Process Inquiry: Analysis of Oral Problem-Solving Skills in Mathematics of Engineering Students

ERIC Educational Resources Information Center

Trance, Naci John C.

2013-01-01

This paper presents another effort in determining the difficulty of engineering students in terms of solving word problems. Students were presented with word problems in algebra. Then, they were asked to solve the word problems orally; that is, before they presented their written solutions, they were required to explain how they understood the…

Gender Differences in Solution of Algebraic Word Problems Containing Irrelevant Information.

ERIC Educational Resources Information Center

Low, Renae; Over, Ray

1993-01-01

Female tenth graders (n=217) were less likely than male tenth graders (n=219) to identify missing or irrelevant information in algebra problems. Female eleventh graders (n=234) were less likely than male eleventh graders (n=287) to solve problems with irrelevant information. Results indicate sex differences in knowledge of problem structure. (SLD)

Remediating Number Combination and Word Problem Deficits Among Students With Mathematics Difficulties: A Randomized Control Trial

PubMed Central

Fuchs, Lynn S.; Powell, Sarah R.; Seethaler, Pamela M.; Cirino, Paul T.; Fletcher, Jack M.; Fuchs, Douglas; Hamlett, Carol L.; Zumeta, Rebecca O.

2009-01-01

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students’ math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20–30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students’ mathematics difficulty status. The tutoring protocols proved transportable across sites. PMID:19865600

Embedding Number-Combinations Practice Within Word-Problem Tutoring

PubMed Central

Powell, Sarah R.; Fuchs, Lynn S.; Fuchs, Douglas

2012-01-01

Two aspects of mathematics with which students with mathematics learning difficulty (MLD) often struggle are word problems and number-combination skills. This article describes a math program in which students receive instruction on using algebraic equations to represent the underlying problem structure for three word-problem types. Students also learn counting strategies for answering number combinations that they cannot retrieve from memory. Results from randomized-control trials indicated that embedding the counting strategies for number combinations produces superior word-problem and number-combination outcomes for students with MLD beyond tutoring programs that focus exclusively on number combinations or word problems. PMID:22661880

Embedding Number-Combinations Practice Within Word-Problem Tutoring.

PubMed

Powell, Sarah R; Fuchs, Lynn S; Fuchs, Douglas

2010-09-01

Two aspects of mathematics with which students with mathematics learning difficulty (MLD) often struggle are word problems and number-combination skills. This article describes a math program in which students receive instruction on using algebraic equations to represent the underlying problem structure for three word-problem types. Students also learn counting strategies for answering number combinations that they cannot retrieve from memory. Results from randomized-control trials indicated that embedding the counting strategies for number combinations produces superior word-problem and number-combination outcomes for students with MLD beyond tutoring programs that focus exclusively on number combinations or word problems.

Algebra Word Problem Solving Approaches in a Chemistry Context: Equation Worked Examples versus Text Editing

ERIC Educational Resources Information Center

Ngu, Bing Hiong; Yeung, Alexander Seeshing

2013-01-01

Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…

Reflective thinking in solving an algebra problem: a case study of field independent-prospective teacher

NASA Astrophysics Data System (ADS)

Agustan, S.; Juniati, Dwi; Yuli Eko Siswono, Tatag

2017-10-01

Nowadays, reflective thinking is one of the important things which become a concern in learning mathematics, especially in solving a mathematical problem. The purpose of this paper is to describe how the student used reflective thinking when solved an algebra problem. The subject of this research is one female student who has field independent cognitive style. This research is a descriptive exploratory study with data analysis using qualitative approach to describe in depth reflective thinking of prospective teacher in solving an algebra problem. Four main categories are used to analyse the reflective thinking in solving an algebra problem: (1) formulation and synthesis of experience, (2) orderliness of experience, (3) evaluating the experience and (4) testing the selected solution based on the experience. The results showed that the subject described the problem by using another word and the subject also found the difficulties in making mathematical modelling. The subject analysed two concepts used in solving problem. For instance, geometry related to point and line while algebra is related to algebra arithmetic operation. The subject stated that solution must have four aspect to get effective solution, specifically the ability to (a) understand the meaning of every words; (b) make mathematical modelling; (c) calculate mathematically; (d) interpret solution obtained logically. To test the internal consistency or error in solution, the subject checked and looked back related procedures and operations used. Moreover, the subject tried to resolve the problem in a different way to compare the answers which had been obtained before. The findings supported the assertion that reflective thinking provides an opportunity for the students in improving their weakness in mathematical problem solving. It can make a grow accuracy and concentration in solving a mathematical problem. Consequently, the students will get the right and logic answer by reflective thinking.

On alphabetic presentations of Clifford algebras and their possible applications

NASA Astrophysics Data System (ADS)

Toppan, Francesco; Verbeek, Piet W.

2009-12-01

In this paper, we address the problem of constructing a class of representations of Clifford algebras that can be named "alphabetic (re)presentations." The Clifford algebra generators are expressed as m-letter words written with a three-character or a four-character alphabet. We formulate the problem of the alphabetic presentations, deriving the main properties and some general results. At the end, we briefly discuss the motivations of this work and outline some possible applications.

Effects of Modified Schema-Based Instruction on Real-World Algebra Problem Solving of Students with Autism Spectrum Disorder and Moderate Intellectual Disability

ERIC Educational Resources Information Center

Root, Jenny Rose

2016-01-01

The current study evaluated the effects of modified schema-based instruction (SBI) on the algebra problem solving skills of three middle school students with autism spectrum disorder and moderate intellectual disability (ASD/ID). Participants learned to solve two types of group word problems: missing-whole and missing-part. The themes of the word…

Effects of Preventative Tutoring on the Mathematical Problem Solving of Third-Grade Students With Math and Reading Difficulties.

PubMed

Fuchs, Lynn S; Seethaler, Pamela M; Powell, Sarah R; Fuchs, Douglas; Hamlett, Carol L; Fletcher, Jack M

2008-01-01

This study assessed the effects of preventative tutoring on the math problem solving of third-grade students with math and reading difficulties. Students (n = 35) were assigned randomly to continue in their general education math program or to receive secondary preventative tutoring 3 times per week, 30 min per session, for 12 weeks. Schema-broadening tutoring taught students to (a) focus on the mathematical structure of 3 problem types; (b) recognize problems as belonging to those 3 problem-type schemas; (c) solve the 3 word-problem types; and (d) transfer solution methods to problems that include irrelevant information, 2-digit operands, missing information in the first or second positions in the algebraic equation, or relevant information in charts, graphs, and pictures. Also, students were taught to perform the calculation and algebraic skills foundational for problem solving. Analyses of variance revealed statistically significant effects on a wide range of word problems, with large effect sizes. Findings support the efficacy of the tutoring protocol for preventing word-problem deficits among third-grade students with math and reading deficits.

Effects of Preventative Tutoring on the Mathematical Problem Solving of Third-Grade Students With Math and Reading Difficulties

PubMed Central

Fuchs, Lynn S.; Seethaler, Pamela M.; Powell, Sarah R.; Fuchs, Douglas; Hamlett, Carol L.; Fletcher, Jack M.

2009-01-01

This study assessed the effects of preventative tutoring on the math problem solving of third-grade students with math and reading difficulties. Students (n = 35) were assigned randomly to continue in their general education math program or to receive secondary preventative tutoring 3 times per week, 30 min per session, for 12 weeks. Schema-broadening tutoring taught students to (a) focus on the mathematical structure of 3 problem types; (b) recognize problems as belonging to those 3 problem-type schemas; (c) solve the 3 word-problem types; and (d) transfer solution methods to problems that include irrelevant information, 2-digit operands, missing information in the first or second positions in the algebraic equation, or relevant information in charts, graphs, and pictures. Also, students were taught to perform the calculation and algebraic skills foundational for problem solving. Analyses of variance revealed statistically significant effects on a wide range of word problems, with large effect sizes. Findings support the efficacy of the tutoring protocol for preventing word-problem deficits among third-grade students with math and reading deficits. PMID:20209074

Longer Bars for Bigger Numbers? Children's Usage and Understanding of Graphical Representations of Algebraic Problems

ERIC Educational Resources Information Center

Lee, Kerry; Khng, Kiat Hui; Ng, Swee Fong; Ng Lan Kong, Jeremy

2013-01-01

In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students' understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also…

Challenges in Math.

ERIC Educational Resources Information Center

Feng, Chengde

1992-01-01

Fourteen mathematics problems from the 1987 Chinese Primary School Mathematics Examination for fifth and sixth grade students are presented. The word problems, accompanied by answers, involve algebra, division, ratios, areas, and other mathematical processes. (JDD)

Meta-Representation in an Algebra I Classroom

ERIC Educational Resources Information Center

Izsak, Andrew; Caglayan, Gunhan; Olive, John

2009-01-01

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different…

A Comparison of Two Mathematics Problem-Solving Strategies: Facilitate Algebra-Readiness

ERIC Educational Resources Information Center

Xin, Yan Ping; Zhang, Dake; Park, Joo Young; Tom, Kinsey; Whipple, Amanda; Si, Luo

2011-01-01

The authors compared a conceptual model-based problem-solving (COMPS) approach with a general heuristic instructional approach for teaching multiplication-division word-problem solving to elementary students with learning problems (LP). The results indicate that only the COMPS group significantly improved, from pretests to posttests, their…

Are Patterns Important? An Investigation of the Relationships between Proficiencies in Patterns, Computation, Executive Functioning, and Algebraic Word Problems

ERIC Educational Resources Information Center

Lee, Kerry; Ng, Swee Fong; Bull, Rebecca; Pe, Madeline Lee; Ho, Ringo Ho Moon

2011-01-01

Although mathematical pattern tasks are often found in elementary school curricula and are deemed a building block for algebra, a recent report (National Mathematics Advisory Panel, 2008) suggests the resources devoted to its teaching and assessment need to be rebalanced. We examined whether children's developing proficiency in solving algebraic…

Fundamentals of the Design and the Operation of an Intelligent Tutoring System for the Learning of the Arithmetical and Algebraic Way of Solving Word Problems

ERIC Educational Resources Information Center

Arnau, David; Arevalillo-Herraez, Miguel; Puig, Luis; Gonzalez-Calero, Jose Antonio

2013-01-01

Designers of interactive learning environments with a focus on word problem solving usually have to compromise between the amount of resolution paths that a user is allowed to follow and the quality of the feedback provided. We have built an intelligent tutoring system (ITS) that is able to both track the user's actions and provide adequate…

The Effect of Strategy on Problem Solving: An FMRI Study

ERIC Educational Resources Information Center

Newman, Sharlene D.; Pruce, Benjamin; Rusia, Akash; Burns, Thomas, Jr.

2010-01-01

fMRI was used to examine the differential effect of two problem-solving strategies. Participants were trained to use both a pictorial/spatial and a symbolic/algebraic strategy to solve word problems. While these two strategies activated similar cortical regions, a number of differences were noted in the level of activation. These differences…

Kleene Algebra and Bytecode Verification

DTIC Science & Technology

2016-04-27

computing the star (Kleene closure) of a matrix of transfer functions. In this paper we show how this general framework applies to the problem of Java ...bytecode verification. We show how to specify transfer functions arising in Java bytecode verification in such a way that the Kleene algebra operations...potentially improve the performance over the standard worklist algorithm when a small cutset can be found. Key words: Java , bytecode, verification, static

Transition Mathematics. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2007

2007-01-01

"Transition Mathematics" aims to increase 7th- through 12th-grade students' skills in applied arithmetic, pre-algebra, and pre-geometry. This one-year curriculum also addresses general application to different wordings of problems, types of numbers, and contexts for problems and aims to promote mathematical reading skills. The curriculum…

Using Verbal Protocol Data to Reflect the Quality of Problem Representation in Solving Algebra Word Problems.

ERIC Educational Resources Information Center

Bull, Elizabeth Kay

The goal of this study was to find a way to quantify three criteria of representational quality, described by Greeno, so that it would be possible to examine statistically the relationship between representational quality and other variables related to problem solution. The sample consisted of 18 college students, 84 percent of whom had…

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

NASA Astrophysics Data System (ADS)

Bokut, L. A.; Chen, Yuqun; Li, Yu

2009-03-01

One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962).

Successfully Transitioning to Linear Equations

ERIC Educational Resources Information Center

Colton, Connie; Smith, Wendy M.

2014-01-01

The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

Improving the Fraction Word Problem Solving of Students with Mathematics Learning Disabilities: Interactive Computer Application

ERIC Educational Resources Information Center

Shin, Mikyung; Bryant, Diane P.

2017-01-01

Students with mathematics learning disabilities (MLD) have a weak understanding of fraction concepts and skills, which are foundations of algebra. Such students might benefit from computer-assisted instruction that utilizes evidence-based instructional components (cognitive strategies, feedback, virtual manipulatives). As a pilot study using a…

Types of Mathematics-Language Reading Interactions that Unnecessarily Hinder Algebra Learning and Assessment

ERIC Educational Resources Information Center

Lager, Carl A.

2006-01-01

English language learners (ELLs)--one of the lowest-achieving and fastest growing middle-school subpopulations--are challenged in the classroom by language components which could potentially jumpstart real mathematical growth for these and, eventually, all other students as well. Going beyond traditional word problems, this study documented and…

Assessing Translation Misconceptions inside the Classroom: A Presentation of an Instrument and Its Results

ERIC Educational Resources Information Center

Mangulabnan, Pauline Anne Therese M.

2013-01-01

This is a descriptive research on the difficulties of Filipino high school students in translating algebraic word problems into mathematical equations. This research is composed of three parts: (1) development of an 11-page "Filipinized" questionnaire; (2) analysis of the mathematical thinking processes of the respondents based on the answers to…

Investigating a Link between Pre-Calculus Students' Uses of Graphing Calculators and Their Understanding of Mathematical Symbols

ERIC Educational Resources Information Center

Kenney, Rachael H.

2014-01-01

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…

Exploration of the Factors That Support Learning: Web-Based Activity and Testing Systems in Community College Algebra

ERIC Educational Resources Information Center

Hauk, Shandy; Matlen, Bryan

2016-01-01

A variety of computerized interactive learning platforms exist. Most include instructional supports in the form of problem sets. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially to fully worked examples. Behind-the-scenes design of such systems varies as well --from static dictionaries of…

Exploration of the Factors That Support Learning: Web-Based Activity and Testing Systems in Community College Algebra [Contributed Report

ERIC Educational Resources Information Center

Hauk, Shandy; Matlen, Bryan; Thomas, Larry

2017-01-01

A variety of computerized interactive learning platforms exist. Most include instructional supports in the form of problem sets. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially- to fully-worked examples. Behind-the-scenes design of systems varies as well--from static dictionaries of problems…

Using Cognitive Tutor Software in Learning Linear Algebra Word Concept

ERIC Educational Resources Information Center

Yang, Kai-Ju

2015-01-01

This paper reports on a study of twelve 10th grade students using Cognitive Tutor, a math software program, to learn linear algebra word concept. The study's purpose was to examine whether students' mathematics performance as it is related to using Cognitive Tutor provided evidence to support Koedlinger's (2002) four instructional principles used…

Analysis of Secondary School Students’ Algebraic Thinking and Math-Talk Learning Community to Help Students Learn

NASA Astrophysics Data System (ADS)

Nurhayati, D. M.; Herman, T.; Suhendra, S.

2017-09-01

This study aims to determine the difficulties of algebraic thinking ability of students in one of secondary school on quadrilateral subject and to describe Math-Talk Learning Community as the alternative way that can be done to overcome the difficulties of the students’ algebraic thinking ability. Research conducted by using quantitative approach with descriptive method. The population in this research was all students of that school and twenty three students as the sample that was chosen by purposive sampling technique. Data of algebraic thinking were collected through essay test. The results showed the percentage of achievement of students’ algebraic thinking’s indicators on three aspects: a) algebra as generalized arithmetic with the indicators (conceptually based computational strategies and estimation); b) algebra as the language of mathematics (meaning of variables, variable expressions and meaning of solution); c) algebra as a tool for functions and mathematical modelling (representing mathematical ideas using equations, tables, or words and generalizing patterns and rules in real-world contexts) is still low. It is predicted that because the secondary school students was not familiar with the abstract problem and they are still at a semi-concrete stage where the stage of cognitive development is between concrete and abstract. Based on the percentage achievement of each indicators, it can be concluded that the level of achievement of student’s mathematical communication using conventional learning is still low, so students’ algebraic thinking ability need to be improved.

Exploration of the Factors That Support Learning: Web-Based Activity and Testing Systems in Community College Algebra [Conference Long Paper

ERIC Educational Resources Information Center

Hauk, Shandy; Matlen, Bryan J.

2017-01-01

This is an extended conference proceedings report [Long Paper] based on a shorter contributed report at the same conference. A variety of computerized learning platforms exist. In mathematics, most include sets of problems to complete. Feedback to users ranges from a single word like "Correct!" to offers of hints and partially- to…

Developing "Algebraic Thinking": Two Key Ways to Establish Some Early Algebraic Ideas in Primary Classrooms

ERIC Educational Resources Information Center

Ormond, Christine

2012-01-01

Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…

An Evaluation of Words in Color or Morphologico-Algebraic Approach to Teaching Reading to Functionally Illiterate Adults.

ERIC Educational Resources Information Center

Hinds, Lillian R.

Seventy Cleveland, Ohio, inner city adult illiterates, 33 from an experimental group and 37 from a contrast group, were studied to determine the efficiency and effectiveness of Words in Color or the Morphologico-Algebraic approach to teaching reading. Results indicated that the reading achievement gain of functionally illiterate adults taught by…

Pre-Algebra Lexicon.

ERIC Educational Resources Information Center

Hayden, Dunstan; Cuevas, Gilberto

The pre-algebra lexicon is a set of classroom exercises designed to teach the technical words and phrases of pre-algebra mathematics, and includes the terms most commonly found in related mathematics courses. The lexicon has three parts, each with its own introduction. The first introduces vocabulary items in three groups forming a learning…

An Early Algebra Approach to Pattern Generalisation: Actualising the Virtual through Words, Gestures and Toilet Paper

ERIC Educational Resources Information Center

Ferrara, Francesca; Sinclair, Nathalie

2016-01-01

This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based…

Chinese Algebra: Using Historical Problems to Think about Current Curricula

ERIC Educational Resources Information Center

Tillema, Erik

2005-01-01

The Chinese used the idea of generating equivalent expressions for solving problems where the problems from a historical Chinese text are studied to understand the ways in which the ideas can lead into algebraic calculations and help students to learn algebra. The texts unify algebraic problem solving through complex algebraic thought and afford…

Gender differences in algebraic thinking ability to solve mathematics problems

NASA Astrophysics Data System (ADS)

Kusumaningsih, W.; Darhim; Herman, T.; Turmudi

2018-05-01

This study aimed to conduct a gender study on students' algebraic thinking ability in solving a mathematics problem, polyhedron concept, for grade VIII. This research used a qualitative method. The data was collected using: test and interview methods. The subjects in this study were eight male and female students with different level of abilities. It was found that the algebraic thinking skills of male students reached high group of five categories. They were superior in terms of reasoning and quick understanding in solving problems. Algebraic thinking ability of high-achieving group of female students also met five categories of algebraic thinking indicators. They were more diligent, tenacious and thorough in solving problems. Algebraic thinking ability of male students in medium category only satisfied three categories of algebraic thinking indicators. They were sufficient in terms of reasoning and understanding in solving problems. Algebraic thinking ability group of female students in medium group also satisfied three categories of algebraic thinking indicators. They were fairly diligent, tenacious and meticulous on working on the problems.

Extended gauge theory and gauged free differential algebras

NASA Astrophysics Data System (ADS)

Salgado, P.; Salgado, S.

2018-01-01

Recently, Antoniadis, Konitopoulos and Savvidy introduced, in the context of the so-called extended gauge theory, a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by higher degree forms. In this article it is shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can be constructed from an algebraic structure known as free differential algebra. In other words, we show that the above mentioned non-abelian gauge theory, where the gauge fields are described by p-forms with p ≥ 2, can be obtained by gauging free differential algebras.

Students’ Algebraic Reasonsing In Solving Mathematical Problems With Adversity Quotient

NASA Astrophysics Data System (ADS)

Aryani, F.; Amin, S. M.; Sulaiman, R.

2018-01-01

Algebraic reasoning is a process in which students generalize mathematical ideas from a set of particular instances and express them in increasingly formal and age-appropriate ways. Using problem solving approach to develop algebraic reasoning of mathematics may enhace the long-term learning trajectory of the majority students. The purpose of this research was to describe the algebraic reasoning of quitter, camper, and climber junior high school students in solving mathematical problems. This research used qualitative descriptive method. Subjects were determined by purposive sampling. The technique of collecting data was done by task-based interviews.The results showed that the algebraic reasoning of three students in the process of pattern seeking by identifying the things that are known and asked in a similar way. But three students found the elements of pattern recognition in different ways or method. So, they are generalize the problem of pattern formation with different ways. The study of algebraic reasoning and problem solving can be a learning paradigm in the improve students’ knowledge and skills in algebra work. The goal is to help students’ improve academic competence, develop algebraic reasoning in problem solving.

