relating to the data in it. We describe in detail the quantitative diagram types produced by these students. We have identified four kinds of quantitative diagrams that the students used to represent the multiplicative comparison problems with inconsistent statements, and these diagrams correspond to the four strategies for tackling the construction of the diagram.

Automatic Generation of Amharic Math Word Problem and Equation

Math word problem uses a real word story to present basic arithmetic operations using textual narration. It is used to develop student’s comprehension skill in conjunction with the ability to generate a solution that agrees with the story given in the problem. To master math word problem solving, students need to be given fresh and enormous amount of problems, which normal textbooks as well as teachers fail to provide most of the time. To fill the gap, a few research works have been proposed on techniques to automatically generate math word problems and equations mainly for English speaking community. Amharic is a Semitic language spoken by more than hundred million Ethiopians and is a language of instruction in elementary schools in Ethiopia. And yet it belongs to one of a less resourced language in the field of linguistics and natural language processing (NLP). Hence, in this paper, a strategy for automatic generation of Amharic Math Word (AMW) problem and equation is proposed, which is a first attempt to introduce the use template based shallow NLP approach to generate math word problem for Amharic language as a step towards enabling comprehension and learning problem solving in mathematics for primary school students. The proposed novel technique accepts a sample AMW problem as user input to form a template. A template provides AMW problem with placeholders, type of problem and equation template. It is used as a pattern to generate semantically equivalent AMW problems with their equations. To validate the reality of the proposed approach, a prototype was developed and used as a testing platform. Experimental results have shown 93.84% overall efficiency on the core task of forming templates from a given corpus containing AMW problems collected from elementary school mathematics textbooks and other school worksheets. Human judges have also found generated AMW problem and equation as solvable as the textbook problems.

Solving Arithmetic Word Problems of Entailing Deep Implicit Relations by Qualia Syntax-Semantic Model

Solving arithmetic word problems that entail deep implicit relations is still a challenging problem.However,significant progress has been made in solving Arithmetic Word Problems(AWP)over the past six decades.This paper proposes to discover deep implicit relations by qualia inference to solve Arithmetic Word Problems entailing Deep Implicit Relations(DIR-AWP),such as entailing commonsense or subject-domain knowledge involved in the problem-solving process.This paper proposes to take three steps to solve DIR-AWPs,in which the first three steps are used to conduct the qualia inference process.The first step uses the prepared set of qualia-quantity models to identify qualia scenes from the explicit relations extracted by the Syntax-Semantic(S2)method from the given problem.The second step adds missing entities and deep implicit relations in order using the identified qualia scenes and the qualia-quantity models,respectively.The third step distills the relations for solving the given problem by pruning the spare branches of the qualia dependency graph of all the acquired relations.The research contributes to the field by presenting a comprehensive approach combining explicit and implicit knowledge to enhance reasoning abilities.The experimental results on Math23K demonstrate hat the proposed algorithm is superior to the baseline algorithms in solving AWPs requiring deep implicit relations.

Lp-Blaschke-Minkowski同态的Shephard问题

Schuster定义了Blaschke-Minkowski同态的概念之后,汪卫又将此概念推广到了Lp形式.在本文中,结合Lp仿射表面积,主要研究了Lp-Blaschke-Minkowski同态的Shephard问题.

Word Problems about Circles

In primary school maths, CIRCLE is one of the most important units. Some of the students would think it would be okay to be able to understand the relationship between the diameter or the radius and the circumference or the area of a circle.Here is a word problem:Choose the correct option A, B or C.Inthesquare,thereisaquartercircleandasmallcircle.

基于外部知识检测的数学几何单词问题求解

随着人工智能技术的不断发展,深度学习方法在数学单词问题智能求解方面得到了广泛应用。基于图到树的编解码网络表现出良好的性能,但外部知识缺乏使得模型求解准确率提升受到限制,构建一种基于几何知识库的检测模块,将其融入编解码器网络模型,提高了模型的预测能力。通过在中文数学问题数据集GeometryQA上进行验证,模型表现出更高的准确率,具有一定的优越性。

基于类比学习的数学应用题求解模型

目前基于类比学习的数学应用题(MWP)求解的研究多从语义相似度或浅层逻辑来筛选样本,存在样本匹配度不足以及样本选取局限于数据集的问题。针对以上问题,提出一种新的基于类比学习的数学应用题求解(MWP-AL)模型。该模型主要从2个角度对数学应用题进行类比学习。从文本编码的角度进行样本筛选,从余弦相似度、树解顶节点以及树深度3个维度对样本进行限制。该方法从语义层面以及深层逻辑方面对样本进行选取,得到的样本与原题的匹配度更高。从解方程的角度进行样本构建,从方程本身出发,针对不同类型的方程在逻辑方面对其进行变体从而构建样本。该方法不局限于从数据集中选取样本,具有较强的泛化性。通过计算交叉熵损失函数对这2种样本进行类比学习。实验结果表明,在2个基线模型上加入MWP-AL模型后,其在英文数据集Math QA和中文数据集Math23K上的准确率分别提升了1.8、2.5和2.8、1.3个百分点,同时较其他基线模型均有所提升。