Algebra from Chips and Chopsticks

ERIC Educational Resources Information Center

Yun, Jeong Oak; Flores, Alfinio

2012-01-01

Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

Mathematical Designs for Teaching and Learning Composition.

ERIC Educational Resources Information Center

Laque, Carol Feiser

Algebraic equations and geometric forms are useful in teaching and learning composition. Algebraic equations can illustrate the modular nature of paragraph structures and can be refined by students to describe types of paragraphs. Discussion of the "slippery" nature of words and their power of transformation can be a lecture topic as the class…

Assessing non-uniqueness: An algebraic approach

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vasco, Don W.

Geophysical inverse problems are endowed with a rich mathematical structure. When discretized, most differential and integral equations of interest are algebraic (polynomial) in form. Techniques from algebraic geometry and computational algebra provide a means to address questions of existence and uniqueness for both linear and non-linear inverse problem. In a sense, the methods extend ideas which have proven fruitful in treating linear inverse problems.

Automated speech understanding: the next generation

NASA Astrophysics Data System (ADS)

Picone, J.; Ebel, W. J.; Deshmukh, N.

1995-04-01

Modern speech understanding systems merge interdisciplinary technologies from Signal Processing, Pattern Recognition, Natural Language, and Linguistics into a unified statistical framework. These systems, which have applications in a wide range of signal processing problems, represent a revolution in Digital Signal Processing (DSP). Once a field dominated by vector-oriented processors and linear algebra-based mathematics, the current generation of DSP-based systems rely on sophisticated statistical models implemented using a complex software paradigm. Such systems are now capable of understanding continuous speech input for vocabularies of several thousand words in operational environments. The current generation of deployed systems, based on small vocabularies of isolated words, will soon be replaced by a new technology offering natural language access to vast information resources such as the Internet, and provide completely automated voice interfaces for mundane tasks such as travel planning and directory assistance.

Continuity in Representation between Children and Adults: Arithmetic Knowledge Hinders Undergraduates' Algebraic Problem Solving

ERIC Educational Resources Information Center

McNeil, Nicole M.; Rittle-Johnson, Bethany; Hattikudur, Shanta; Petersen, Lori A.

2010-01-01

This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184)…

Algebraic reasoning and bat-and-ball problem variants: Solving isomorphic algebra first facilitates problem solving later.

PubMed

Hoover, Jerome D; Healy, Alice F

2017-12-01

The classic bat-and-ball problem is used widely to measure biased and correct reasoning in decision-making. University students overwhelmingly tend to provide the biased answer to this problem. To what extent might reasoners be led to modify their judgement, and, more specifically, is it possible to facilitate problem solution by prompting participants to consider the problem from an algebraic perspective? One hundred ninety-seven participants were recruited to investigate the effect of algebraic cueing as a debiasing strategy on variants of the bat-and-ball problem. Participants who were cued to consider the problem algebraically were significantly more likely to answer correctly relative to control participants. Most of this cueing effect was confined to a condition that required participants to solve isomorphic algebra equations corresponding to the structure of bat-and-ball question types. On a subsequent critical question with differing item and dollar amounts presented without a cue, participants were able to generalize the learned information to significantly reduce overall bias. Math anxiety was also found to be significantly related to bat-and-ball problem accuracy. These results suggest that, under specific conditions, algebraic reasoning is an effective debiasing strategy on bat-and-ball problem variants, and provide the first documented evidence for the influence of math anxiety on Cognitive Reflection Test performance.

A Proposed Algebra Assessment for Use in a Problem-Analysis Framework

ERIC Educational Resources Information Center

Walick, Christopher M.; Burns, Matthew K.

2017-01-01

Algebra is critical to high school graduation and college success, but student achievement in algebra frequently falls significantly below expected proficiency levels. While existing research emphasizes the importance of quality algebra instruction, there is little research about how to conduct problem analysis for struggling secondary students.…

Contributions of Domain-General Cognitive Resources and Different Forms of Arithmetic Development to Pre-Algebraic Knowledge

ERIC Educational Resources Information Center

Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Powell, Sarah R.; Schumacher, Robin F.; Hamlett, Carol L.; Vernier, Emily; Namkung, Jessica M.; Vukovic, Rose K.

2012-01-01

The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n = 279, mean age = 7.59 years) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems…

Meanings Given to Algebraic Symbolism in Problem-Posing

ERIC Educational Resources Information Center

Cañadas, María C.; Molina, Marta; del Río, Aurora

2018-01-01

Some errors in the learning of algebra suggest that students might have difficulties giving meaning to algebraic symbolism. In this paper, we use problem posing to analyze the students' capacity to assign meaning to algebraic symbolism and the difficulties that students encounter in this process, depending on the characteristics of the algebraic…

Working memory, worry, and algebraic ability.

PubMed

Trezise, Kelly; Reeve, Robert A

2014-05-01

Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship. Copyright © 2013 Elsevier Inc. All rights reserved.

Just-in-Time Algebra: A Problem Solving Approach Including Multimedia and Animation.

ERIC Educational Resources Information Center

Hofmann, Roseanne S.; Hunter, Walter R.

2003-01-01

Describes a beginning algebra course that places stronger emphasis on learning to solve problems and introduces topics using real world applications. Students learn estimating, graphing, and algebraic algorithms for the purpose of solving problems. Indicates that applications motivate students by appearing to be a more relevant topic as well as…

Abstract numeric relations and the visual structure of algebra.

PubMed

Landy, David; Brookes, David; Smout, Ryan

2014-09-01

Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic.

Primary School Students' Strategies in Early Algebra Problem Solving Supported by an Online Game

ERIC Educational Resources Information Center

van den Heuvel-Panhuizen, Marja; Kolovou, Angeliki; Robitzsch, Alexander

2013-01-01

In this study we investigated the role of a dynamic online game on students' early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10-12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the…

Knowledge of Morphologically Complex Words: A Developmental Study of Older Children and Young Adolescents

ERIC Educational Resources Information Center

Nippold, Marilyn A.; Sun, Lei

2008-01-01

Purpose: This study examined knowledge of derived nominals (e.g., measurement, prediction) and derived adjectives (e.g., algebraic, molecular) in older children and young adolescents. Little was known about students' comprehension of these morphologically complex words that occur in textbooks that are used in public schools to teach challenging…

DOE Fundamentals Handbook: Mathematics, Volume 1

DOE Office of Scientific and Technical Information (OSTI.GOV)

Not Available

1992-06-01

The Mathematics Fundamentals Handbook was developed to assist nuclear facility operating contractors provide operators, maintenance personnel, and the technical staff with the necessary fundamentals training to ensure a basic understanding of mathematics and its application to facility operation. The handbook includes a review of introductory mathematics and the concepts and functional use of algebra, geometry, trigonometry, and calculus. Word problems, equations, calculations, and practical exercises that require the use of each of the mathematical concepts are also presented. This information will provide personnel with a foundation for understanding and performing basic mathematical calculations that are associated with various DOE nuclearmore » facility operations.« less

DOE Fundamentals Handbook: Mathematics, Volume 2

DOE Office of Scientific and Technical Information (OSTI.GOV)

Not Available

1992-06-01

The Mathematics Fundamentals Handbook was developed to assist nuclear facility operating contractors provide operators, maintenance personnel, and the technical staff with the necessary fundamentals training to ensure a basic understanding of mathematics and its application to facility operation. The handbook includes a review of introductory mathematics and the concepts and functional use of algebra, geometry, trigonometry, and calculus. Word problems, equations, calculations, and practical exercises that require the use of each of the mathematical concepts are also presented. This information will provide personnel with a foundation for understanding and performing basic mathematical calculations that are associated with various DOE nuclearmore » facility operations.« less

Numerical methods on some structured matrix algebra problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1996-06-01

This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was tomore » translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools.« less

What Students Choose to Do and Have to Say about Use of Multiple Representations in College Algebra

ERIC Educational Resources Information Center

Herman, Marlena

2007-01-01

This report summarizes findings on strategies chosen by students (n=38) when solving algebra problems related to various functions with the freedom to use a TI-83 graphing calculator, influences on student problem-solving strategy choices, student ability to approach algebra problems with use of multiple representations, and student beliefs on how…

Assessing Algebraic Solving Ability: A Theoretical Framework

ERIC Educational Resources Information Center

Lian, Lim Hooi; Yew, Wun Thiam

2012-01-01

Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…

STUDIES IN THE USE OF COLOR.

ERIC Educational Resources Information Center

HINDS, LILLIAN R.

STUDIES RELATED TO WORDS IN COLOR, THE MORPHOLOGICO-ALGEBRAIC APPROACH TO TEACHING READING, ARE DISCUSSED. ADULT CLASSES IN MILWAUKEE TAUGHT TO READ BY THIS METHOD ACHIEVED A MEAN GAIN OF .93 OF A YEAR IN 30 HOURS OF INSTRUCTION. IN EUCLID, OHIO, KINDERGARTENERS WHOSE PROGRESS WAS FOLLOWED THROUGH THE SECOND GRADE WERE TAUGHT BY WORDS IN COLOR AND…

Literal algebra for satellite dynamics. [perturbation analysis

NASA Technical Reports Server (NTRS)

Gaposchkin, E. M.

1975-01-01

A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.

Learning Algebra from Worked Examples

ERIC Educational Resources Information Center

Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.

2014-01-01

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…

Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report

DOE Office of Scientific and Technical Information (OSTI.GOV)

NONE

1997-12-31

Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that peoplemore » from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.« less

Control and stabilization of decentralized systems

NASA Technical Reports Server (NTRS)

Byrnes, Christopher I.; Gilliam, David; Martin, Clyde F.

1989-01-01

Proceeding from the problem posed by the need to stabilize the motion of two helicopters maneuvering a single load, a methodology is developed for the stabilization of classes of decentralized systems based on a more algebraic approach, which involves the external symmetries of decentralized systems. Stabilizing local-feedback laws are derived for any class of decentralized systems having a semisimple algebra of symmetries; the helicopter twin-lift problem, as well as certain problems involving the stabilization of discretizations of distributed parameter problems, have just such algebras of symmetries.

Form in Algebra: Reflecting, with Peacock, on Upper Secondary School Teaching.

ERIC Educational Resources Information Center

Menghini, Marta

1994-01-01

Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)

Analyzing Algebraic Thinking Using "Guess My Number" Problems

ERIC Educational Resources Information Center

Patton, Barba; De Los Santos, Estella

2012-01-01

The purpose of this study was to assess student knowledge of numeric, visual and algebraic representations. A definite gap between arithmetic and algebra has been documented in the research. The researchers' goal was to identify a link between the two. Using four "Guess My Number" problems, seventh and tenth grade students were asked to write…

Analysis of junior high school students' attempt to solve a linear inequality problem

NASA Astrophysics Data System (ADS)

Taqiyuddin, Muhammad; Sumiaty, Encum; Jupri, Al

2017-08-01

Linear inequality is one of fundamental subjects within junior high school mathematics curricula. Several studies have been conducted to asses students' perform on linear inequality. However, it can hardly be found that linear inequality problems are in the form of "ax + b < dx + e" with "a, d ≠ 0", and "a ≠ d" as it can be seen on the textbook used by Indonesian students and several studies. This condition leads to the research questions concerning students' attempt on solving a simple linear inequality problem in this form. In order to do so, the written test was administered to 58 students from two schools in Bandung followed by interviews. The other sources of the data are from teachers' interview and mathematics books used by students. After that, the constant comparative method was used to analyse the data. The result shows that the majority approached the question by doing algebraic operations. Interestingly, most of them did it incorrectly. In contrast, algebraic operations were correctly used by some of them. Moreover, the others performed expected-numbers solution, rewriting the question, translating the inequality into words, and blank answer. Furthermore, we found that there is no one who was conscious of the existence of all-numbers solution. It was found that this condition is reasonably due to how little the learning components concern about why a procedure of solving a linear inequality works and possibilities of linear inequality solution.

Tracking problem solving by multivariate pattern analysis and Hidden Markov Model algorithms.

PubMed

Anderson, John R

2012-03-01

Multivariate pattern analysis can be combined with Hidden Markov Model algorithms to track the second-by-second thinking as people solve complex problems. Two applications of this methodology are illustrated with a data set taken from children as they interacted with an intelligent tutoring system for algebra. The first "mind reading" application involves using fMRI activity to track what students are doing as they solve a sequence of algebra problems. The methodology achieves considerable accuracy at determining both what problem-solving step the students are taking and whether they are performing that step correctly. The second "model discovery" application involves using statistical model evaluation to determine how many substates are involved in performing a step of algebraic problem solving. This research indicates that different steps involve different numbers of substates and these substates are associated with different fluency in algebra problem solving. Copyright © 2011 Elsevier Ltd. All rights reserved.

Geometry of quantum state manifolds generated by the Lie algebra operators

NASA Astrophysics Data System (ADS)

Kuzmak, A. R.

2018-03-01

The Fubini-Study metric of quantum state manifold generated by the operators which satisfy the Heisenberg Lie algebra is calculated. The similar problem is studied for the manifold generated by the so(3) Lie algebra operators. Using these results, we calculate the Fubini-Study metrics of state manifolds generated by the position and momentum operators. Also the metrics of quantum state manifolds generated by some spin systems are obtained. Finally, we generalize this problem for operators of an arbitrary Lie algebra.

Problems Relating Mathematics and Science in the High School.

ERIC Educational Resources Information Center

Morrow, Richard; Beard, Earl

This document contains various science problems which require a mathematical solution. The problems are arranged under two general areas. The first (algebra I) contains biology, chemistry, and physics problems which require solutions related to linear equations, exponentials, and nonlinear equations. The second (algebra II) contains physics…

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

NASA Technical Reports Server (NTRS)

Tsai, Wen-Lang; Kikuchi, Noboru

1993-01-01

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

Algebraic theory of molecules

NASA Technical Reports Server (NTRS)

Iachello, Franco

1995-01-01

An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.

The Dixmier Map for Nilpotent Super Lie Algebras

NASA Astrophysics Data System (ADS)

Herscovich, Estanislao

2012-07-01

In this article we prove that there exists a Dixmier map for nilpotent super Lie algebras. In other words, if we denote by {Prim({U}({g}))} the set of (graded) primitive ideals of the enveloping algebra {{U}({g})} of a nilpotent Lie superalgebra {{g}} and {{A}d0} the adjoint group of {{g}0}, we prove that the usual Dixmier map for nilpotent Lie algebras can be naturally extended to the context of nilpotent super Lie algebras, i.e. there exists a bijective map I : {g}0^{*}/{A}d0 rightarrow Prim({U}({g})) defined by sending the equivalence class [ λ] of a functional λ to a primitive ideal I( λ) of {{U}({g})}, and which coincides with the Dixmier map in the case of nilpotent Lie algebras. Moreover, the construction of the previous map is explicit, and more or less parallel to the one for Lie algebras, a major difference with a previous approach ( cf. [18]). One key fact in the construction is the existence of polarizations for super Lie algebras, generalizing the concept defined for Lie algebras. As a corollary of the previous description, we obtain the isomorphism {{U}({g})/I(λ) ˜eq Cliffq(k) ⊗ Ap(k)}, where {(p,q) = (dim({g}0/{g}0^{λ})/2,dim({g}1/{g}1^{λ}))}, we get a direct construction of the maximal ideals of the underlying algebra of {{U}({g})} and also some properties of the stabilizers of the primitive ideals of {{U}({g})}.

Finite-dimensional integrable systems: A collection of research problems

NASA Astrophysics Data System (ADS)

Bolsinov, A. V.; Izosimov, A. M.; Tsonev, D. M.

2017-05-01

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan-Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry.

Factors Related to Problem Solving by College Students in Developmental Algebra.

ERIC Educational Resources Information Center

Schonberger, Ann K.

A study was conducted to contrast the characteristics of three groups of college students who completed a developmental algebra course at the University of Maine at Orono during 1980-81. On the basis of a two-part final examination, involving a multiple-choice test of algebraic concepts and skills and a free-response test of problem-solving…

Identities of Finitely Generated Algebras Over AN Infinite Field

NASA Astrophysics Data System (ADS)

Kemer, A. R.

1991-02-01

It is proved that for each finitely generated associative PI-algebra U over an infinite field F, there is a finite-dimensional F-algebra C such that the ideals of identities of the algebras U and C coincide. This yields a positive solution to the local problem of Specht for algebras over an infinite field: A finitely generated free associative algebra satisfies the maximum condition for T-ideals.

Lincos: An interplanetary language. [mathematical method for cosmic radio contact with extraterrestrial life

NASA Technical Reports Server (NTRS)

Freudenthal, H.

1974-01-01

A language for cosmic contacts is envisioned that utilizes radio signals of different wavelengths as sounds to form words. These words are in most cases abbreviations of Latin words understood from their English and French cognates. The logistic syntax uses pauses for punctuation in a binary system; pairs of algebraic formulas are transmitted where in a such pair the second element is always derived from the first; between them is transmitted a word that is understood as -follows- by the listener. The concepts of difference in position, of motion, of space, and of mass can be mathematically described by this language.

Algebraic criteria for positive realness relative to the unit circle.

NASA Technical Reports Server (NTRS)

Siljak, D. D.

1973-01-01

A definition is presented of the circle positive realness of real rational functions relative to the unit circle in the complex variable plane. The problem of testing this kind of positive reality is reduced to the algebraic problem of determining the distribution of zeros of a real polynomial with respect to and on the unit circle. Such reformulation of the problem avoids the search for explicit information about imaginary poles of rational functions. The stated algebraic problem is solved by applying the polynomial criteria of Marden (1966) and Jury (1964), and a completely recursive algorithm for circle positive realness is obtained.

Stability Analysis of Finite Difference Schemes for Hyperbolic Systems, and Problems in Applied and Computational Linear Algebra.

DTIC Science & Technology

FINITE DIFFERENCE THEORY, * LINEAR ALGEBRA , APPLIED MATHEMATICS, APPROXIMATION(MATHEMATICS), BOUNDARY VALUE PROBLEMS, COMPUTATIONS, HYPERBOLAS, MATHEMATICAL MODELS, NUMERICAL ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS, STABILITY.

Rupture or Continuity: The Arithmetico-Algebraic Thinking as an Alternative in a Modelling Process in a Paper and Pencil and Technology Environment

ERIC Educational Resources Information Center

Hitt, Fernando; Saboya, Mireille; Zavala, Carlos Cortés

2017-01-01

Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which…

FOURTH SEMINAR TO THE MEMORY OF D.N. KLYSHKO: Algebraic solution of the synthesis problem for coded sequences

NASA Astrophysics Data System (ADS)

Leukhin, Anatolii N.

2005-08-01

The algebraic solution of a 'complex' problem of synthesis of phase-coded (PC) sequences with the zero level of side lobes of the cyclic autocorrelation function (ACF) is proposed. It is shown that the solution of the synthesis problem is connected with the existence of difference sets for a given code dimension. The problem of estimating the number of possible code combinations for a given code dimension is solved. It is pointed out that the problem of synthesis of PC sequences is related to the fundamental problems of discrete mathematics and, first of all, to a number of combinatorial problems, which can be solved, as the number factorisation problem, by algebraic methods by using the theory of Galois fields and groups.

Using Student Work to Develop Teachers' Knowledge of Algebra

ERIC Educational Resources Information Center

Herbel-Eisenmann, Beth A.; Phillips, Elizabeth Difanis

2005-01-01

This article describes a set of learning activities that use algebraic problems and written student work to help preservice and in-service teachers understand students' algebraic thinking. (Contains 4 figures.)

Classical versus Computer Algebra Methods in Elementary Geometry

ERIC Educational Resources Information Center

Pech, Pavel

2005-01-01

Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…

Using CAS to Solve Classical Mathematics Problems

ERIC Educational Resources Information Center

Burke, Maurice J.; Burroughs, Elizabeth A.

2009-01-01

Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…

Redesigning the Quantum Mechanics Curriculum to Incorporate Problem Solving Using a Computer Algebra System

NASA Astrophysics Data System (ADS)

Roussel, Marc R.

1999-10-01

One of the traditional obstacles to learning quantum mechanics is the relatively high level of mathematical proficiency required to solve even routine problems. Modern computer algebra systems are now sufficiently reliable that they can be used as mathematical assistants to alleviate this difficulty. In the quantum mechanics course at the University of Lethbridge, the traditional three lecture hours per week have been replaced by two lecture hours and a one-hour computer-aided problem solving session using a computer algebra system (Maple). While this somewhat reduces the number of topics that can be tackled during the term, students have a better opportunity to familiarize themselves with the underlying theory with this course design. Maple is also available to students during examinations. The use of a computer algebra system expands the class of feasible problems during a time-limited exercise such as a midterm or final examination. A modern computer algebra system is a complex piece of software, so some time needs to be devoted to teaching the students its proper use. However, the advantages to the teaching of quantum mechanics appear to outweigh the disadvantages.

Solving Optimization Problems with Spreadsheets

ERIC Educational Resources Information Center

Beigie, Darin

2017-01-01

Spreadsheets provide a rich setting for first-year algebra students to solve problems. Individual spreadsheet cells play the role of variables, and creating algebraic expressions for a spreadsheet to perform a task allows students to achieve a glimpse of how mathematics is used to program a computer and solve problems. Classic optimization…

Inverse Modelling Problems in Linear Algebra Undergraduate Courses

ERIC Educational Resources Information Center

Martinez-Luaces, Victor E.

2013-01-01

This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…

Inequalities, Assessment and Computer Algebra

ERIC Educational Resources Information Center

Sangwin, Christopher J.

2015-01-01

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary…

A tensor Banach algebra approach to abstract kinetic equations

NASA Astrophysics Data System (ADS)

Greenberg, W.; van der Mee, C. V. M.

The study deals with a concrete algebraic construction providing the existence theory for abstract kinetic equation boundary-value problems, when the collision operator A is an accretive finite-rank perturbation of the identity operator in a Hilbert space H. An algebraic generalization of the Bochner-Phillips theorem is utilized to study solvability of the abstract boundary-value problem without any regulatory condition. A Banach algebra in which the convolution kernel acts is obtained explicitly, and this result is used to prove a perturbation theorem for bisemigroups, which then plays a vital role in solving the initial equations.

A rigorous approach to investigating common assumptions about disease transmission: Process algebra as an emerging modelling methodology for epidemiology.

PubMed

McCaig, Chris; Begon, Mike; Norman, Rachel; Shankland, Carron

2011-03-01

Changing scale, for example, the ability to move seamlessly from an individual-based model to a population-based model, is an important problem in many fields. In this paper, we introduce process algebra as a novel solution to this problem in the context of models of infectious disease spread. Process algebra allows us to describe a system in terms of the stochastic behaviour of individuals, and is a technique from computer science. We review the use of process algebra in biological systems, and the variety of quantitative and qualitative analysis techniques available. The analysis illustrated here solves the changing scale problem: from the individual behaviour we can rigorously derive equations to describe the mean behaviour of the system at the level of the population. The biological problem investigated is the transmission of infection, and how this relates to individual interactions.

Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.

PubMed

Sialaros, Michalis; Christianidis, Jean

2016-06-01

Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

Contextualizing symbol, symbolizing context

NASA Astrophysics Data System (ADS)

Maudy, Septiani Yugni; Suryadi, Didi; Mulyana, Endang

2017-08-01

When students learn algebra for the first time, inevitably they are experiencing transition from arithmetic to algebraic thinking. Once students could apprehend this essential mathematical knowledge, they are cultivating their ability in solving daily life problems by applying algebra. However, as we dig into this transitional stage, we identified possible students' learning obstacles to be dealt with seriously in order to forestall subsequent hindrance in studying more advance algebra. We come to realize this recurring problem as we undertook the processes of re-personalization and re-contextualization in which we scrutinize the very basic questions: 1) what is variable, linear equation with one variable and their relationship with the arithmetic-algebraic thinking? 2) Why student should learn such concepts? 3) How to teach those concepts to students? By positioning ourselves as a seventh grade student, we address the possibility of children to think arithmetically when confronted with the problems of linear equation with one variable. To help them thinking algebraically, Bruner's modes of representation developed contextually from concrete to abstract were delivered to enhance their interpretation toward the idea of variables. Hence, from the outset we designed the context for student to think symbolically initiated by exploring various symbols that could be contextualized in order to bridge student traversing the arithmetic-algebraic fruitfully.

Math Ties: Problem Solving, Logic Teasers, and Math Puzzles All "Tied" To the Math Curriculum. Book B1.

ERIC Educational Resources Information Center

Santi, Terri

This book contains a classroom-tested approach to the teaching of problem solving to all students in Grades 6-8, regardless of ability. Information on problem solving in general is provided, then mathematical problems on logic, exponents, fractions, pre-algebra, algebra, geometry, number theory, set theory, ratio, proportion, percent, probability,…

High-School Students' Approaches to Solving Algebra Problems that Are Posed Symbolically: Results from an Interview Study

ERIC Educational Resources Information Center

Huntley, Mary Ann; Davis, Jon D.

2008-01-01

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are…

Ermakov's Superintegrable Toy and Nonlocal Symmetries

NASA Astrophysics Data System (ADS)

Leach, P. G. L.; Karasu Kalkanli, A.; Nucci, M. C.; Andriopoulos, K.

2005-11-01

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.

Associations of Students' Beliefs with Self-Regulated Problem Solving in College Algebra

ERIC Educational Resources Information Center

Cifarelli, Victor; Goodson-Espy, Tracy; Chae, Jeong-Lim

2010-01-01

This paper reports results from a study of self-regulated problem solving actions of students enrolled in College Algebra (N = 139). The study examined the associations between the expressed mathematical beliefs of students and the students' self-regulated actions in solving mathematics problems. The research questions are: (a) What are some…

Grade 11 Students' Interconnected Use of Conceptual Knowledge, Procedural Skills, and Strategic Competence in Algebra: A Mixed Method Study of Error Analysis

ERIC Educational Resources Information Center

Egodawatte, Gunawardena; Stoilescu, Dorian

2015-01-01

The purpose of this mixed-method study was to investigate grade 11 university/college stream mathematics students' difficulties in applying conceptual knowledge, procedural skills, strategic competence, and algebraic thinking in solving routine (instructional) algebraic problems. A standardized algebra test was administered to thirty randomly…

Asymptotic identity in min-plus algebra: a report on CPNS.

PubMed

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions.

Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

PubMed Central

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions. PMID:21822446

Numerical algebraic geometry: a new perspective on gauge and string theories

NASA Astrophysics Data System (ADS)

Mehta, Dhagash; He, Yang-Hui; Hauensteine, Jonathan D.

2012-07-01

There is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called `embarrassing parallelizability' allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

Realization theory and quadratic optimal controllers for systems defined over Banach and Frechet algebras

NASA Technical Reports Server (NTRS)

Byrnes, C. I.

1980-01-01

It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.

Solving Our Algebra Problem: Getting All Students through Algebra I to Improve Graduation Rates

ERIC Educational Resources Information Center

Schachter, Ron

2013-01-01

graduation as well as admission to most colleges. But taking algebra also can turn into a pathway for failure, from which some students never recover. In 2010, a national U.S. Department of Education study…

Contractions and deformations of quasiclassical Lie algebras preserving a nondegenerate quadratic Casimir operator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Campoamor-Stursberg, R., E-mail: rutwig@mat.ucm.e

2008-05-15

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

Graphs and matroids weighted in a bounded incline algebra.

PubMed

Lu, Ling-Xia; Zhang, Bei

2014-01-01

Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied.

Numerical Problem Solving Using Mathcad in Undergraduate Reaction Engineering

ERIC Educational Resources Information Center

Parulekar, Satish J.

2006-01-01

Experience in using a user-friendly software, Mathcad, in the undergraduate chemical reaction engineering course is discussed. Example problems considered for illustration deal with simultaneous solution of linear algebraic equations (kinetic parameter estimation), nonlinear algebraic equations (equilibrium calculations for multiple reactions and…

Brain activity associated with translation from a visual to a symbolic representation in algebra and geometry.

PubMed

Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza

2014-03-01

This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.

Genetic algorithms in teaching artificial intelligence (automated generation of specific algebras)

NASA Astrophysics Data System (ADS)

Habiballa, Hashim; Jendryscik, Radek

2017-11-01

The problem of teaching essential Artificial Intelligence (AI) methods is an important task for an educator in the branch of soft-computing. The key focus is often given to proper understanding of the principle of AI methods in two essential points - why we use soft-computing methods at all and how we apply these methods to generate reasonable results in sensible time. We present one interesting problem solved in the non-educational research concerning automated generation of specific algebras in the huge search space. We emphasize above mentioned points as an educational case study of an interesting problem in automated generation of specific algebras.

Effects of Argumentation on Group Micro-Creativity: Statistical Discourse Analyses of Algebra Students' Collaborative Problem Solving

ERIC Educational Resources Information Center

Chiu, Ming Ming

2008-01-01

The micro-time context of group processes (such as argumentation) can affect a group's micro-creativity (new ideas). Eighty high school students worked in groups of four on an algebra problem. Groups with higher mathematics grades showed greater micro-creativity, and both were linked to better problem solving outcomes. Dynamic multilevel analyses…

W-algebra for solving problems with fuzzy parameters

NASA Astrophysics Data System (ADS)

Shevlyakov, A. O.; Matveev, M. G.

2018-03-01

A method of solving the problems with fuzzy parameters by means of a special algebraic structure is proposed. The structure defines its operations through operations on real numbers, which simplifies its use. It avoids deficiencies limiting applicability of the other known structures. Examples for solution of a quadratic equation, a system of linear equations and a network planning problem are given.

Characterizing the Nature of Students' Feature Noticing-and-Using with Respect to Mathematical Symbols across Different Levels of Algebra Exposure

ERIC Educational Resources Information Center

Sullivan, Patrick

2013-01-01

The purpose of this study is to examine the nature of what students notice about symbols and use as they solve unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions. The researcher conducted a study of students at three levels of algebra exposure: (a) students enrolled in a high school pre-calculus…

Change in Peer Ability as a Mediator and Moderator of the Effect of the Algebra-For-All Policy on Ninth Graders' Math Outcomes

ERIC Educational Resources Information Center

Hong, Guanglei; Nomi, Takako

2011-01-01

A recent report by the Mathematics Advisory Panel referred to algebra as a "gateway" to later achievement (National Mathematics Advisory Panel, 2008). To address the problem of low academic performance in algebra, an increasing number of states and districts have started to implement a policy of requiring algebra for all students in…

Undecidability of the elementary theory of the semilattice of GLP-words

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pakhomov, Fedor N

The Lindenbaum algebra of Peano PA can be enriched by the n-consistency operators which assign, to a given formula, the statement that the formula is compatible with the theory PA extended by the set of all true {Pi}{sub n}-sentences. In the Lindenbaum algebra of PA, a lower semilattice is generated from 1 by the n-consistency operators. We prove the undecidability of the elementary theory of this semilattice and the decidability of the elementary theory of the subsemilattice (of this semilattice) generated by the 0-consistency and 1-consistency operators only. Bibliography: 16 titles.

Implementing the Curriculum and Evaluation Standards: First-Year Algebra.

ERIC Educational Resources Information Center

Kysh, Judith

1991-01-01

Described is an alternative first year algebra program developed to bridge the gap between the NCTM's Curriculum and Evaluation Standards and institutional demands of schools. Increased attention is given to graphing as a context for algebra, calculator use, solving "memorable problems," and incorporating geometry concepts, while…

Formula recollection through a WORLDLY recognized mnemonic technique

NASA Astrophysics Data System (ADS)

Schunicht, Shannon

2009-10-01

Physics may be made fun, and encourage further learning through ease of recollection of complicated formulas; allthewhile increasing a student's confortability with their algebraic skills. Examples will be shown how ANY complicated formula will be made into a memorable acronym using this author's mnemonic technique, i.e. allowing each vowel to represent a mathematical operation: ``a'' multiplication implying ``@''; ``o''-division implying ``over''; ``i''-subtraction to imply ``minus''; ``u''-addition to imply ``plus''; and ``e'' implying ``equals''. Most constants and variables are indeed consonants; ``c'' = ``speed of light'' & ``z'' = ``altitude''. With this mnemonic technique ANY formula may be algebraically manipulated into a word, or series of words for ease of recollection. Additional letters may be added to enhance the intelligibility of such a letter combination, but these additional letters need be consonants ONLY. This mnemonic technique was developed as a compensatory memory method when taking physics at Texas A&M University following a severe head injury (19 days unconsciousness!) suffered by this author.

A Linear Algebraic Approach to Teaching Interpolation

ERIC Educational Resources Information Center

Tassa, Tamir

2007-01-01

A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…

Effect of Worked Examples and Cognitive Tutor Training on Constructing Equations

ERIC Educational Resources Information Center

Reed, Stephen K.; Corbett, Albert; Hoffman, Bob; Wagner, Angela; MacLaren, Ben

2013-01-01

Algebra students studied either static-table, static-graphics, or interactive-graphics instructional worked examples that alternated with Algebra Cognitive Tutor practice problems. A control group did not study worked examples but solved both the instructional and practice problems on the Cognitive Tutor (CT). Students in the control group…

Digital Maps, Matrices and Computer Algebra

ERIC Educational Resources Information Center

Knight, D. G.

2005-01-01

The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…

A new application of algebraic geometry to systems theory

NASA Technical Reports Server (NTRS)

Martin, C. F.; Hermann, R.

1976-01-01

Following an introduction to algebraic geometry, the dominant morphism theorem is stated, and the application of this theorem to systems-theoretic problems, such as the feedback problem, is discussed. The Gaussian elimination method used for solving linear equations is shown to be an example of a dominant morphism.

Alternative Representations for Algebraic Problem Solving: When Are Graphs Better than Equations?

ERIC Educational Resources Information Center

Mielicki, Marta K.; Wiley, Jennifer

2016-01-01

Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation…

Algebraic Reasoning in Solving Mathematical Problem Based on Learning Style

NASA Astrophysics Data System (ADS)

Indraswari, N. F.; Budayasa, I. K.; Ekawati, R.

2018-01-01

This study aimed to describe algebraic reasoning of secondary school’s pupils with different learning styles in solving mathematical problem. This study begins by giving the questionnaire to find out the learning styles and followed by mathematical ability test to get three subjects of 8th-grade whereas the learning styles of each pupil is visual, auditory, kinesthetic and had similar mathematical abilities. Then it continued with given algebraic problems and interviews. The data is validated using triangulation of time. The result showed that in the pattern of seeking indicator, subjects identified the things that were known and asked based on them observations. The visual and kinesthetic learners represented the known information in a chart, whereas the auditory learner in a table. In addition, they found the elements which makes the pattern and made a relationship between two quantities. In the pattern recognition indicator, they created conjectures on the relationship between two quantities and proved it. In the generalization indicator, they were determining the general rule of pattern found on each element of pattern using algebraic symbols and created a mathematical model. Visual and kinesthetic learners determined the general rule of equations which was used to solve problems using algebraic symbols, but auditory learner in a sentence.

A differential operator realisation approach for constructing Casimir operators of non-semisimple Lie algebras

NASA Astrophysics Data System (ADS)

Alshammari, Fahad; Isaac, Phillip S.; Marquette, Ian

2018-02-01

We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and (2) the Schrödinger Lie algebras. We find that an abstract form of dimensional analysis assists us in our algorithm, and greatly reduces the complexity of the problem.

Class and Homework Problems: The Break-Even Radius of Insulation Computed Using Excel Solver and WolframAlpha

ERIC Educational Resources Information Center

Foley, Greg

2014-01-01

A problem that illustrates two ways of computing the break-even radius of insulation is outlined. The problem is suitable for students who are taking an introductory module in heat transfer or transport phenomena and who have some previous knowledge of the numerical solution of non- linear algebraic equations. The potential for computer algebra,…

The Algebra of Complex Numbers.

ERIC Educational Resources Information Center

LePage, Wilbur R.

This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…

Teaching Linear Algebra: Must the Fog Always Roll In?

ERIC Educational Resources Information Center

Carlson, David

1993-01-01

Proposes methods to teach the more difficult concepts of linear algebra. Examines features of the Linear Algebra Curriculum Study Group Core Syllabus, and presents problems from the core syllabus that utilize the mathematical process skills of making conjectures, proving the results, and communicating the results to colleagues. Presents five…

Promoting Quantitative Literacy in an Online College Algebra Course

ERIC Educational Resources Information Center

Tunstall, Luke; Bossé, Michael J.

2016-01-01

College algebra (a university freshman level algebra course) fulfills the quantitative literacy requirement of many college's general education programs and is a terminal course for most who take it. An online problem-based learning environment provides a unique means of engaging students in quantitative discussions and research. This article…

Algebraic Concepts: What's Really New in New Curricula?

ERIC Educational Resources Information Center

Star, Jon R.; Herbel-Eisenmann, Beth A.; Smith, John P., III

2000-01-01

Examines 8th grade units from the Connected Mathematics Project (CMP). Identifies differences in older and newer conceptions, fundamental objects of study, typical problems, and typical solution methods in algebra. Also discusses where the issue of what is new in algebra is relevant to many other innovative middle school curricula. (KHR)

Algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jiang, Tongsong, E-mail: jiangtongsong@sina.com; Department of Mathematics, Heze University, Heze, Shandong 274015; Jiang, Ziwu

In the study of the relation between complexified classical and non-Hermitian quantum mechanics, physicists found that there are links to quaternionic and split quaternionic mechanics, and this leads to the possibility of employing algebraic techniques of split quaternions to tackle some problems in complexified classical and quantum mechanics. This paper, by means of real representation of a split quaternion matrix, studies the problem of diagonalization of a split quaternion matrix and gives algebraic techniques for diagonalization of split quaternion matrices in split quaternionic mechanics.

Multiple shooting algorithms for jump-discontinuous problems in optimal control and estimation

NASA Technical Reports Server (NTRS)

Mook, D. J.; Lew, Jiann-Shiun

1991-01-01

Multiple shooting algorithms are developed for jump-discontinuous two-point boundary value problems arising in optimal control and optimal estimation. Examples illustrating the origin of such problems are given to motivate the development of the solution algorithms. The algorithms convert the necessary conditions, consisting of differential equations and transversality conditions, into algebraic equations. The solution of the algebraic equations provides exact solutions for linear problems. The existence and uniqueness of the solution are proved.

Technology Focus: Multi-Representational Approaches to Equation Solving

ERIC Educational Resources Information Center

Garofalo, Joe; Trinter, Christine

2009-01-01

Most mathematical functions can be represented in numerous ways. The main representations typically addressed in school, often referred to as "the big three," are graphical, algebraic, and numerical representations, but there are others as well (e.g., diagrams, words, simulations). These different types of representations "often illuminate…

A Double-Minded Fractal

ERIC Educational Resources Information Center

Simoson, Andrew J.

2009-01-01

This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed. (Contains 5 figures.)

"Playing the Game" of Story Problems: Coordinating Situation-Based Reasoning with Algebraic Representation

ERIC Educational Resources Information Center

Walkington, Candace; Sherman, Milan; Petrosino, Anthony

2012-01-01

This study critically examines a key justification used by educational stakeholders for placing mathematics in context--the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were…

Hidden algebra method (quasi-exact-solvability in quantum mechanics)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Turbiner, Alexander; Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apartado, Postal 70-543, 04510 Mexico, D. F.

1996-02-20

A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland N-body problems ass ociated with an existence of the hidden algebra slN is discussed extensively.

Bicycles, Birds, Bats and Balloons: New Applications for Algebra Classes.

ERIC Educational Resources Information Center

Yoshiwara, Bruce; Yoshiwara, Kathy

This collection of activities is intended to enhance the teaching of college algebra through the use of modeling. The problems use real data and involve the representation and interpretation of the data. The concepts addressed include rates of change, linear and quadratic regression, and functions. The collection consists of eight problems, four…

A Method for the Microanalysis of Pre-Algebra Transfer

ERIC Educational Resources Information Center

Pavlik, Philip I., Jr.; Yudelson, Michael; Koedinger, Kenneth R.

2011-01-01

The objective of this research was to better understand the transfer of learning between different variations of pre-algebra problems. While the authors could have addressed a specific variation that might address transfer, they were interested in developing a general model of transfer, so we gathered data from multiple problem types and their…

Visual, Algebraic and Mixed Strategies in Visually Presented Linear Programming Problems.

ERIC Educational Resources Information Center

Shama, Gilli; Dreyfus, Tommy

1994-01-01

Identified and classified solution strategies of (n=49) 10th-grade students who were presented with linear programming problems in a predominantly visual setting in the form of a computerized game. Visual strategies were developed more frequently than either algebraic or mixed strategies. Appendix includes questionnaires. (Contains 11 references.)…

Activities for Students: Biology as a Source for Algebra Equations--The Heart

ERIC Educational Resources Information Center

Horak, Virginia M.

2005-01-01

The high school course that integrated first year algebra with an introductory environmental biology/anatomy and physiology course, in order to solve algebra problems is discussed. Lessons and activities for the course were taken by identifying the areas where mathematics and biology content intervenes may help students understand biology concepts…

Catching Up on Algebra

ERIC Educational Resources Information Center

Cavanagh, Sean

2008-01-01

A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…

Using the Internet To Investigate Algebra.

ERIC Educational Resources Information Center

Sherwood, Walter

The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…

A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry

ERIC Educational Resources Information Center

Kalainoff, Melinda; Lachance, Russ; Riegner, Dawn; Biaglow, Andrew

2012-01-01

In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general…

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

NASA Astrophysics Data System (ADS)

Nazarov, Anton

2012-11-01

In this paper we present Affine.m-a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Catalogue identifier: AENA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 844 No. of bytes in distributed program, including test data, etc.: 1 045 908 Distribution format: tar.gz Programming language: Mathematica. Computer: i386-i686, x86_64. Operating system: Linux, Windows, Mac OS, Solaris. RAM: 5-500 Mb Classification: 4.2, 5. Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems. Solution method: We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases. Restrictions: Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical. Unusual features: We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface. Running time: From seconds to days depending on the rank of the algebra and the complexity of the representation.