小学2-4年级儿童数学应用题表征策略对其解决不规则问题影响的研究

本研究运用实验法和临床访谈法对某普通小学的123名2-4年级学生进行了不规则数学应用题测验,以考察数学学优生和学差生在解决这些应用题时表征策略的差异。结果表明:⑴从2-4年级儿童解答条件多余和条件不足应用题上看,学优生的成绩要好于学差生,口头报告分析显示学优生较多地使用问题模型策略对问题进行表征,而学差生较多使用直接转换策略对问题进行表征;⑵在三年级时,条件多余和条件不足应用题的解题正确率有所提高,但到四年级时,正确率却有所下降;⑶在解决条件多余和条件不足应用题中,性别差异不显著。

小学2～4年级儿童数学应用题表征策略差异的研究

本研究运用实验法和临床访谈法对某普通小学的123名2～4年级学生进行了数学应用题测验,以考察数学学优生和学差生在解决比较应用题时表征策略的差异。结果表明:(1)从2～4年级儿童解答一致和不一致应用题上看,学优生较多地使用问题模型策略对问题进行表征,学差生较多地使用直接转换策略对问题进行表征;(2)除学差女生的解题正确率低于学差男生的正确率,学差女生自我报告中直接转换策略的使用多于学差男生外,在其它方面,性别差异并不显著;(3)随着年级的升高,学优生在使用问题模型策略上越来越成熟,学差生并没有学会使用更加有效的问题模型表征策略,仍然停留在直接转换策略上,但他们在关于策略使用的认识上有所提高。

加减文字题解决研究概述

加减文字题指应用加减法运算解答的简单数学应用题。基本类型有合并题、变化题和比较题。人们主要采用四种方法研究解题过程:解答问题、回忆和构造问题、建立计算机模型和眼动记录。过去研究发现语义类型、年龄、难以理解的词句、问题陈述的简约性、题材个人化、问题陈述结构、数量大小、未知集类型和解答问题的方式等因素显著影响解题过程。人们对解题过程提出了两种理论模型,一是数学知识应用模型,一是语言理解模型。

小学数学真实性问题解决的调查研究

在常规课堂上对来自北京市两所小学 1 48位 1 0 -1 2岁的小学生进行真实性问题的测验 ,研究结果表明 :1 )被试中对真实数学问题作出真实性解答的人数比例约为四分之一 ,高于国外同类研究中比例。 2 )作出常规解答的人数比例约为一半 ,显著高于作出真实解答的人数比例。 3 )六年级被试作出真实解答的人数比例普遍高于四年级 ,但是在某些问题上 ,六年级被试中作出常规解答的人数比例反而高于四年级。 4)被试对不同真实问题的反应存在一定的差异 ,这可能与真实问题的特点有关。

外部表征和问题呈现方式对小学生数学应用题解决的影响

运用实验法对208名小学四年级学生进行了应用题测验,以考察问题呈现方式、外部表征方式对小学数学应用题解决的影响.结果表明:学生在静态呈现方式下解答应用题的成绩要优于动态呈现方式下的应用题成绩;无论是静态呈现还是动态呈现,信息图表征的成绩都优于文字描述表征和装饰图表征成绩.

数学应用题图式层次性研究

数学应用题图式具有层次性,这种层次性主要集中在思维图式范围内,而且随着层次的递进,图式抽象度越来越高;数学应用题的图式层次性与徐利治先生提出的抽象度理论具有高度一致性.利用图式的层次性可以使数学应用题课程设计更趋于科学化.

阅读水平不同的儿童解决比较型算术题研究

以不同阅读水平的30名小学三年级儿童为被试,以比较型算术应用题为材料,对解决算术应用题的能力进行研究,结果如下:1 )在总得分上,阅读能力优秀组的得分显著高于阅读能力不良组,其它各组之间差异不显著。在解决问题的总时间上,阅读理解主效应显著,各组之间差异均显著,阅读能力越好,解题时间越少。2 )题目类型主效应显著,一致型题得分显著高于不一致型题。在解题时间上,题目类型主效应不显著。

小学生表征数学应用题策略的实验研究

通过一个 2 (成功与否 )× 2 (提示与否 )× 2 (题型 )的混合实验设计 ,对小学五年级学生解决和差应用题的表征策略进行了研究。结果表明 :( 1 )与比较应用题的表征相类似 ,小学生对和差应用题的表征也存在着直译策略和问题模型策略 ;( 2 )不成功组解题者在表征和差应用题时倾向于运用直译策略 ,而成功组的解题者更倾向于运用问题模型策略 ,这导致了成功者与不成功者在列式上的差异 ,特别是在不一致题型上表现得更明显 ;( 3 )在读题前给以“请注意理解这道题的意思”这样简单的提示 ,对不成功的解题者对和差问题的正确表征并不能起到作用 ;( 4 )成功的和差应用题解题者和不成功的解题者在列式正确性的自我评价上存在显著差异。

儿童数学应用题表征水平的特点研究

本研究运用实验法对某普通小学的161名4-6年级学生进行了长方形面积任务和MPI测验，以考察学生的应用题表征水平及其对问题解决的影响。结果表明：（1）表征水平随着年级的升高而不断提高，学生在表征水平上性别差异不显著；（2）无论哪个年级，表征水平都是优等生好于中等生，中等生好于差等生；表征水平越高，学生在问题解决上的成绩越好；（3）随着题目难度的不断加大，各表征水平的学生在解题正确率上的差距也在不断拉大。也就是说，题目越难，表征水平在解题中的作用也就越明显。