Diagrams benefit symbolic problem-solving.

PubMed

Chu, Junyi; Rittle-Johnson, Bethany; Fyfe, Emily R

2017-06-01

The format of a mathematics problem often influences students' problem-solving performance. For example, providing diagrams in conjunction with story problems can benefit students' understanding, choice of strategy, and accuracy on story problems. However, it remains unclear whether providing diagrams in conjunction with symbolic equations can benefit problem-solving performance as well. We tested the impact of diagram presence on students' performance on algebra equation problems to determine whether diagrams increase problem-solving success. We also examined the influence of item- and student-level factors to test the robustness of the diagram effect. We worked with 61 seventh-grade students who had received 2 months of pre-algebra instruction. Students participated in an experimenter-led classroom session. Using a within-subjects design, students solved algebra problems in two matched formats (equation and equation-with-diagram). The presence of diagrams increased equation-solving accuracy and the use of informal strategies. This diagram benefit was independent of student ability and item complexity. The benefits of diagrams found previously for story problems generalized to symbolic problems. The findings are consistent with cognitive models of problem-solving and suggest that diagrams may be a useful additional representation of symbolic problems. © 2017 The British Psychological Society.

Optical systolic solutions of linear algebraic equations

NASA Technical Reports Server (NTRS)

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

The Effect of Worked Examples on Student Learning and Error Anticipation in Algebra

ERIC Educational Resources Information Center

Booth, Julie L.; Begolli, Kreshnik N.; McCann, Nicholas

2016-01-01

The present study examines the effectiveness of incorporating worked examples with prompts for self-explanation into a middle school math textbook. Algebra 1 students (N = 75) completed an equation-solving unit with reform textbooks either containing the original practice problems or in which a portion of those problems were converted into…

Middle School Students' Conceptual Understanding of Equations: Evidence From Writing Story Problems. WCER Working Paper No. 2009-3

ERIC Educational Resources Information Center

Alibali, Martha W.; Kao, Yvonne S.; Brown, Alayna N.; Nathan, Mitchell J.; Stephens, Ana C.

2009-01-01

This study investigated middle school students' conceptual understanding of algebraic equations. Participants in the study--257 sixth- and seventh-grade students--were asked to solve one set of algebraic equations and to generate story problems corresponding with another set of equations. Structural aspects of the equations, including the number…

Interactions between Language and Mathematics with Deaf Students: Defining the "Language-Mathematics" Equation.

ERIC Educational Resources Information Center

Hillegeist, Eleanor; Epstein, Kenneth

The study examined the relationship between language and mathematics with 11 classes of deaf students taking Algebra 1 or Algebra 2 at the Gallaudet University School of Preparatory Studies. Specifically, the study attempted to predict the difficulty of a variety of relatively simple algebra problems based on the abstractness of the math and the…

Using Example Problems to Improve Student Learning in Algebra: Differentiating between Correct and Incorrect Examples

ERIC Educational Resources Information Center

Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.

2013-01-01

In a series of two "in vivo" experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly…

Using Example Problems to Improve Student Learning in Algebra: Differentiating between Correct and Incorrect Examples

ERIC Educational Resources Information Center

Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.

2013-01-01

In a series of two in vivo experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly assigned…

A Study to Determine Differences in the Level of Perceived Preparedness in Teaching Algebra to Eighth Graders between Teachers in the United States and Teachers in Lebanon

ERIC Educational Resources Information Center

Khajarian, Seta

2011-01-01

Algebra is a branch in mathematics and taking Algebra in middle school is often a gateway to advanced courses in high school. The problem is that the United States and Lebanon had low scores in Algebra in the 2007 Trends in Mathematics and Sciences Study (TIMSS), an international assessment administered to 4th and 8th graders every 4 years. On the…

Maximizing algebraic connectivity in air transportation networks

NASA Astrophysics Data System (ADS)

Wei, Peng

In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the weight assignment can not be studied separately for the problem with operating cost constraint. Therefore a relaxed SDP method with golden section search is developed to solve both at the same time. The cluster decomposition is utilized to solve large scale networks.

The roles of prefrontal and posterior parietal cortex in algebra problem solving: a case of using cognitive modeling to inform neuroimaging data.

PubMed

Danker, Jared F; Anderson, John R

2007-04-15

In naturalistic algebra problem solving, the cognitive processes of representation and retrieval are typically confounded, in that transformations of the equations typically require retrieval of mathematical facts. Previous work using cognitive modeling has associated activity in the prefrontal cortex with the retrieval demands of algebra problems and activity in the posterior parietal cortex with the transformational demands of algebra problems, but these regions tend to behave similarly in response to task manipulations (Anderson, J.R., Qin, Y., Sohn, M.-H., Stenger, V.A., Carter, C.S., 2003. An information-processing model of the BOLD response in symbol manipulation tasks. Psychon. Bull. Rev. 10, 241-261; Qin, Y., Carter, C.S., Silk, E.M., Stenger, A., Fissell, K., Goode, A., Anderson, J.R., 2004. The change of brain activation patterns as children learn algebra equation solving. Proc. Natl. Acad. Sci. 101, 5686-5691). With this study we attempt to isolate activity in these two regions by using a multi-step algebra task in which transformation (parietal) is manipulated in the first step and retrieval (prefrontal) is manipulated in the second step. Counter to our initial predictions, both brain regions were differentially active during both steps. We designed two cognitive models, one encompassing our initial assumptions and one in which both processes were engaged during both steps. The first model provided a poor fit to the behavioral and neural data, while the second model fit both well. This simultaneously emphasizes the strong relationship between retrieval and representation in mathematical reasoning and demonstrates that cognitive modeling can serve as a useful tool for understanding task manipulations in neuroimaging experiments.

Hidden algebra method (quasi-exact-solvability in quantum mechanics)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Turbiner, A.

1996-02-01

A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland {ital N}-body problems ass ociated with an existence of the hidden algebra {ital sl}{sub {ital N}} is discussed extensively. {copyright} {ital 1996 American Institute of Physics.}

Particle-like structure of coaxial Lie algebras

NASA Astrophysics Data System (ADS)

Vinogradov, A. M.

2018-01-01

This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over R can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.

General Algebraic Modeling System Tutorial | High-Performance Computing |

Science.gov Websites

power generation from two different fuels. The goal is to minimize the cost for one of the fuels while Here's a basic tutorial for modeling optimization problems with the General Algebraic Modeling System (GAMS). Overview The GAMS (General Algebraic Modeling System) package is essentially a compiler for a

The noncommutative Poisson bracket and the deformation of the family algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wei, Zhaoting, E-mail: zhaotwei@indiana.edu

The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra C{sub τ}(g). We show that P controls the first-order 1-parameter formal deformation from C{sub τ}(g) to Q{sub τ}(g) where the latter is the quantum family algebra. Moreover, we will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary, and therefore, the deformation is infinitesimally trivial. In the last part of this paper, we discuss the relation between Mackey’s analogue and the quantization problem of the family algebras.

Selecting reusable components using algebraic specifications

NASA Technical Reports Server (NTRS)

Eichmann, David A.

1992-01-01

A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.

Packing a Box with Bricks.

ERIC Educational Resources Information Center

Jepsen, Charles H.

1991-01-01

Presented are solutions to variations of a combinatorics problem from a recent International Mathematics Olympiad. In particular, the matrix algebra solution illustrates an interaction among the undergraduate areas of geometry, combinatorics, linear algebra, and group theory. (JJK)

Proceedings of the Tenth Annual National Conference on Ada Technology. Held in Arlington, VA, on February 24-28, 1992

DTIC Science & Technology

1992-02-01

Newsletter, Vol. 5, No. 1, January 1983 be translated from HAL’S. 4. Klumpp, Allan R., An Ada Linear Algebra Software development costs for using the...a linear algebra approach to As noted above, the concept of the problem and address the problem of unitdimensional analysis extends beyond problems...you will join us again next year. The 11th Annual Conference on Ada Technology (1993) will be held here at the Hyatt Regency - Crystal City

Combinatorial Formulas for Characteristic Classes, and Localization of Secondary Topological Invariants.

NASA Astrophysics Data System (ADS)

Smirnov, Mikhail

1995-01-01

The problems solved in this thesis originated from combinatorial formulas for characteristic classes. This thesis deals with Chern-Simons classes, their generalizations and related algebraic and analytic problems. (1) In this thesis, I describe a new class of algebras whose elements contain Chern and generalized Chern -Simons classes. There is a Poisson bracket in these algebras, similar to the bracket in Kontsevich's noncommutative symplectic geometry (Kon). I prove that the Poisson bracket gives rise to a graded Lie algebra containing differential forms representing Chern and Chern-Simons classes. This is a new result. I describe algebraic analogs of the dilogarithm and higher polylogarithms in the algebra corresponding to Chern-Simons classes. (2) I study the properties of this bracket. It is possible to write the exterior differential and other operations in the algebra using this bracket. The bracket of any two Chern classes is zero and the bracket of a Chern class and a Chern-Simons class is d-closed. The construction developed here easily gives explicit formulas for known secondary classes and makes it possible to construct new ones. (3) I develop an algebraic model for the action of the gauge group and describe how elements of algebra corresponding to the secondary characteristic classes change under this action (see theorem 3 page xi). (4) It is possible give new explicit formulas for cocycles on a gauge group of a bundle and for the corresponding cocycles on the Lie algebra of the gauge group. I use formulas for secondary characteristic classes and an algebraic approach developed in chapter 1. I also use the work of Faddeev, Reiman and Semyonov-Tian-Shanskii (FRS) on cocycles as quantum anomalies. (5) I apply the methods of differential geometry of formal power series to construct universal characteristic and secondary characteristic classes. Given a pair of gauge equivalent connections using local formulas I obtain dilogarithmic and trilogarithmic analogs of Chern-Simons classes.

Parallel Element Agglomeration Algebraic Multigrid and Upscaling Library

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barker, Andrew T.; Benson, Thomas R.; Lee, Chak Shing

ParELAG is a parallel C++ library for numerical upscaling of finite element discretizations and element-based algebraic multigrid solvers. It provides optimal complexity algorithms to build multilevel hierarchies and solvers that can be used for solving a wide class of partial differential equations (elliptic, hyperbolic, saddle point problems) on general unstructured meshes. Additionally, a novel multilevel solver for saddle point problems with divergence constraint is implemented.

Space Mathematics: A Resource for Secondary School Teachers

NASA Technical Reports Server (NTRS)

Kastner, Bernice

1985-01-01

A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.

Solution Strategies, Modes of Representation and Justifications of Primary Five Pupils in Solving Pre Algebra Problems: An Experience of Using Task-Based Interview and Verbal Protocol Analysis

ERIC Educational Resources Information Center

Ling, Gan We; Ghazali, Munirah

2007-01-01

This descriptive study was aimed at looking into how Primary 5 pupils solve pre-algebra problems concerning patterns and unknown quantities. Specifically, objectives of this study were to describe Primary 5 pupils' solution strategies, modes of representations and justifications in: (a) discovering, describing and using numerical and geometrical…

Some Issues about the Introduction of First Concepts in Linear Algebra during Tutorial Sessions at the Beginning of University

ERIC Educational Resources Information Center

Grenier-Boley, Nicolas

2014-01-01

Certain mathematical concepts were not introduced to solve a specific open problem but rather to solve different problems with the same tools in an economic formal way or to unify several approaches: such concepts, as some of those of linear algebra, are presumably difficult to introduce to students as they are potentially interwoven with many…

An Example of Competence-Based Learning: Use of Maxima in Linear Algebra for Engineers

ERIC Educational Resources Information Center

Diaz, Ana; Garcia, Alfonsa; de la Villa, Agustin

2011-01-01

This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…

Applied Algebra: The Modeling Technique of Least Squares

ERIC Educational Resources Information Center

Zelkowski, Jeremy; Mayes, Robert

2008-01-01

The article focuses on engaging students in algebra through modeling real-world problems. The technique of least squares is explored, encouraging students to develop a deeper understanding of the method. (Contains 2 figures and a bibliography.)

On Correspondence of BRST-BFV, Dirac, and Refined Algebraic Quantizations of Constrained Systems

NASA Astrophysics Data System (ADS)

Shvedov, O. Yu.

2002-11-01

The correspondence between BRST-BFV, Dirac, and refined algebraic (group averaging, projection operator) approaches to quantizing constrained systems is analyzed. For the closed-algebra case, it is shown that the component of the BFV wave function corresponding to maximal (minimal) value of number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the refined algebraic (Dirac) quantization approach. The Giulini-Marolf group averaging formula for the inner product in the refined algebraic quantization approach is obtained from the Batalin-Marnelius prescription for the BRST-BFV inner product, which should be generally modified due to topological problems. The considered prescription for the correspondence of states is observed to be applicable to the open-algebra case. The refined algebraic quantization approach is generalized then to the case of nontrivial structure functions. A simple example is discussed. The correspondence of observables for different quantization methods is also investigated.

Connes' embedding problem and Tsirelson's problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Junge, M.; Palazuelos, C.; Navascues, M.

2011-01-15

We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C{sup *}-algebras. Connes' embedding problem asks whether any separable II{sub 1} factor is a subfactor of the ultrapower of the hyperfinite II{sub 1} factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely,more » a positive answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem.« less

Quantum cluster algebras and quantum nilpotent algebras.

PubMed

Goodearl, Kenneth R; Yakimov, Milen T

2014-07-08

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein-Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405-455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337-397] for the case of symmetric Kac-Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1-52] associated with double Bruhat cells coincide with the corresponding cluster algebras.

Quantum cluster algebras and quantum nilpotent algebras

PubMed Central

Goodearl, Kenneth R.; Yakimov, Milen T.

2014-01-01

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197

Influence of Additive and Multiplicative Structure and Direction of Comparison on the Reversal Error

ERIC Educational Resources Information Center

González-Calero, José Antonio; Arnau, David; Laserna-Belenguer, Belén

2015-01-01

An empirical study has been carried out to evaluate the potential of word order matching and static comparison as explanatory models of reversal error. Data was collected from 214 undergraduate students who translated a set of additive and multiplicative comparisons expressed in Spanish into algebraic language. In these multiplicative comparisons…

Solving Geometric Problems by Using Algebraic Representation for Junior High School Level 3 in Van Hiele at Geometric Thinking Level

ERIC Educational Resources Information Center

Suwito, Abi; Yuwono, Ipung; Parta, I. Nengah; Irawati, Santi; Oktavianingtyas, Ervin

2016-01-01

This study aims to determine the ability of algebra students who have 3 levels van Hiele levels. Follow its framework Dindyal framework (2007). Students are required to do 10 algebra shaped multiple choice, then students work 15 about the geometry of the van Hiele level in the form of multiple choice questions. The question has been tested levels…

Cognitive and Linguistic Predictors of Mathematical Word Problems With and Without Irrelevant Information.

PubMed

Wang, Amber Y; Fuchs, Lynn S; Fuchs, Douglas

2016-12-01

The purpose of this study was to identify cognitive and linguistic predictors of word problems with versus without irrelevant information. The sample was 701 2nd-grade students who received no specialized intervention on word problems. In the fall, they were assessed on initial arithmetic and word-problem skill as well as language ability, working memory capacity, and processing speed; in the spring, they were tested on a word-problem measure that included items with versus without irrelevant information. Significant predictors common to both forms of word problems were initial arithmetic and word problem-solving skill as well as language and working memory. Nonverbal reasoning predicted word problems with irrelevant information, but not word problems without irrelevant information. Findings are discussed in terms of implications for intervention and future research.

Quantization, Frobenius and Bi algebras from the Categorical Framework of Quantum Mechanics to Natural Language Semantics

NASA Astrophysics Data System (ADS)

Sadrzadeh, Mehrnoosh

2017-07-01

Compact Closed categories and Frobenius and Bi algebras have been applied to model and reason about Quantum protocols. The same constructions have also been applied to reason about natural language semantics under the name: ``categorical distributional compositional'' semantics, or in short, the ``DisCoCat'' model. This model combines the statistical vector models of word meaning with the compositional models of grammatical structure. It has been applied to natural language tasks such as disambiguation, paraphrasing and entailment of phrases and sentences. The passage from the grammatical structure to vectors is provided by a functor, similar to the Quantization functor of Quantum Field Theory. The original DisCoCat model only used compact closed categories. Later, Frobenius algebras were added to it to model long distance dependancies such as relative pronouns. Recently, bialgebras have been added to the pack to reason about quantifiers. This paper reviews these constructions and their application to natural language semantics. We go over the theory and present some of the core experimental results.

Cognitive and Linguistic Predictors of Mathematical Word Problems With and Without Irrelevant Information

PubMed Central

Fuchs, Lynn S.; Fuchs, Douglas

2016-01-01

The purpose of this study was to identify cognitive and linguistic predictors of word problems with versus without irrelevant information. The sample was 701 2nd-grade students who received no specialized intervention on word problems. In the fall, they were assessed on initial arithmetic and word-problem skill as well as language ability, working memory capacity, and processing speed; in the spring, they were tested on a word-problem measure that included items with versus without irrelevant information. Significant predictors common to both forms of word problems were initial arithmetic and word problem-solving skill as well as language and working memory. Nonverbal reasoning predicted word problems with irrelevant information, but not word problems without irrelevant information. Findings are discussed in terms of implications for intervention and future research. PMID:28190942

Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training

PubMed Central

Boonen, Anton J. H.; de Koning, Björn B.; Jolles, Jelle; van der Schoot, Menno

2016-01-01

Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME. PMID:26925012

Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training.

PubMed

Boonen, Anton J H; de Koning, Björn B; Jolles, Jelle; van der Schoot, Menno

2016-01-01

Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME.

Computers and the Multiplicity of Polynomial Roots.

ERIC Educational Resources Information Center

Wavrik, John J.

1982-01-01

Described are stages in the development of a computer program to solve a particular algebra problem and the nature of algebraic computation is presented. A program in BASIC is provided to give ideas to others for developing their own programs. (MP)

Algebraic methods in system theory

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Willems, J. C.; Willsky, A. S.

1975-01-01

Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.

Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Morozov, Oleg I.

2018-06-01

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDE s under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

GENERAL: Application of Symplectic Algebraic Dynamics Algorithm to Circular Restricted Three-Body Problem

NASA Astrophysics Data System (ADS)

Lu, Wei-Tao; Zhang, Hua; Wang, Shun-Jin

2008-07-01

Symplectic algebraic dynamics algorithm (SADA) for ordinary differential equations is applied to solve numerically the circular restricted three-body problem (CR3BP) in dynamical astronomy for both stable motion and chaotic motion. The result is compared with those of Runge-Kutta algorithm and symplectic algorithm under the fourth order, which shows that SADA has higher accuracy than the others in the long-term calculations of the CR3BP.

C-semiring Frameworks for Minimum Spanning Tree Problems

NASA Astrophysics Data System (ADS)

Bistarelli, Stefano; Santini, Francesco

In this paper we define general algebraic frameworks for the Minimum Spanning Tree problem based on the structure of c-semirings. We propose general algorithms that can compute such trees by following different cost criteria, which must be all specific instantiation of c-semirings. Our algorithms are extensions of well-known procedures, as Prim or Kruskal, and show the expressivity of these algebraic structures. They can deal also with partially-ordered costs on the edges.

Simplifications for hydronic system models in modelica

DOE PAGES

Jorissen, F.; Wetter, M.; Helsen, L.

2018-01-12

Building systems and their heating, ventilation and air conditioning flow networks, are becoming increasingly complex. Some building energy simulation tools simulate these flow networks using pressure drop equations. These flow network models typically generate coupled algebraic nonlinear systems of equations, which become increasingly more difficult to solve as their sizes increase. This leads to longer computation times and can cause the solver to fail. These problems also arise when using the equation-based modelling language Modelica and Annex 60-based libraries. This may limit the applicability of the library to relatively small problems unless problems are restructured. This paper discusses two algebraicmore » loop types and presents an approach that decouples algebraic loops into smaller parts, or removes them completely. The approach is applied to a case study model where an algebraic loop of 86 iteration variables is decoupled into smaller parts with a maximum of five iteration variables.« less

A Nonlinear, Multiinput, Multioutput Process Control Laboratory Experiment

ERIC Educational Resources Information Center

Young, Brent R.; van der Lee, James H.; Svrcek, William Y.

2006-01-01

Experience in using a user-friendly software, Mathcad, in the undergraduate chemical reaction engineering course is discussed. Example problems considered for illustration deal with simultaneous solution of linear algebraic equations (kinetic parameter estimation), nonlinear algebraic equations (equilibrium calculations for multiple reactions and…

A New Approach to an Old Order.

ERIC Educational Resources Information Center

Rambhia, Sanjay

2002-01-01

Explains the difficulties middle school students face in algebra regarding the order of operations. Describes a more visual approach to teaching the order of operations so that students can better solve complex problems and be better prepared for the rigors of algebra. (YDS)

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

DOE PAGES

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul; ...

2017-12-20

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul

We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

Four Proofs of the Converse of the Chinese Remainder Theorem

ERIC Educational Resources Information Center

Dobbs, D. E.

2008-01-01

Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [double-struck z][subscript m] [direct sum] [double-struck…

Topology of the conceptual network of language

NASA Astrophysics Data System (ADS)

Motter, Adilson E.; de Moura, Alessandro P.; Lai, Ying-Cheng; Dasgupta, Partha

2002-06-01

We define two words in a language to be connected if they express similar concepts. The network of connections among the many thousands of words that make up a language is important not only for the study of the structure and evolution of languages, but also for cognitive science. We study this issue quantitatively, by mapping out the conceptual network of the English language, with the connections being defined by the entries in a Thesaurus dictionary. We find that this network presents a small-world structure, with an amazingly small average shortest path, and appears to exhibit an asymptotic scale-free feature with algebraic connectivity distribution.

Mathematics Unit Plans. PACE '94.

ERIC Educational Resources Information Center

Wiles, Clyde A., Ed.; Schoon, Kenneth J., Ed.

This booklet contains mathematics unit plans for Algebra 1, Geometry, Math for Technology, Mathematical Problem Solving, and Pre-Algebra developed by PACE (Promoting Academic Excellence In Mathematics, Science & Technology for Workers of the 21st Century). Each unit plan contains suggested timing, objectives, skills to be acquired, workplace…

The Golden Ratio: A Golden Opportunity to Investigate Multiple Representations of a Problem.

ERIC Educational Resources Information Center

Dickey, Edwin M.

1993-01-01

This article explores the multiple representations (verbal, algebraic, graphical, and numerical) that can be used to study the golden ratio. Emphasis is placed on using technology (both calculators and computers) to investigate the algebraic, graphical, and numerical representations. (JAF)

Heterogeneous Software System Interoperability Through Computer-Aided Resolution of Modeling Differences

DTIC Science & Technology

2002-06-01

techniques for addressing the software component retrieval problem. Steigerwald [Ste91] introduced the use of algebraic specifications for defining the...provided in terms of a specification written using Luqi’s Prototype Specification Description Language (PSDL) [LBY88] augmented with an algebraic

Algebraic Thinking in Solving Linier Program at High School Level: Female Student’s Field Independent Cognitive Style

NASA Astrophysics Data System (ADS)

Hardiani, N.; Budayasa, I. K.; Juniati, D.

2018-01-01

The aim of this study was to describe algebraic thinking of high school female student’s field independent cognitive style in solving linier program problem by revealing deeply the female students’ responses. Subjects in this study were 7 female students having field independent cognitive style in class 11. The type of this research was descriptive qualitative. The method of data collection used was observation, documentation, and interview. Data analysis technique was by reduction, presentation, and conclusion. The results of this study showed that the female students with field independent cognitive style in solving the linier program problem had the ability to represent algebraic ideas from the narrative question that had been read by manipulating symbols and variables presented in tabular form, creating and building mathematical models in two variables linear inequality system which represented algebraic ideas, and interpreting the solutions as variables obtained from the point of intersection in the solution area to obtain maximum benefit.

Algebraic and geometric structures of analytic partial differential equations

NASA Astrophysics Data System (ADS)

Kaptsov, O. V.

2016-11-01

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

The development and nature of problem-solving among first-semester calculus students

NASA Astrophysics Data System (ADS)

Dawkins, Paul Christian; Mendoza Epperson, James A.

2014-08-01

This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students' problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem-solving as well as multiple mathematical registers and quantitative modelling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students' ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students' performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviours that characterized top performance or attrition in the course. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of US calculus reform. We provide explanations for these patterns in students' problem-solving performance in view of both their preparation for university calculus and the courses' assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students' problem-solving performance, we observe that current instruction requires ongoing refinement to help students develop multi-register fluency and the ability to model quantitatively, as is called for in current US standards for mathematical instruction.

Difficulties in initial algebra learning in Indonesia

NASA Astrophysics Data System (ADS)

Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja

2014-12-01

Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.

Robot Control Based On Spatial-Operator Algebra

NASA Technical Reports Server (NTRS)

Rodriguez, Guillermo; Kreutz, Kenneth K.; Jain, Abhinandan

1992-01-01

Method for mathematical modeling and control of robotic manipulators based on spatial-operator algebra providing concise representation and simple, high-level theoretical frame-work for solution of kinematical and dynamical problems involving complicated temporal and spatial relationships. Recursive algorithms derived immediately from abstract spatial-operator expressions by inspection. Transition from abstract formulation through abstract solution to detailed implementation of specific algorithms to compute solution greatly simplified. Complicated dynamical problems like two cooperating robot arms solved more easily.

Analysis of endomorphisms

NASA Astrophysics Data System (ADS)

Conti, Roberto; Hong, Jeong Hee; Szymański, Wojciech

2012-02-01

In this expository article, we discuss the recent progress in the study of endomorphisms and automorphisms of the Cuntz algebras and, more generally graph C* -algebras (or Cuntz-Krieger algebras). In particular, we discuss the definition and properties of both the full and the restricted Weyl group of such an algebra. Then we outline a powerful combinatorial approach to analysis of endomorphisms arising from permutation unitaries. The restricted Weyl group consists of automorphisms of this type. We also discuss the action of the restricted Weyl group on the diagonal MASA and its relationship with the automorphism group of the full two-sided n-shift. Finally, several open problems are presented.

Algebraic multigrid methods applied to problems in computational structural mechanics

NASA Technical Reports Server (NTRS)

Mccormick, Steve; Ruge, John

1989-01-01

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Dolan Grady relations and noncommutative quasi-exactly solvable systems

NASA Astrophysics Data System (ADS)

Klishevich, Sergey M.; Plyushchay, Mikhail S.

2003-11-01

We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the {\\mathfrak{su}}(2) and {\\mathfrak{sl}}(2,{\\bb {R}}) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization of the spectrum of the corresponding systems on the fuzzy sphere and noncommutative hyperbolic plane. A completely covariant method based on the notion of intrinsic algebra is proposed to deal with the spectral problem of such systems.

Variational data assimilation system "INM RAS - Black Sea"

NASA Astrophysics Data System (ADS)

Parmuzin, Eugene; Agoshkov, Valery; Assovskiy, Maksim; Giniatulin, Sergey; Zakharova, Natalia; Kuimov, Grigory; Fomin, Vladimir

2013-04-01

Development of Informational-Computational Systems (ICS) for Data Assimilation Procedures is one of multidisciplinary problems. To study and solve these problems one needs to apply modern results from different disciplines and recent developments in: mathematical modeling; theory of adjoint equations and optimal control; inverse problems; numerical methods theory; numerical algebra and scientific computing. The problems discussed above are studied in the Institute of Numerical Mathematics of the Russian Academy of Science (INM RAS) in ICS for Personal Computers (PC). Special problems and questions arise while effective ICS versions for PC are being developed. These problems and questions can be solved with applying modern methods of numerical mathematics and by solving "parallelism problem" using OpenMP technology and special linear algebra packages. In this work the results on the ICS development for PC-ICS "INM RAS - Black Sea" are presented. In the work the following problems and questions are discussed: practical problems that can be studied by ICS; parallelism problems and their solutions with applying of OpenMP technology and the linear algebra packages used in ICS "INM - Black Sea"; Interface of ICS. The results of ICS "INM RAS - Black Sea" testing are presented. Efficiency of technologies and methods applied are discussed. The work was supported by RFBR, grants No. 13-01-00753, 13-05-00715 and by The Ministry of education and science of Russian Federation, project 8291, project 11.519.11.1005 References: [1] V.I. Agoshkov, M.V. Assovskii, S.A. Lebedev, Numerical simulation of Black Sea hydrothermodynamics taking into account tide-forming forces. Russ. J. Numer. Anal. Math. Modelling (2012) 27, No.1, 5-31 [2] E.I. Parmuzin, V.I. Agoshkov, Numerical solution of the variational assimilation problem for sea surface temperature in the model of the Black Sea dynamics. Russ. J. Numer. Anal. Math. Modelling (2012) 27, No.1, 69-94 [3] V.B. Zalesny, N.A. Diansky, V.V. Fomin, S.N. Moshonkin, S.G. Demyshev, Numerical model of the circulation of Black Sea and Sea of Azov. Russ. J. Numer. Anal. Math. Modelling (2012) 27, No.1, 95-111 [4] V.I. Agoshkov, S.V. Giniatulin, G.V. Kuimov. OpenMP technology and linear algebra packages in the variation data assimilation systems. - Abstracts of the 1-st China-Russia Conference on Numerical Algebra with Applications in Radiactive Hydrodynamics, Beijing, China, October 16-18, 2012. [5] Zakharova N.B., Agoshkov V.I., Parmuzin E.I., The new method of ARGO buoys system observation data interpolation. Russian Journal of Numerical Analysis and Mathematical Modelling. Vol. 28, Issue 1, 2013.

Mathematics in the Early Grades: Operations & Algebraic Thinking. Interactive STEM Research + Practice Brief

ERIC Educational Resources Information Center

Education Development Center, Inc., 2016

2016-01-01

In the domain of "Operations & Algebraic Thinking," Common Core State Standards indicate that in kindergarten, first grade, and second grade, children should demonstrate and expand their ability to understand, represent, and solve problems using the operations of addition and subtraction, laying the foundation for operations using…

Teaching Linear Algebra: Proceeding More Efficiently by Staying Comfortably within Z

ERIC Educational Resources Information Center

Beaver, Scott

2015-01-01

For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram-Schmidt process.

Excel Spreadsheets for Algebra: Improving Mental Modeling for Problem Solving

ERIC Educational Resources Information Center

Engerman, Jason; Rusek, Matthew; Clariana, Roy

2014-01-01

This experiment investigates the effectiveness of Excel spreadsheets in a high school algebra class. Students in the experiment group convincingly outperformed the control group on a post lesson assessment. The student responses, teacher observations involving Excel spreadsheet revealed that it operated as a mindtool, which formed the users'…

Fixing Ganache: Another Real-Life Use for Algebra

ERIC Educational Resources Information Center

Kalman, Adam M.

2011-01-01

This article presents a real-world application of proportional reasoning and equation solving. The author describes how students adjust ingredient amounts in a recipe for chocolate ganache. Using this real-world scenario provided students an opportunity to solve a difficult and nonstandard algebra problem, a lot of practice with fractions, a…

Using Technology to Facilitate Reasoning: Lifting the Fog from Linear Algebra

ERIC Educational Resources Information Center

Berry, John S.; Lapp, Douglas A.; Nyman, Melvin A.

2008-01-01

This article discusses student difficulties in grasping concepts from linear algebra. Using an example from an interview with a student, we propose changes that might positively impact student understanding of concepts within a problem-solving context. In particular, we illustrate barriers to student understanding and suggest technological…

Algebra 2u, Mathematics (Experimental): 5216.26.

ERIC Educational Resources Information Center

Crawford, Glenda

The sixth in a series of six guidebooks on minimum course content for second-year algebra, this booklet presents an introduction to sequences, series, permutation, combinations, and probability. Included are arithmetic and geometric progressions and problems solved by counting and factorials. Overall course goals are specified, a course outline is…

An Evaluation of Interventions to Facilitate Algebra Problem Solving

ERIC Educational Resources Information Center

Mayfield, Kristin H.; Glenn, Irene M.

2008-01-01

Three participants were trained on 6 target algebra skills and subsequently received a series of 5 instructional interventions (cumulative practice, tiered feedback, feedback plus solution sequence instruction, review practice, and transfer training) in a multiple baseline across skills design. The effects of the interventions on the performance…

BRST Exactness of Stress-Energy Tensors

NASA Astrophysics Data System (ADS)

Miyata, Hideo; Sugimoto, Hiroshi

BRST commutators in the topological conformal field theories obtained by twisting N=2 theories are evaluated explicitly. By our systematic calculations of the multiple integrals which contain screening operators, the BRST exactness of the twisted stress-energy tensors is deduced for classical simple Lie algebras and general level k. We can see that the paths of integrations do not affect the result, and further, the N=2 coset theories are obtained by deleting two simple roots with Kac-label 1 from the extended Dynkin diagram; in other words, by not performing the integrations over the variables corresponding to the two simple roots of Kac-Moody algebras. It is also shown that a series of N=1 theories are generated in the same way by deleting one simple root with Kac-label 2.

Students’ Algebraic Thinking Process in Context of Point and Line Properties

NASA Astrophysics Data System (ADS)

Nurrahmi, H.; Suryadi, D.; Fatimah, S.

2017-09-01

Learning of schools algebra is limited to symbols and operating procedures, so students are able to work on problems that only require the ability to operate symbols but unable to generalize a pattern as one of part of algebraic thinking. The purpose of this study is to create a didactic design that facilitates students to do algebraic thinking process through the generalization of patterns, especially in the context of the property of point and line. This study used qualitative method and includes Didactical Design Research (DDR). The result is students are able to make factual, contextual, and symbolic generalization. This happen because the generalization arises based on facts on local terms, then the generalization produced an algebraic formula that was described in the context and perspective of each student. After that, the formula uses the algebraic letter symbol from the symbol t hat uses the students’ language. It can be concluded that the design has facilitated students to do algebraic thinking process through the generalization of patterns, especially in the context of property of the point and line. The impact of this study is this design can use as one of material teaching alternative in learning of school algebra.

From Quantum Fields to Local Von Neumann Algebras

NASA Astrophysics Data System (ADS)

Borchers, H. J.; Yngvason, Jakob

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

A matrix dependent/algebraic multigrid approach for extruded meshes with applications to ice sheet modeling

DOE PAGES

Tuminaro, Raymond S.; Perego, Mauro; Tezaur, Irina Kalashnikova; ...

2016-10-06

A multigrid method is proposed that combines ideas from matrix dependent multigrid for structured grids and algebraic multigrid for unstructured grids. It targets problems where a three-dimensional mesh can be viewed as an extrusion of a two-dimensional, unstructured mesh in a third dimension. Our motivation comes from the modeling of thin structures via finite elements and, more specifically, the modeling of ice sheets. Extruded meshes are relatively common for thin structures and often give rise to anisotropic problems when the thin direction mesh spacing is much smaller than the broad direction mesh spacing. Within our approach, the first few multigridmore » hierarchy levels are obtained by applying matrix dependent multigrid to semicoarsen in a structured thin direction fashion. After sufficient structured coarsening, the resulting mesh contains only a single layer corresponding to a two-dimensional, unstructured mesh. Algebraic multigrid can then be employed in a standard manner to create further coarse levels, as the anisotropic phenomena is no longer present in the single layer problem. The overall approach remains fully algebraic, with the minor exception that some additional information is needed to determine the extruded direction. Furthermore, this facilitates integration of the solver with a variety of different extruded mesh applications.« less

Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

NASA Astrophysics Data System (ADS)

Shevchenko, I. I.

2008-05-01

The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could be solved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability.

Minimal models of compact symplectic semitoric manifolds

NASA Astrophysics Data System (ADS)

Kane, D. M.; Palmer, J.; Pelayo, Á.

2018-02-01

A symplectic semitoric manifold is a symplectic 4-manifold endowed with a Hamiltonian (S1 × R) -action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into account the effects of the monodromy near focus-focus singularities. We give two applications of the helix: first, we use it to give a classification of the minimal models of symplectic semitoric manifolds, where "minimal" is in the sense of not admitting any blowdowns. The second application is an extension to the compact case of a well known result of Vũ Ngọc about the constraints posed on a symplectic semitoric manifold by the existence of focus-focus singularities. The helix permits to translate a symplectic geometric problem into an algebraic problem, and the paper describes a method to solve this type of algebraic problem.

xPerm: fast index canonicalization for tensor computer algebra

NASA Astrophysics Data System (ADS)

Martín-García, José M.

2008-10-01

We present a very fast implementation of the Butler-Portugal algorithm for index canonicalization with respect to permutation symmetries. It is called xPerm, and has been written as a combination of a Mathematica package and a C subroutine. The latter performs the most demanding parts of the computations and can be linked from any other program or computer algebra system. We demonstrate with tests and timings the effectively polynomial performance of the Butler-Portugal algorithm with respect to the number of indices, though we also show a case in which it is exponential. Our implementation handles generic tensorial expressions with several dozen indices in hundredths of a second, or one hundred indices in a few seconds, clearly outperforming all other current canonicalizers. The code has been already under intensive testing for several years and has been essential in recent investigations in large-scale tensor computer algebra. Program summaryProgram title: xPerm Catalogue identifier: AEBH_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBH_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 93 582 No. of bytes in distributed program, including test data, etc.: 1 537 832 Distribution format: tar.gz Programming language: C and Mathematica (version 5.0 or higher) Computer: Any computer running C and Mathematica (version 5.0 or higher) Operating system: Linux, Unix, Windows XP, MacOS RAM:: 20 Mbyte Word size: 64 or 32 bits Classification: 1.5, 5 Nature of problem: Canonicalization of indexed expressions with respect to permutation symmetries. Solution method: The Butler-Portugal algorithm. Restrictions: Multiterm symmetries are not considered. Running time: A few seconds with generic expressions of up to 100 indices. The xPermDoc.nb notebook supplied with the distribution takes approximately one and a half hours to execute in full.

Radioactivity Calculations

ERIC Educational Resources Information Center

Onega, Ronald J.

1969-01-01

Three problems in radioactive buildup and decay are presented and solved. Matrix algebra is used to solve the second problem. The third problem deals with flux depression and is solved by the use of differential equations. (LC)

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Kreutz, Kenneth; Jain, Abhinandan

1989-01-01

A recently developed spatial operator algebra, useful for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanizaton of specific algorithms is greatly simplified. The analytical formulation of the operator algebra, as well as its implementation in the Ada programming language are discussed.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei

The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl{sub -1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q{yields}-1 limit of the dual q-Hahn polynomials. The Hopf algebra sl{sub -1}(2) has four generators including an involution, it is also a q{yields}-1 limit of the quantum algebra sl{sub q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence relation of themore » -1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl{sub -1}(2) algebras, so that the Clebsch-Gordan coefficients of sl{sub -1}(2) are dual -1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual -1 Hahn polynomials is constructed.« less

A framework for modeling and optimizing dynamic systems under uncertainty

DOE PAGES

Nicholson, Bethany; Siirola, John

2017-11-11

Algebraic modeling languages (AMLs) have drastically simplified the implementation of algebraic optimization problems. However, there are still many classes of optimization problems that are not easily represented in most AMLs. These classes of problems are typically reformulated before implementation, which requires significant effort and time from the modeler and obscures the original problem structure or context. In this work we demonstrate how the Pyomo AML can be used to represent complex optimization problems using high-level modeling constructs. We focus on the operation of dynamic systems under uncertainty and demonstrate the combination of Pyomo extensions for dynamic optimization and stochastic programming.more » We use a dynamic semibatch reactor model and a large-scale bubbling fluidized bed adsorber model as test cases.« less

A framework for modeling and optimizing dynamic systems under uncertainty

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nicholson, Bethany; Siirola, John

Algebraic modeling languages (AMLs) have drastically simplified the implementation of algebraic optimization problems. However, there are still many classes of optimization problems that are not easily represented in most AMLs. These classes of problems are typically reformulated before implementation, which requires significant effort and time from the modeler and obscures the original problem structure or context. In this work we demonstrate how the Pyomo AML can be used to represent complex optimization problems using high-level modeling constructs. We focus on the operation of dynamic systems under uncertainty and demonstrate the combination of Pyomo extensions for dynamic optimization and stochastic programming.more » We use a dynamic semibatch reactor model and a large-scale bubbling fluidized bed adsorber model as test cases.« less

Encryption and decryption algorithm using algebraic matrix approach

NASA Astrophysics Data System (ADS)

Thiagarajan, K.; Balasubramanian, P.; Nagaraj, J.; Padmashree, J.

2018-04-01

Cryptographic algorithms provide security of data against attacks during encryption and decryption. However, they are computationally intensive process which consume large amount of CPU time and space at time of encryption and decryption. The goal of this paper is to study the encryption and decryption algorithm and to find space complexity of the encrypted and decrypted data by using of algorithm. In this paper, we encrypt and decrypt the message using key with the help of cyclic square matrix provides the approach applicable for any number of words having more number of characters and longest word. Also we discussed about the time complexity of the algorithm. The proposed algorithm is simple but difficult to break the process.

Graphing as a Problem-Solving Strategy.

ERIC Educational Resources Information Center

Cohen, Donald

1984-01-01

The focus is on how line graphs can be used to approximate solutions to rate problems and to suggest equations that offer exact algebraic solutions to the problem. Four problems requiring progressively greater graphing sophistication are presented plus four exercises. (MNS)

Geometric and Algebraic Approaches in the Concept of Complex Numbers

ERIC Educational Resources Information Center

Panaoura, A.; Elia, I.; Gagatsis, A.; Giatilis, G.-P.

2006-01-01

This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from…

Students' Use of Computational Thinking in Linear Algebra

ERIC Educational Resources Information Center

Bagley, Spencer; Rabin, Jeffrey M.

2016-01-01

In this work, we examine students' ways of thinking when presented with a novel linear algebra problem. Our intent was to explore how students employ and coordinate three modes of thinking, which we call computational, abstract, and geometric, following similar frameworks proposed by Hillel (2000) and Sierpinska (2000). However, the undergraduate…

High Level Technology in a Low Level Mathematics Course.

ERIC Educational Resources Information Center

Schultz, James E.; Noguera, Norma

2000-01-01

Describes a teaching experiment in which spreadsheets and computer algebra systems were used to teach a low-level college consumer mathematics course. Students were successful in using different types of functions to solve a variety of problems drawn from real-world situations. Provides an existence proof that computer algebra systems can assist…

Undergraduate Mathematics Students' Emotional Experiences in Linear Algebra Courses

ERIC Educational Resources Information Center

Martínez-Sierra, Gustavo; García-González, María del Socorro

2016-01-01

Little is known about students' emotions in the field of Mathematics Education that go beyond students' emotions in problem solving. To start filling this gap this qualitative research has the aim to identify emotional experiences of undergraduate mathematics students in Linear Algebra courses. In order to obtain data, retrospective focus group…

Algebraic Functions, Computer Programming, and the Challenge of Transfer

ERIC Educational Resources Information Center

Schanzer, Emmanuel Tanenbaum

2015-01-01

Students' struggles with algebra are well documented. Prior to the introduction of functions, mathematics is typically focused on applying a set of arithmetic operations to compute an answer. The introduction of functions, however, marks the point at which mathematics begins to focus on building up abstractions as a way to solve complex problems.…

Support for Struggling Students in Algebra: Contributions of Incorrect Worked Examples

ERIC Educational Resources Information Center

Barbieri, Christina; Booth, Julie L.

2016-01-01

Middle school algebra students (N = 125) randomly assigned within classroom to a Problem-solving control group, a Correct worked examples control group, or an Incorrect worked examples group, completed an experimental classroom study to assess the differential effects of incorrect examples versus the two control groups on students' algebra…

Computer Algebra Systems in Education Newsletter[s].

ERIC Educational Resources Information Center

Computer Algebra Systems in Education Newsletter, 1990

1990-01-01

Computer Algebra Systems (CAS) are computer systems for the exact solution of problems in symbolic form. The newspaper is designed to serve as a conduit for information and ideas on the use of CAS in education, especially in lower division college and university courses. Articles included are about CAS programs in several colleges, experiences…

Teachers' Instructional Practices within Connected Classroom Technology Environments to Support Representational Fluency

ERIC Educational Resources Information Center

Gunpinar, Yasemin; Pape, Stephen

2018-01-01

The purpose of this study was to investigate the ways that teachers use connected classroom technology (CCT) in conjunction with the Texas Instruments Nspire calculator to potentially support achievement on Algebra problems that require translation between representations (i.e., symbolic to graphical). Four Algebra I classrooms that initially…

Designing Tasks for Math Modeling in College Algebra: A Critical Review

ERIC Educational Resources Information Center

Staats, Susan; Robertson, Douglas

2014-01-01

Over the last decade, the pedagogical approach known as mathematical modeling has received increased interest in college algebra classes in the United States. Math modeling assignments ask students to develop their own problem-solving tools to address non-routine, realistic scenarios. The open-ended quality of modeling activities creates dilemmas…

University of Chicago School Mathematics Project 6-12 Curriculum. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2011

2011-01-01

The "University of Chicago School Mathematics Project ("UCSMP") 6-12 Curriculum" is a series of yearlong courses--(1) Transition Mathematics; (2) Algebra; (3) Geometry; (4) Advanced Algebra; (5) Functions, Statistics, and Trigonometry; and (6) Precalculus and Discrete Mathematics--emphasizing problem solving, real-world applications, and the use…

Scratch Your Brain Where It Itches: Math Games, Tricks and Quick Activities, Book D-1 Algebra.

ERIC Educational Resources Information Center

Brumbaugh, Doug

This resource book for algebra contains games, tricks, and quick activities for the classroom. Categories of activities include puzzlers, patterns, manipulatives, measurement, graphing, and a section that contains reproducible statement and value cards. Twenty one puzzle problems, four pattern activities, and 11 quick activities that engage…

Design Research on Personalized Problem Posing in Algebra

ERIC Educational Resources Information Center

Walkington, Candace

2017-01-01

Algebra is an area of pressing national concern around issues of equity and access in education. Recent theories and research suggest that personalization of instruction can allow students to activate their funds of knowledge and can elicit interest in the content to be learned. This paper examines the results of a large-scale teaching experiment…

Enhancing Mathematical Communication for Virtual Math Teams

ERIC Educational Resources Information Center

Stahl, Gerry; Çakir, Murat Perit; Weimar, Stephen; Weusijana, Baba Kofi; Ou, Jimmy Xiantong

2010-01-01

The Math Forum is an online resource center for pre-algebra, algebra, geometry and pre-calculus. Its Virtual Math Teams (VMT) service provides an integrated web-based environment for small teams of people to discuss math and to work collaboratively on math problems or explore interesting mathematical micro-worlds together. The VMT Project studies…

Strengthening Grade 3-5 Students' Foundational Knowledge of Rational Numbers

ERIC Educational Resources Information Center

Good, Thomas L.; Wood, Marcy B.; Sabers, Darrell; Olson, Amy M.; Lavigne, Alyson Leah; Sun, Huaping; Kalinec-Craig, Crystal

2013-01-01

Background: American students have done poorly in algebra and that has generated policy concerns about preparing students for STEM careers. There has been growing recognition that the algebra problem may begin in earlier grades when students do not adequately master rational numbers. Purpose: The study provided a series of workshops organized…

Superitem Test: An Alternative Assessment Tool to Assess Students' Algebraic Solving Ability

ERIC Educational Resources Information Center

Lian, Lim Hooi; Yew, Wun Thiam; Idris, Noraini

2010-01-01

Superitem test based on the SOLO model (Structure of the Observing Learning Outcome) has become a powerful alternative assessment tool for monitoring the growth of students' cognitive ability in solving mathematics problems. This article focused on developing a superitem test to assess students' algebraic solving ability through interview method.…

Stability of Linear Equations--Algebraic Approach

ERIC Educational Resources Information Center

Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.

2012-01-01

This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…

Sensitivity calculations for iteratively solved problems

NASA Technical Reports Server (NTRS)

Haftka, R. T.

1985-01-01

The calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated. A modified finite difference procedure is presented which improves the accuracy of the calculated derivatives. The procedure is demonstrated for a simple algebraic example as well as an element-by-element preconditioned conjugate gradient iterative solution technique applied to truss examples.

Comparative study of original recover and recover KL in separable non-negative matrix factorization for topic detection in Twitter

NASA Astrophysics Data System (ADS)

Prabandari, R. D.; Murfi, H.

2017-07-01

An increasing amount of information on social media such as Twitter requires an efficient way to find the topics so that the information can be well managed. One of an automated method for topic detection is separable non-negative matrix factorization (SNMF). SNMF assumes that each topic has at least one word that does not appear on other topics. This method uses the direct approach and gives polynomial-time complexity, while the previous methods are iterative approaches and have NP-hard complexity. There are three steps of SNMF algorithm, i.e. constructing word co-occurrences, finding anchor words, and recovering topics. In this paper, we examine two topic recover methods, namely original recover that is using algebraic manipulation and recover KL that using probability approach with Kullback-Leibler divergence. Our simulations show that recover KL provides better accuracies in term of topic recall than original recover.

Algebraic model checking for Boolean gene regulatory networks.

PubMed

Tran, Quoc-Nam

2011-01-01

We present a computational method in which modular and Groebner bases (GB) computation in Boolean rings are used for solving problems in Boolean gene regulatory networks (BN). In contrast to other known algebraic approaches, the degree of intermediate polynomials during the calculation of Groebner bases using our method will never grow resulting in a significant improvement in running time and memory space consumption. We also show how calculation in temporal logic for model checking can be done by means of our direct and efficient Groebner basis computation in Boolean rings. We present our experimental results in finding attractors and control strategies of Boolean networks to illustrate our theoretical arguments. The results are promising. Our algebraic approach is more efficient than the state-of-the-art model checker NuSMV on BNs. More importantly, our approach finds all solutions for the BN problems.

Type II superstring field theory: geometric approach and operadic description

NASA Astrophysics Data System (ADS)

Jurčo, Branislav; Münster, Korbinian

2013-04-01

We outline the construction of type II superstring field theory leading to a geometric and algebraic BV master equation, analogous to Zwiebach's construction for the bosonic string. The construction uses the small Hilbert space. Elementary vertices of the non-polynomial action are described with the help of a properly formulated minimal area problem. They give rise to an infinite tower of superstring field products defining a {N} = 1 generalization of a loop homotopy Lie algebra, the genus zero part generalizing a homotopy Lie algebra. Finally, we give an operadic interpretation of the construction.

Classical integrable many-body systems disconnected with semi-simple Lie algebras

NASA Astrophysics Data System (ADS)

Inozemtsev, V. I.

2017-05-01

The review of the results in the theory of integrable many-body systems disconnected with semisimple Lie algebras is done. The one-dimensional systems of light Calogero-Sutherland-Moser particles interacting with one particle of infinite mass located at the origin are described in detail. In some cases the exact solutions of the equations of motion are obtained. The general theory of integration of the equations of motion needs the methods of algebraic geometry. The Lax pairs with spectral parameter are constructed for this purpose. The theory still contains many unsolved problems.

High-Speed, Low-Cost Workstation for Computation-Intensive Statistics. Phase 1

DTIC Science & Technology

1990-06-20

routine implementation and performance. 5 The two compiled versions given in the table were coded in an attempt to obtain an optimized compiled version...level statistics and linear algebra routines (BSAS and BLAS) that have been prototyped in this study. For each routine, both the C code ( Turbo C...OISTRIBUTION /AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unlimited distribution 13. ABSTRACT (Maximum 200 words) High-performance and low-cost

Solving Word Problems using Schemas: A Review of the Literature

PubMed Central

Powell, Sarah R.

2011-01-01

Solving word problems is a difficult task for students at-risk for or with learning disabilities (LD). One instructional approach that has emerged as a valid method for helping students at-risk for or with LD to become more proficient at word-problem solving is using schemas. A schema is a framework for solving a problem. With a schema, students are taught to recognize problems as falling within word-problem types and to apply a problem solution method that matches that problem type. This review highlights two schema approaches for 2nd- and 3rd-grade students at-risk for or with LD: schema-based instruction and schema-broadening instruction. A total of 12 schema studies were reviewed and synthesized. Both types of schema approaches enhanced the word-problem skill of students at-risk for or with LD. Based on the review, suggestions are provided for incorporating word-problem instruction using schemas. PMID:21643477

Some Applications Of Semigroups And Computer Algebra In Discrete Structures

NASA Astrophysics Data System (ADS)

Bijev, G.

2009-11-01

An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. A map (p) is defined, which is similar to an orthogonal projection in linear vector spaces. Some other important maps with properties similar to those of the generalized inverses (pseudoinverses) of linear transformations and matrices corresponding to them are also defined and investigated. Let Ax = b be an equation with matrix A and vectors x and b Boolean. Stochastic experiments for solving the equation, which involves the maps defined and use computer algebra methods, have been made. As a result, the Hamming distance between vectors Ax = p(b) and b is equal or close to the least possible. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra and research. Some examples for computations with binary relations using Maple are given.

Culturally and Linguistically Responsive Schema Intervention: Improving Word Problem Solving for English Language Learners with Mathematics Difficulty

ERIC Educational Resources Information Center

Driver, Melissa K.; Powell, Sarah R.

2017-01-01

Word problems are prevalent on high-stakes assessments, and success on word problems has implications for grade promotion and graduation. Unfortunately, English Language Learners (ELLs) continue to perform significantly below their native English-speaking peers on mathematics assessments featuring word problems. Little is known about the…

Contribution of Equal-Sign Instruction beyond Word-Problem Tutoring for Third-Grade Students with Mathematics Difficulty.

PubMed

Powell, Sarah R; Fuchs, Lynn S

2010-05-01

Elementary school students often misinterpret the equal sign (=) as an operational rather than a relational symbol. Such misunderstanding is problematic because solving equations with missing numbers may be important for higher-order mathematics skills including word problems. Research indicates equal-sign instruction can alter how typically-developing students use the equal sign, but no study has examined effects for students with mathematics difficulty (MD) or how equal-sign instruction contributes to word-problem skill for students with or without MD. The present study assessed the efficacy of equal-sign instruction within word-problem tutoring. Third-grade students with MD (n = 80) were assigned to word-problem tutoring, word-problem tutoring plus equal-sign instruction (combined) tutoring, or no-tutoring control. Combined tutoring produced better improvement on equal sign tasks and open equations compared to the other 2 conditions. On certain forms of word problems, combined tutoring but not word-problem tutoring alone produced better improvement than control. When compared at posttest to 3(rd)-grade students without MD on equal sign tasks and open equations, only combined tutoring students with MD performed comparably.

Solution of Inverse Kinematics for 6R Robot Manipulators With Offset Wrist Based on Geometric Algebra.

PubMed

Fu, Zhongtao; Yang, Wenyu; Yang, Zhen

2013-08-01

In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators.

The Effect of Graphing Calculators on Student Achievement in College Algebra and Pre-Calculus Mathematics Courses

ERIC Educational Resources Information Center

Hatem, Neil

2010-01-01

This study investigates the relationship between the use of graphing calculators employed as Type II technology and student achievement, as determined by assessing students' problem solving skills associated with the concept of function, at the college algebra and pre-calculus level. In addition, this study explores the integration of graphing…

Redesigning College Algebra: Combining Educational Theory and Web-Based Learning to Improve Student Attitudes and Performance

ERIC Educational Resources Information Center

Hagerty, Gary; Smith, Stanley; Goodwin, Danielle

2010-01-01

In 2001, Black Hills State University (BHSU) redesigned college algebra to use the computer-based mastery learning program, Assessment and Learning in Knowledge Spaces [1], historical development of concepts modules, whole class discussions, cooperative activities, relevant applications problems, and many fewer lectures. This resulted in a 21%…

An Intervention Including an Online Game to Improve Grade 6 Students' Performance in Early Algebra

ERIC Educational Resources Information Center

Kolovou, Angeliki; van den Heuvel-Panhuizen, Marja; Koller, Olaf

2013-01-01

This study investigated whether an intervention including an online game contributed to 236 Grade 6 students' performance in early algebra, that is, solving problems with covarying quantities. An exploratory quasi-experimental study was conducted with a pretest-posttest-control-group design. Students in the experimental group were asked to solve…

Algebraic Reasoning: Professor Arbegla Introduces Variables and Functions. GEMS Teacher's Guide for Grades 3-5.

ERIC Educational Resources Information Center

Kopp, Jaine; Bergman, Lincoln

This teacher guide helps build a solid foundation in algebra for students in grades 3-5 in which students gain essential understanding of properties of numbers, variables, functions, equations, and formulas. Throughout the problem solving activities, students use computational skills and gain a deeper understanding of the number system. Students…

Algebra and Problem-Solving in Down Syndrome: A Study with 15 Teenagers

ERIC Educational Resources Information Center

Martinez, Elisabetta Monari; Pellegrini, Katia

2010-01-01

There is a common opinion that mathematics is difficult for persons with Down syndrome, because of a weakness in numeracy and in abstract thinking. Since 1996, some single case studies have suggested that new opportunities in mathematics are possible for these students: some of them learned algebra and also learned to use equations in…

Solution of Algebraic Equations in the Analysis, Design, and Optimization of Continuous Ultrafiltration

ERIC Educational Resources Information Center

Foley, Greg

2011-01-01

Continuous feed and bleed ultrafiltration, modeled with the gel polarization model for the limiting flux, is shown to provide a rich source of non-linear algebraic equations that can be readily solved using numerical and graphical techniques familiar to undergraduate students. We present a variety of numerical problems in the design, analysis, and…

A Comparison of Equality in Computer Algebra and Correctness in Mathematical Pedagogy (II)

ERIC Educational Resources Information Center

Bradford, Russell; Davenport, James H.; Sangwin, Chris

2010-01-01

A perennial problem in computer-aided assessment is that "a right answer", pedagogically speaking, is not the same thing as "a mathematically correct expression", as verified by a computer algebra system, or indeed other techniques such as random evaluation. Paper I in this series considered the difference in cases where there was "the right…

Understanding the Equals Sign as a Gateway to Algebraic Thinking

ERIC Educational Resources Information Center

Matthews, Percival G.; Rittle-Johnson, Bethany; Taylor, Roger S.; McEldoon, Katherine L.

2010-01-01

In this study, the authors wanted to examine whether success on items testing basic equivalence knowledge, such as the meaning of the equal sign and ability to solve problems such as 3 + 5 = 4 + _, predicted success on items testing more advanced algebraic thinking (i.e. principles of equality and solving equations that use letter variables). This…

ABILITIES OF FIRST-GRADE PUPILS TO LEARN MATERIALS IN TERMS OF ALGEBRAIC STRUCTURES BY TEACHING MACHINES.

ERIC Educational Resources Information Center

CRAWFORD, ROBERT C.; KEISLAR, EVAN R.

THE MAJOR PROBLEM OF THIS INVESTIGATION WAS TO DETERMINE TO WHAT EXTENT FIRST-GRADE PUPILS ARE CAPABLE OF LEARNING ALGEBRAIC STRUCTURES THROUGH PROGRAMED INSTRUCTION. IN THE EXPERIMENT APPROXIMATELY 130 FIRST-GRADERS WERE INSTRUCTED THROUGH AUDIOVISUAL TEACHING MACHINES FOR APPROXIMATELY 15 WEEKS. AT THE END OF THE PROGRAM, THE CHILDREN WERE…

Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

NASA Astrophysics Data System (ADS)

Connes, Alain; Kreimer, Dirk

This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

Analysis of algebraic reasoning ability of cognitive style perspectives on field dependent field independent and gender

NASA Astrophysics Data System (ADS)

Rosita, N. T.

2018-03-01

The purpose of this study is to analyse algebraic reasoning ability using the SOLO model as a theoretical framework to assess students’ algebraic reasoning abilities of Field Dependent cognitive (FD), Field Independent (FI) and Gender perspectives. The method of this study is a qualitative research. The instrument of this study is the researcher himself assisted with algebraic reasoning tests, the problems have been designed based on NCTM indicators and algebraic reasoning according to SOLO model. While the cognitive style of students is determined using Group Embedded Figure Test (GEFT), as well as interviews on the subject as triangulation. The subjects are 15 female and 15 males of the sixth semester students of mathematics education, STKIP Sebelas April. The results of the qualitative data analysis is that most subjects are at the level of unistructural and multi-structural, subjects at the relational level have difficulty in forming a new linear pattern. While the subjects at the extended abstract level are able to meet all the indicators of algebraic reasoning ability even though some of the answers are not perfect yet. Subjects of FI tend to have higher algebraic reasoning abilities than of the subject of FD.

Eigenmode computation of cavities with perturbed geometry using matrix perturbation methods applied on generalized eigenvalue problems

NASA Astrophysics Data System (ADS)

Gorgizadeh, Shahnam; Flisgen, Thomas; van Rienen, Ursula

2018-07-01

Generalized eigenvalue problems are standard problems in computational sciences. They may arise in electromagnetic fields from the discretization of the Helmholtz equation by for example the finite element method (FEM). Geometrical perturbations of the structure under concern lead to a new generalized eigenvalue problems with different system matrices. Geometrical perturbations may arise by manufacturing tolerances, harsh operating conditions or during shape optimization. Directly solving the eigenvalue problem for each perturbation is computationally costly. The perturbed eigenpairs can be approximated using eigenpair derivatives. Two common approaches for the calculation of eigenpair derivatives, namely modal superposition method and direct algebraic methods, are discussed in this paper. Based on the direct algebraic methods an iterative algorithm is developed for efficiently calculating the eigenvalues and eigenvectors of the perturbed geometry from the eigenvalues and eigenvectors of the unperturbed geometry.

Individualized Math Problems in Algebra. Oregon Vo-Tech Mathematics Problem Sets.

ERIC Educational Resources Information Center

Cosler, Norma, Ed.

This is one of eighteen sets of individualized mathematics problems developed by the Oregon Vo-Tech Math Project. Each of these problem packages is organized around a mathematical topic, and contains problems related to diverse vocations. Solutions are provided for all problems. Problems presented in this package concern ratios used in food…

How Can One Learn Mathematical Word Problems in a Second Language? A Cognitive Load Perspective

ERIC Educational Resources Information Center

Moussa-Inaty, Jase; Causapin, Mark; Groombridge, Timothy

2015-01-01

Language may ordinarily account for difficulties in solving word problems and this is particularly true if mathematical word problems are taught in a language other than one's native language. Research into cognitive load may offer a clear theoretical framework when investigating word problems because memory, specifically working memory, plays a…

Do word-problem features differentially affect problem difficulty as a function of students' mathematics difficulty with and without reading difficulty?

PubMed

Powell, Sarah R; Fuchs, Lynn S; Fuchs, Douglas; Cirino, Paul T; Fletcher, Jack M

2009-01-01

This study examined whether and, if so, how word-problem features differentially affect problem difficulty as a function of mathematics difficulty (MD) status: no MD (n = 109), MD only (n = 109), or MD in combination with reading difficulties (MDRD; n = 109). The problem features were problem type (total, difference, or change) and position of missing information in the number sentence representing the word problem (first, second, or third position). Students were assessed on 14 word problems near the beginning of third grade. Consistent with the hypothesis that mathematical cognition differs as a function of MD subtype, problem type affected problem difficulty differentially for MDRD versus MD-only students; however, the position of missing information in word problems did not. Implications for MD subtyping and for instruction are discussed.

Convergence of the standard RLS method and UDUT factorisation of covariance matrix for solving the algebraic Riccati equation of the DLQR via heuristic approximate dynamic programming

NASA Astrophysics Data System (ADS)

Moraes Rêgo, Patrícia Helena; Viana da Fonseca Neto, João; Ferreira, Ernesto M.

2015-08-01

The main focus of this article is to present a proposal to solve, via UDUT factorisation, the convergence and numerical stability problems that are related to the covariance matrix ill-conditioning of the recursive least squares (RLS) approach for online approximations of the algebraic Riccati equation (ARE) solution associated with the discrete linear quadratic regulator (DLQR) problem formulated in the actor-critic reinforcement learning and approximate dynamic programming context. The parameterisations of the Bellman equation, utility function and dynamic system as well as the algebra of Kronecker product assemble a framework for the solution of the DLQR problem. The condition number and the positivity parameter of the covariance matrix are associated with statistical metrics for evaluating the approximation performance of the ARE solution via RLS-based estimators. The performance of RLS approximators is also evaluated in terms of consistence and polarisation when associated with reinforcement learning methods. The used methodology contemplates realisations of online designs for DLQR controllers that is evaluated in a multivariable dynamic system model.

Quantization and Superselection Sectors I:. Transformation Group C*-ALGEBRAS

NASA Astrophysics Data System (ADS)

Landsman, N. P.

Quantization is defined as the act of assigning an appropriate C*-algebra { A} to a given configuration space Q, along with a prescription mapping self-adjoint elements of { A} into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here { A} is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of { A}. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of { A}.

Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

NASA Astrophysics Data System (ADS)

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

Cognitive and Linguistic Predictors of Mathematical Word Problems with and without Irrelevant Information

ERIC Educational Resources Information Center

Wang, Amber Y.; Fuchs, Lynn S.; Fuchs, Douglas

2016-01-01

The purpose of this study was to identify cognitive and linguistic predictors of word problems with versus without irrelevant information. The sample was 701 2nd-grade students who received no specialized intervention on word problems. In the fall, they were assessed on initial arithmetic and word-problem skill as well as language ability, working…

Contribution of Equal-Sign Instruction beyond Word-Problem Tutoring for Third-Grade Students with Mathematics Difficulty

PubMed Central

Powell, Sarah R.; Fuchs, Lynn S.

2010-01-01

Elementary school students often misinterpret the equal sign (=) as an operational rather than a relational symbol. Such misunderstanding is problematic because solving equations with missing numbers may be important for higher-order mathematics skills including word problems. Research indicates equal-sign instruction can alter how typically-developing students use the equal sign, but no study has examined effects for students with mathematics difficulty (MD) or how equal-sign instruction contributes to word-problem skill for students with or without MD. The present study assessed the efficacy of equal-sign instruction within word-problem tutoring. Third-grade students with MD (n = 80) were assigned to word-problem tutoring, word-problem tutoring plus equal-sign instruction (combined) tutoring, or no-tutoring control. Combined tutoring produced better improvement on equal sign tasks and open equations compared to the other 2 conditions. On certain forms of word problems, combined tutoring but not word-problem tutoring alone produced better improvement than control. When compared at posttest to 3rd-grade students without MD on equal sign tasks and open equations, only combined tutoring students with MD performed comparably. PMID:20640240

Exact solution of some linear matrix equations using algebraic methods

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1979-01-01

Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

Using Self-Generated Drawings to Solve Arithmetic Word Problems.

ERIC Educational Resources Information Center

Van Essen, Gerard; Hamaker, Christiaan

1990-01-01

Results are presented from two intervention studies which investigate whether encouraging elementary students to generate drawings of arithmetic word problems facilitates problem-solving performance. Findings indicate that fifth graders (N=50) generated many drawings of word problems and improved problem solutions after the intervention, whereas…

Perceptual learning modules in mathematics: enhancing students' pattern recognition, structure extraction, and fluency.

PubMed

Kellman, Philip J; Massey, Christine M; Son, Ji Y

2010-04-01

Learning in educational settings emphasizes declarative and procedural knowledge. Studies of expertise, however, point to other crucial components of learning, especially improvements produced by experience in the extraction of information: perceptual learning (PL). We suggest that such improvements characterize both simple sensory and complex cognitive, even symbolic, tasks through common processes of discovery and selection. We apply these ideas in the form of perceptual learning modules (PLMs) to mathematics learning. We tested three PLMs, each emphasizing different aspects of complex task performance, in middle and high school mathematics. In the MultiRep PLM, practice in matching function information across multiple representations improved students' abilities to generate correct graphs and equations from word problems. In the Algebraic Transformations PLM, practice in seeing equation structure across transformations (but not solving equations) led to dramatic improvements in the speed of equation solving. In the Linear Measurement PLM, interactive trials involving extraction of information about units and lengths produced successful transfer to novel measurement problems and fraction problem solving. Taken together, these results suggest (a) that PL techniques have the potential to address crucial, neglected dimensions of learning, including discovery and fluent processing of relations; (b) PL effects apply even to complex tasks that involve symbolic processing; and (c) appropriately designed PL technology can produce rapid and enduring advances in learning. Copyright © 2009 Cognitive Science Society, Inc.

Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nataf, J.M.; Winkelmann, F.

We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK's symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of thesemore » methods to solving the partial differential equations for two-dimensional heat flow is illustrated.« less

Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nataf, J.M.; Winkelmann, F.

We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of thesemore » methods to solving the partial differential equations for two-dimensional heat flow is illustrated.« less

On superintegrable monopole systems

NASA Astrophysics Data System (ADS)

Fazlul Hoque, Md; Marquette, Ian; Zhang, Yao-Zhong

2018-02-01

Superintegrable systems with monopole interactions in flat and curved spaces have attracted much attention. For example, models in spaces with a Taub-NUT metric are well-known to admit the Kepler-type symmetries and provide non-trivial generalizations of the usual Kepler problems. In this paper, we overview new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We present their higher-order, algebraically independent integrals of motion via the direct and constructive approaches which prove the superintegrability of the models. The integrals form symmetry polynomial algebras of the systems with structure constants involving Casimir operators of certain Lie algebras. Such algebraic approaches provide a deeper understanding to the degeneracies of the energy spectra and connection between wave functions and differential equations and geometry.

Solving a System of Nonlinear Algebraic Equations You Only Get Error Messages--What to Do Next?

ERIC Educational Resources Information Center

Shacham, Mordechai; Brauner, Neima

2017-01-01

Chemical engineering problems often involve the solution of systems of nonlinear algebraic equations (NLE). There are several software packages that can be used for solving NLE systems, but they may occasionally fail, especially in cases where the mathematical model contains discontinuities and/or regions where some of the functions are undefined.…

Effects of Multimedia-Based Instructional Technology on African American Ninth Grade Students' Mastery of Algebra Concepts

ERIC Educational Resources Information Center

Malik, Ishan Z.

2011-01-01

Urban African American students lack an abstract understanding of algebra and are below their academic level in comparison to other ethnic groups, and this is a pervasive problem (McKinney, Chappell, Berry, & Hickman, 2009). The purpose of this quantitative study using a quasi-experimental design was to determine whether the use of…

CENTER CONDITIONS AND CYCLICITY FOR A FAMILY OF CUBIC SYSTEMS: COMPUTER ALGEBRA APPROACH.

PubMed

Ferčec, Brigita; Mahdi, Adam

2013-01-01

Using methods of computational algebra we obtain an upper bound for the cyclicity of a family of cubic systems. We overcame the problem of nonradicality of the associated Bautin ideal by moving from the ring of polynomials to a coordinate ring. Finally, we determine the number of limit cycles bifurcating from each component of the center variety.

Lie-algebraic Approach to Dynamics of Closed Quantum Systems and Quantum-to-Classical Correspondence

NASA Astrophysics Data System (ADS)

Galitski, Victor

2012-02-01

I will briefly review our recent work on a Lie-algebraic approach to various non-equilibrium quantum-mechanical problems, which has been motivated by continuous experimental advances in the field of cold atoms. First, I will discuss non-equilibrium driven dynamics of a generic closed quantum system. It will be emphasized that mathematically a non-equilibrium Hamiltonian represents a trajectory in a Lie algebra, while the evolution operator is a trajectory in a Lie group generated by the underlying algebra via exponentiation. This turns out to be a constructive statement that establishes, in particular, the fact that classical and quantum unitary evolutions are two sides of the same coin determined uniquely by the same dynamic generators in the group. An equation for these generators - dubbed dual Schr"odinger-Bloch equation - will be derived and analyzed for a few of specific examples. This non-linear equation allows one to construct new exact non-linear solutions to quantum-dynamical systems. An experimentally-relevant example of a family of exact solutions to the many-body Landau-Zener problem will be presented. One practical application of the latter result includes dynamical means to optimize molecular production rate following a quench across the Feshbach resonance.

Does understanding relational terminology mediate effects of intervention on compare word problems?

PubMed

Schumacher, Robin F; Fuchs, Lynn S

2012-04-01

The purpose of this study was to assess whether understanding relational terminology (i.e., more, less, and fewer) mediates the effects of intervention on compare word problems. Second-grade classrooms (N=31) were randomly assigned to one of three conditions: researcher-designed word-problem intervention, researcher-designed calculation intervention, or business-as-usual (teacher-designed) control. Students in word-problem intervention classrooms received instruction on the compare problem type, which included a focus on understanding relational terminology within compare word problems. Analyses, which accounted for variance associated with classroom clustering, indicated that (a) compared with the calculation intervention and business-as-usual conditions, word-problem intervention significantly increased performance on all three subtypes of compare problems and on understanding relational terminology, and (b) the intervention effect was fully mediated by students' understanding of relational terminology for one subtype of compare problems and partially mediated by students' understanding of relational terminology for the other two subtypes. Copyright © 2011 Elsevier Inc. All rights reserved.

On ``Overestimation-free Computational Version of Interval Analysis''

NASA Astrophysics Data System (ADS)

Popova, Evgenija D.

2013-10-01

The transformation of interval parameters into trigonometric functions, proposed in Int. J. Comput. Meth. Eng. Sci. Mech., vol. 13, pp. 319-328 (2012), is not motivated in comparison to the infinitely many equivalent algebraic transformations. The conclusions about the efficacy of the methodology used are based on incorrect comparisons between solutions of different problems. We show theoretically, and in the examples considered in the commented article, that changing the number of the parameters in a system of linear algebraic equations may change the initial problem, respectively, its solution set. We also correct various misunderstandings and bugs that appear in the article noted above.

Implicit Runge-Kutta Methods with Explicit Internal Stages

NASA Astrophysics Data System (ADS)

Skvortsov, L. M.

2018-03-01

The main computational costs of implicit Runge-Kutta methods are caused by solving a system of algebraic equations at every step. By introducing explicit stages, it is possible to increase the stage (or pseudo-stage) order of the method, which makes it possible to increase the accuracy and avoid reducing the order in solving stiff problems, without additional costs of solving algebraic equations. The paper presents implicit methods with an explicit first stage and one or two explicit internal stages. The results of solving test problems are compared with similar methods having no explicit internal stages.

A linear programming manual

NASA Technical Reports Server (NTRS)

Tuey, R. C.

1972-01-01

Computer solutions of linear programming problems are outlined. Information covers vector spaces, convex sets, and matrix algebra elements for solving simultaneous linear equations. Dual problems, reduced cost analysis, ranges, and error analysis are illustrated.

Discovering Steiner Triple Systems through Problem Solving

ERIC Educational Resources Information Center

Sriraman, Bharath

2004-01-01

An attempt to implement problem solving as a teacher of ninth grade algebra is described. The problems selected were not general ones, they involved combinations and represented various situations and were more complex which lead to the discovery of Steiner triple systems.

Solving of Clock Problems Using An Algebraic Approach And Developing An Application For Automatic Conversion

NASA Astrophysics Data System (ADS)

Lakshmi Devaraj, Shanmuga

2018-04-01

The recent trend in learning Mathematics is through android apps like Byju’s. The clock problems asked in aptitude tests could be learnt using such computer applications. The Clock problems are of four categories namely: 1. What is the angle between the hands of a clock at a particular time 2. When the hands of a clock will meet after a particular time 3. When the hands of a clock will be at right angle after a particular time 4. When the hands of a clock will be in a straight line but not together after a particular time The aim of this article is to convert the clock problems which were solved using the traditional approach to algebraic equations and solve them. Shortcuts are arrived which help in solving the questions in just a few seconds. Any aptitude problem could be converted to an algebraic equation by tracing the way the problem proceeds by applying our analytical skills. Solving of equations would be the easiest part in coming up with the solution. Also a computer application could be developed by using the equations that were arrived at in the analysis part. The computer application aims at solving the four different problems in Clocks. The application helps the learners of aptitude for CAT and other competitive exams to know the approach of the problem. Learning Mathematics with a gaming tool like this would be interesting to the learners. This paper provides a path to creating gaming apps to learn Mathematics.

Three-M in Word Problem Solving

ERIC Educational Resources Information Center

Hajra, Sayonita Ghosh; Kofman, Victoria

2018-01-01

We describe three activities that help undergraduates (pre-service teachers) to develop scientific vocabulary on measurable attributes and units of measurement. Measurable attributes are important features in understanding a word problem and solving the problem. These activities help students comprehend word problems better by identifying…

A Comparison of Solver Performance for Complex Gastric Electrophysiology Models

PubMed Central

Sathar, Shameer; Cheng, Leo K.; Trew, Mark L.

2016-01-01

Computational techniques for solving systems of equations arising in gastric electrophysiology have not been studied for efficient solution process. We present a computationally challenging problem of simulating gastric electrophysiology in anatomically realistic stomach geometries with multiple intracellular and extracellular domains. The multiscale nature of the problem and mesh resolution required to capture geometric and functional features necessitates efficient solution methods if the problem is to be tractable. In this study, we investigated and compared several parallel preconditioners for the linear systems arising from tetrahedral discretisation of electrically isotropic and anisotropic problems, with and without stimuli. The results showed that the isotropic problem was computationally less challenging than the anisotropic problem and that the application of extracellular stimuli increased workload considerably. Preconditioning based on block Jacobi and algebraic multigrid solvers were found to have the best overall solution times and least iteration counts, respectively. The algebraic multigrid preconditioner would be expected to perform better on large problems. PMID:26736543

TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION

NASA Technical Reports Server (NTRS)

Smith, R. E.

1994-01-01

TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.

Pupils' over-reliance on linearity: a scholastic effect?

PubMed

Van Dooren, Wim; De Bock, Dirk; Janssens, Dirk; Verschaffel, Lieven

2007-06-01

From upper elementary education on, children develop a tendency to over-use linearity. Particularly, it is found that many pupils assume that if a figure enlarges k times, the area enlarges k times too. However, most research was conducted with traditional, school-like word problems. This study examines whether pupils also over-use linearity if non-linear problems are embedded in meaningful, authentic performance tasks instead of traditional, school-like word problems, and whether this experience influences later behaviour. Ninety-three sixth graders from two primary schools in Flanders, Belgium. Pupils received a pre-test with traditional word problems. Those who made a linear error on the non-linear area problem were subjected to individual interviews. They received one new non-linear problem, in the S-condition (again a traditional, scholastic word problem), D-condition (the same word problem with a drawing) or P-condition (a meaningful performance-based task). Shortly afterwards, pupils received a post-test, containing again a non-linear word problem. Most pupils from the S-condition displayed linear reasoning during the interview. Offering drawings (D-condition) had a positive effect, but presenting the problem as a performance task (P-condition) was more beneficial. Linear reasoning was nearly absent in the P-condition. Remarkably, at the post-test, most pupils from all three groups again applied linear strategies. Pupils' over-reliance on linearity seems partly elicited by the school-like word problem format of test items. Pupils perform much better if non-linear problems are offered as performance tasks. However, a single experience does not change performances on a comparable word problem test afterwards.

Properties of coupled-cluster equations originating in excitation sub-algebras

NASA Astrophysics Data System (ADS)

Kowalski, Karol

2018-03-01

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

Facilitating Case Reuse during Problem Solving in Algebra-Based Physics

ERIC Educational Resources Information Center

Mateycik, Frances Ann

2010-01-01

This research project investigates students' development of problem solving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problem solving strategies. Individual…

Error Analysis for Arithmetic Word Problems--A Case Study of Primary Three Students in One Singapore School

ERIC Educational Resources Information Center

Cheng, Lu Pien

2015-01-01

In this study, ways in which 9-year old students from one Singapore school solved 1-step and 2-step word problems based on the three semantic structures were examined. The students' work and diagrams provided insights into the range of errors in word problem solving for 1- step and 2-step word problems. In particular, the errors provided some…

Software Reviews.

ERIC Educational Resources Information Center

Miller, Anne, Ed.; Radziemski, Cathy, Ed.

1988-01-01

Three pieces of computer software are described and reviewed: HyperCard, to build and use varied applications; Iggy's Gnees, for problem solving with shapes in grades kindergarten-two; and Algebra Shop, for practicing skills and problem solving. (MNS)

The classical dynamic symmetry for the U(1) -Kepler problems

NASA Astrophysics Data System (ADS)

Bouarroudj, Sofiane; Meng, Guowu

2018-01-01

For the Jordan algebra of hermitian matrices of order n ≥ 2, we let X be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map πX: T∗ X → X with the canonical map X → CP n - 1 (i.e., the map that sends a given hermitian matrix to its column space), pulls back the Kähler form of the Fubini-Study metric on CP n - 1 to a real closed differential two-form ωK on T∗ X. Let ωX be the canonical symplectic form on T∗ X and μ a real number. A standard fact says that ωμ ≔ωX + 2 μωK turns T∗ X into a symplectic manifold, hence a Poisson manifold with Poisson bracket {,}μ. In this article we exhibit a Poisson realization of the simple real Lie algebra su(n , n) on the Poisson manifold (T∗ X ,{,}μ) , i.e., a Lie algebra homomorphism from su(n , n) to (C∞(T∗ X , R) ,{,}μ). Consequently one obtains the Laplace-Runge-Lenz vector for the classical U(1) -Kepler problem of level n and magnetic charge μ. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the U(1) -Kepler problems of level 2, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin (1971) on the classical dynamic symmetry for the MICZ-Kepler problems.

Communication Avoiding and Overlapping for Numerical Linear Algebra

DTIC Science & Technology

2012-05-08

future exascale systems, communication cost must be avoided or overlapped. Communication-avoiding 2.5D algorithms improve scalability by reducing...linear algebra problems to future exascale systems, communication cost must be avoided or overlapped. Communication-avoiding 2.5D algorithms improve...will continue to grow relative to the cost of computation. With exascale computing as the long-term goal, the community needs to develop techniques

An algebraic program for the states associated with the U(5) ⊃ O(5) ⊃ O(3) chain of groups

NASA Astrophysics Data System (ADS)

Yannouleas, C.; Pacheco, J. M.

1988-12-01

A REDUCE program is presented that calculates algebraically the γ-dependent part of the states associated with the U(5) ⊃ O(5) ⊃ O(3) chain of groups, familiar from nuclear-structure problems. The method of solution is a direct implementation of the analytic expressions given by Chacón and Moshinsky.

Computer Aided Instruction for a Course in Boolean Algebra and Logic Design. Final Report (Revised).

ERIC Educational Resources Information Center

Roy, Rob

The use of computers to prepare deficient college and graduate students for courses that build upon previously acquired information would solve the growing problem of professors who must spend up to one third of their class time in review of material. But examination of students who were taught Boolean Algebra and Logic Design by means of Computer…

Differences in the Processes of Solving Physics Problems between Good Physics Problem Solvers and Poor Physics Problem Solvers.

ERIC Educational Resources Information Center

Finegold, M.; Mass, R.

1985-01-01

Good problem solvers and poor problem solvers in advanced physics (N=8) were significantly different in their ability in translating, planning, and physical reasoning, as well as in problem solving time; no differences in reliance on algebraic solutions and checking problems were noted. Implications for physics teaching are discussed. (DH)

Word Problems: A "Meme" for Our Times.

ERIC Educational Resources Information Center

Leamnson, Robert N.

1996-01-01

Discusses a novel approach to word problems that involves linear relationships between variables. Argues that working stepwise through intermediates is the way our minds actually work and therefore this should be used in solving word problems. (JRH)

Child-Level Predictors of Responsiveness to Evidence-Based Mathematics Intervention.

PubMed

Powell, Sarah R; Cirino, Paul T; Malone, Amelia S

2017-07-01

We identified child-level predictors of responsiveness to 2 types of mathematics (calculation and word-problem) intervention among 2nd-grade children with mathematics difficulty. Participants were 250 children in 107 classrooms in 23 schools pretested on mathematics and general cognitive measures and posttested on mathematics measures. We assigned classrooms randomly assigned to calculation intervention, word-problem intervention, or business-as-usual control. Intervention lasted 17 weeks. Path analyses indicated that scores on working memory and language comprehension assessments moderated responsiveness to calculation intervention. No moderators were identified for responsiveness to word-problem intervention. Across both intervention groups and the control group, attentive behavior predicted both outcomes. Initial calculation skill predicted the calculation outcome, and initial language comprehension predicted word-problem outcomes. These results indicate that screening for calculation intervention should include a focus on working memory, language comprehension, attentive behavior, and calculations. Screening for word-problem intervention should focus on attentive behavior and word problems.

Boundaries for algebras of holomorphic functions on Marcinkiewicz sequence spaces

NASA Astrophysics Data System (ADS)

Choi, Yun Sung; Han, Kwang Hee

2006-11-01

Let be the Banach algebra of all complex-valued bounded continuous functions on the closed unit ball BE of a complex Banach space E and holomorphic in the interior of BE and let be the closed subalgebra of those functions which are uniformly continuous on BE. For the case whose bidual is a Marcinkiewicz sequence space Mw, we describe some sufficient conditions for a set to be a boundary of either or . Moreover, we consider some analogous problems on to those which were studied on the Gowers space Gp of characteristic p by Grados and Moraes [L.R. Grados, L.A. Moraes, Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003) 575-586; L.R. Grados, L.A. Moraes, Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (1) (2004) 231-242].

Mathematical modelling in engineering: an alternative way to teach Linear Algebra

NASA Astrophysics Data System (ADS)

Domínguez-García, S.; García-Planas, M. I.; Taberna, J.

2016-10-01

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic classroom approach in which students modelled real-world problems and turn gain a deeper knowledge of the Linear Algebra subject. Considering that most students are digital natives, we use the e-portfolio as a tool of communication between students and teachers, besides being a good place making the work visible. In this article, we present an overview of the design and implementation of a project-based learning for a Linear Algebra course taught during the 2014-2015 at the 'ETSEIB'of Universitat Politècnica de Catalunya (UPC).

The Performance of Chinese Primary School Students on Realistic Arithmetic Word Problems

ERIC Educational Resources Information Center

Xin, Ziqiang; Lin, Chongde; Zhang, Li; Yan, Rong

2007-01-01

Compared with standard arithmetic word problems demanding only the direct use of number operations and computations, realistic problems are harder to solve because children need to incorporate "real-world" knowledge into their solutions. Using the realistic word problem testing materials developed by Verschaffel, De Corte, and Lasure…

Using Technology to Meet the Developmental Needs of Deaf Students To Improve Their Mathematical Word Problem Solving Skills.

ERIC Educational Resources Information Center

Kelly, Ronald R.

2003-01-01

Presents "Project Solve," a web-based problem-solving instruction and guided practice for mathematical word problems. Discusses implications for college students for whom reading and comprehension of mathematical word problem solving are difficult, especially learning disabled students. (Author/KHR)

BFV-BRST analysis of the classical and quantum q-deformations of the sl(2) algebra

NASA Astrophysics Data System (ADS)

Dayi, O. F.

1994-01-01

BFV--BRST charge for q-deformed algebras is not unique. Different constructions of it in the classical as well as in the quantum phase space for the $q$-deformed algebra sl_q(2) are discussed. Moreover, deformation of the phase space without deforming the generators of sl(2) is considered. $\\hbar$-q-deformation of the phase space is shown to yield the Witten's second deformation. To study the BFV--BRST cohomology problem when both the quantum phase space and the group are deformed, a two parameter deformation of sl(2) is proposed, and its BFV-BRST charge is given.

Exact solution of some linear matrix equations using algebraic methods

NASA Technical Reports Server (NTRS)

Djaferis, T. E.; Mitter, S. K.

1977-01-01

A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.

Yang-Baxter maps, discrete integrable equations and quantum groups

NASA Astrophysics Data System (ADS)

Bazhanov, Vladimir V.; Sergeev, Sergey M.

2018-01-01

For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper we present detailed considerations of the above scheme on the example of the algebra Uq (sl (2)) leading to discrete Liouville equations, however the approach is rather general and can be applied to any quantized Lie algebra.

The Impossibility of "Real-Life" Word Problems (According to Bakhtin, Lacan, Zizek and Baudrillard)

ERIC Educational Resources Information Center

Gerofsky, Susan

2010-01-01

In recent years a great deal of work on mathematical word problems has focused on efforts to bring more of "real life" into the problems themselves and students' uptake of these problems. Following on from earlier studies of the word problem as a pedagogical and literary genre, the author argues that we cannot unproblematically assume an ability…

Improved Linear Algebra Methods for Redshift Computation from Limited Spectrum Data - II

NASA Technical Reports Server (NTRS)

Foster, Leslie; Waagen, Alex; Aijaz, Nabella; Hurley, Michael; Luis, Apolo; Rinsky, Joel; Satyavolu, Chandrika; Gazis, Paul; Srivastava, Ashok; Way, Michael

2008-01-01

Given photometric broadband measurements of a galaxy, Gaussian processes may be used with a training set to solve the regression problem of approximating the redshift of this galaxy. However, in practice solving the traditional Gaussian processes equation is too slow and requires too much memory. We employed several methods to avoid this difficulty using algebraic manipulation and low-rank approximation, and were able to quickly approximate the redshifts in our testing data within 17 percent of the known true values using limited computational resources. The accuracy of one method, the V Formulation, is comparable to the accuracy of the best methods currently used for this problem.

Closedness of orbits in a space with SU(2) Poisson structure

NASA Astrophysics Data System (ADS)

Fatollahi, Amir H.; Shariati, Ahmad; Khorrami, Mohammad

2014-06-01

The closedness of orbits of central forces is addressed in a three-dimensional space in which the Poisson bracket among the coordinates is that of the SU(2) Lie algebra. In particular it is shown that among problems with spherically symmetric potential energies, it is only the Kepler problem for which all bounded orbits are closed. In analogy with the case of the ordinary space, a conserved vector (apart from the angular momentum) is explicitly constructed, which is responsible for the orbits being closed. This is the analog of the Laplace-Runge-Lenz vector. The algebra of the constants of the motion is also worked out.

The hit problem for symmetric polynomials over the Steenrod algebra

NASA Astrophysics Data System (ADS)

Janfada, A. S.; Wood, R. M. W.

2002-09-01

We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [open face F]2[x1, ;…, xn] = [oplus B: plus sign in circle]d[gt-or-equal, slanted]0 Pd(n), viewed as a graded left module over the Steenrod algebra [script A] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn of grading 1. The present article investigates the hit problem for the [script A]-submodule of symmetric polynomials B(n) = P(n)[sum L: summation operator]n , where [sum L: summation operator]n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let [mu](d) denote the smallest value of k for which d = [sum L: summation operator]ki=1(2[lambda]i[minus sign]1), where [lambda]i [gt-or-equal, slanted] 0.

Closed, analytic, boson realizations for Sp(4)

NASA Astrophysics Data System (ADS)

Klein, Abraham; Zhang, Qing-Ying

1986-08-01

The problem of determing a boson realization for an arbitrary irrep of the unitary simplectic algebra Sp(2d) [or of the corresponding discrete unitary irreps of the unbounded algebra Sp(2d,R)] has been solved completely in recent papers by Deenen and Quesne [J. Deenen and C. Quesne, J. Math. Phys. 23, 878, 2004 (1982); 25, 1638 (1984); 26, 2705 (1985)] and by Moshinsky and co-workers [O. Castaños, E. Chacón, M. Moshinsky, and C. Quesne, J. Math. Phys. 26, 2107 (1985); M. Moshinsky, ``Boson realization of symplectic algebras,'' to be published]. This solution is not known in closed analytic form except for d=1 and for special classes of irreps for d>1. A different method of obtaining a boson realization that solves the full problem for Sp(4) is described. The method utilizes the chain Sp(2d)⊇SU(2)×SU(2) ×ṡṡṡ×SU(2) (d times), which, for d≥4, does not provide a complete set of quantum numbers. Though a simple solution of the missing label problem can be given, this solution does not help in the construction of a mapping algorithm for general d.

A Simple Algebraic Grid Adaptation Scheme with Applications to Two- and Three-dimensional Flow Problems

NASA Technical Reports Server (NTRS)

Hsu, Andrew T.; Lytle, John K.

1989-01-01

An algebraic adaptive grid scheme based on the concept of arc equidistribution is presented. The scheme locally adjusts the grid density based on gradients of selected flow variables from either finite difference or finite volume calculations. A user-prescribed grid stretching can be specified such that control of the grid spacing can be maintained in areas of known flowfield behavior. For example, the grid can be clustered near a wall for boundary layer resolution and made coarse near the outer boundary of an external flow. A grid smoothing technique is incorporated into the adaptive grid routine, which is found to be more robust and efficient than the weight function filtering technique employed by other researchers. Since the present algebraic scheme requires no iteration or solution of differential equations, the computer time needed for grid adaptation is trivial, making the scheme useful for three-dimensional flow problems. Applications to two- and three-dimensional flow problems show that a considerable improvement in flowfield resolution can be achieved by using the proposed adaptive grid scheme. Although the scheme was developed with steady flow in mind, it is a good candidate for unsteady flow computations because of its efficiency.

Procedural versus Content-Related Hints for Word Problem Solving: An Exploratory Study

ERIC Educational Resources Information Center

Kock, W. D.; Harskamp, E. G.

2016-01-01

For primary school students, mathematical word problems are often more difficult to solve than straightforward number problems. Word problems require reading and analysis skills, and in order to explain their situational contexts, the proper mathematical knowledge and number operations have to be selected. To improve students' ability in solving…

In Praise of Numerical Computation

NASA Astrophysics Data System (ADS)

Yap, Chee K.

Theoretical Computer Science has developed an almost exclusively discrete/algebraic persona. We have effectively shut ourselves off from half of the world of computing: a host of problems in Computational Science & Engineering (CS&E) are defined on the continuum, and, for them, the discrete viewpoint is inadequate. The computational techniques in such problems are well-known to numerical analysis and applied mathematics, but are rarely discussed in theoretical algorithms: iteration, subdivision and approximation. By various case studies, I will indicate how our discrete/algebraic view of computing has many shortcomings in CS&E. We want embrace the continuous/analytic view, but in a new synthesis with the discrete/algebraic view. I will suggest a pathway, by way of an exact numerical model of computation, that allows us to incorporate iteration and approximation into our algorithms’ design. Some recent results give a peek into how this view of algorithmic development might look like, and its distinctive form suggests the name “numerical computational geometry” for such activities.

On the stabilizability of multivariable systems by minimum order compensation

NASA Technical Reports Server (NTRS)

Byrnes, C. I.; Anderson, B. D. O.

1983-01-01

In this paper, a derivation is provided of the necessary condition, mp equal to or greater than n, for stabilizability by constant gain feedback of the generic degree n, p x m system. This follows from another of the main results, which asserts that generic stabilizability is equivalent to generic solvability of a deadbeat control problem, provided mp equal to or less than n. Taken together, these conclusions make it possible to make some sharp statements concerning minimum order stabilization. The techniques are primarily drawn from decision algebra and classical algebraic geometry and have additional consequences for problems of stabilizability and pole-assignability. Among these are the decidability (by a Sturm test) of the equivalence of generic pole-assignability and generic stabilizability, the semi-algebraic nature of the minimum order, q, of a stabilizing compensator, and the nonexistence of formulae involving rational operations and extraction of square roots for pole-assigning gains when they exist, answering in the negative a question raised by Anderson, Bose, and Jury (1975).

Problem Solving through Paper Folding

ERIC Educational Resources Information Center

Wares, Arsalan

2014-01-01

The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem-solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented…

The Matrix Pencil and its Applications to Speech Processing

DTIC Science & Technology

2007-03-01

Elementary Linear Algebra ” 8th edition, pp. 278, 2000 John Wiley & Sons, Inc., New York [37] Wai C. Chu, “Speech Coding Algorithms”, New Jeresy: John...Ben; Daniel, James W.; “Applied Linear Algebra ”, pp. 342-345, 1988 Prentice Hall, Englewood Cliffs, NJ [35] Haykin, Simon “Applied Linear Adaptive...ABSTRACT Matrix Pencils facilitate the study of differential equations resulting from oscillating systems. Certain problems in linear ordinary

Relational Algebra in Spatial Decision Support Systems Ontologies.

PubMed

Diomidous, Marianna; Chardalias, Kostis; Koutonias, Panagiotis; Magnita, Adrianna; Andrianopoulos, Charalampos; Zimeras, Stelios; Mechili, Enkeleint Aggelos

2017-01-01

Decision Support Systems (DSS) is a powerful tool, for facilitates researchers to choose the correct decision based on their final results. Especially in medical cases where doctors could use these systems, to overcome the problem with the clinical misunderstanding. Based on these systems, queries must be constructed based on the particular questions that doctors must answer. In this work, combination between questions and queries would be presented via relational algebra.

On deformation of complex continuum immersed in a plane space

NASA Astrophysics Data System (ADS)

Kovalev, V. A.; Murashkin, E. V.; Radayev, Y. N.

2018-05-01

The present paper is devoted to mathematical modelling of complex continua deformations considered as immersed in an external plane space. The complex continuum is defined as a differential manifold supplied with metrics induced by the external space. A systematic derivation of strain tensors by notion of isometric immersion of the complex continuum into a plane space of a higher dimension is proposed. Problem of establishing complete systems of irreducible objective strain and extrastrain tensors for complex continuum immersed in an external plane space is resolved. The solution to the problem is obtained by methods of the field theory and the theory of rational algebraic invariants. Strain tensors of the complex continuum are derived as irreducible algebraic invariants of contravariant vectors of the external space emerging as functional arguments in the complex continuum action density. Present analysis is restricted to rational algebraic invariants. Completeness of the considered systems of rational algebraic invariants is established for micropolar elastic continua. Rational syzygies for non-quadratic invariants are discussed. Objective strain tensors (indifferent to frame rotations in the external plane space) for micropolar continuum are alternatively obtained by properly combining multipliers of polar decompositions of deformation and extra-deformation gradients. The latter is realized only for continua immersed in a plane space of the equal mathematical dimension.

Articulation Management for Intelligent Integration of Information

NASA Technical Reports Server (NTRS)

Maluf, David A.; Tran, Peter B.; Clancy, Daniel (Technical Monitor)

2001-01-01

When combining data from distinct sources, there is a need to share meta-data and other knowledge about various source domains. Due to semantic inconsistencies and heterogeneity of representations, problems arise in combining multiple domains when the domains are merged. The knowledge that is irrelevant to the task of interoperation will be included, making the result unnecessarily complex. This heterogeneity problem can be eliminated by mediating the conflicts and managing the intersections of the domains. For interoperation and intelligent access to heterogeneous information, the focus is on the intersection of the knowledge, since intersection will define the required articulation rules. An algebra over domain has been proposed to use articulation rules to support disciplined manipulation of domain knowledge resources. The objective of a domain algebra is to provide the capability for interrogating many domain knowledge resources, which are largely semantically disjoint. The algebra supports formally the tasks of selecting, combining, extending, specializing, and modifying Components from a diverse set of domains. This paper presents a domain algebra and demonstrates the use of articulation rules to link declarative interfaces for Internet and enterprise applications. In particular, it discusses the articulation implementation as part of a production system capable of operating over the domain described by the IDL (interface description language) of objects registered in multiple CORBA servers.

Simple and Accurate Method for Central Spin Problems

NASA Astrophysics Data System (ADS)

Lindoy, Lachlan P.; Manolopoulos, David E.

2018-06-01

We describe a simple quantum mechanical method that can be used to obtain accurate numerical results over long timescales for the spin correlation tensor of an electron spin that is hyperfine coupled to a large number of nuclear spins. This method does not suffer from the statistical errors that accompany a Monte Carlo sampling of the exact eigenstates of the central spin Hamiltonian obtained from the algebraic Bethe ansatz, or from the growth of the truncation error with time in the time-dependent density matrix renormalization group (TDMRG) approach. As a result, it can be applied to larger central spin problems than the algebraic Bethe ansatz, and for longer times than the TDMRG algorithm. It is therefore an ideal method to use to solve central spin problems, and we expect that it will also prove useful for a variety of related problems that arise in a number of different research fields.

Attentional Cuing in Math Word Problems for Girls At-Risk for ADHD and Their Peers in General Education Settings

ERIC Educational Resources Information Center

Kercood, Suneeta; Zentall, Sydney S.; Vinh, Megan; Tom-Wright, Kinsey

2012-01-01

The purpose of this theoretically-based study was to examine the effects of yellow-highlighting "relevant" words and units within math word problems. Initial differences were documented between 10 girls at-risk for ADHD and 10 comparisons on the performance of group and individual assessments of math computations and word problems, as had…

The effects of using diagramming as a representational technique on high school students' achievement in solving math word problems

NASA Astrophysics Data System (ADS)

Banerjee, Banmali

Methods and procedures for successfully solving math word problems have been, and continue to be a mystery to many U.S. high school students. Previous studies suggest that the contextual and mathematical understanding of a word problem, along with the development of schemas and their related external representations, positively contribute to students' accomplishments when solving word problems. Some studies have examined the effects of diagramming on students' abilities to solve word problems that only involved basic arithmetic operations. Other studies have investigated how instructional models that used technology influenced students' problem solving achievements. Still other studies have used schema-based instruction involving students with learning disabilities. No study has evaluated regular high school students' achievements in solving standard math word problems using a diagramming technique without technological aid. This study evaluated students' achievement in solving math word problems using a diagramming technique. Using a quasi-experimental experimental pretest-posttest research design, quantitative data were collected from 172 grade 11 Hispanic English language learners (ELLS) and African American learners whose first language is English (EFLLs) in 18 classes at an inner city high school in Northern New Jersey. There were 88 control and 84 experimental students. The pretest and posttest of each participating student and samples of the experimental students' class assignments provided the qualitative data for the study. The data from this study exhibited that the diagramming method of solving math word problems significantly improved student achievement in the experimental group (p<.01) compared to the control group. The study demonstrated that urban, high school, ELLs benefited from instruction that placed emphasis on the mathematical vocabulary and symbols used in word problems and that both ELLs and EFLLs improved their problem solving success through careful attention to the creation and labeling of diagrams to represent the mathematics involved in standard word problems. Although Learnertype (ELL, EFLL), Classtype (Bilingual and Mixed), and Gender (Female, Male) were not significant indicators of student achievement, there was significant interaction between Treatment and Classtype at the level of the Bilingual students ( p<.01) and between Treatment and Learnertype at the level of the ELLs (p<.01).

A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

PubMed

Benhammouda, Brahim

2016-01-01

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

Selections from Kuang-Ming JIH-PAO (Source Span: 17 May - 26 June 1961), Number 8 Communist China.

DTIC Science & Technology

1961-08-31

to have a feeling of being unaccustomed to a certain new method, much like the feeling they have towards the use of phonetic symbols, Romanization or...seems to me that there is a certain unanimity among those who advocate the checking of Chinese words through phonetic sounds. The differences are...children’s mental development. We could not possibly ask children in kindergarten to learn algebra because natural maturity is also important. We have

Teaching Fifth Grade Mathematical Concepts: Effects of Word Problems Used with Traditional Methods.

ERIC Educational Resources Information Center

Coy, Jessica

The view of the researcher is that students in the upper elementary to middle school range need to increase their problem-solving skills by making logical deductions and organizing and structuring their thoughts through the use of word problems. Giving children a daily word problem challenged and introduced them to the lesson. This activity…

A Joint Probabilistic Classification Model of Relevant and Irrelevant Sentences in Mathematical Word Problems

ERIC Educational Resources Information Center

Cetintas, Suleyman; Si, Luo; Xin, Yan Ping; Zhang, Dake; Park, Joo Young; Tzur, Ron

2010-01-01

Estimating the difficulty level of math word problems is an important task for many educational applications. Identification of relevant and irrelevant sentences in math word problems is an important step for calculating the difficulty levels of such problems. This paper addresses a novel application of text categorization to identify two types of…

Cognition-emotion interactions: patterns of change and implications for math problem solving

PubMed Central

Trezise, Kelly; Reeve, Robert A.

2014-01-01

Surprisingly little is known about whether relationships between cognitive and emotional states remain stable or change over time, or how different patterns of stability and/or change in the relationships affect problem solving abilities. Nevertheless, cross-sectional studies show that anxiety/worry may reduce working memory (WM) resources, and the ability to minimize the effects anxiety/worry is higher in individuals with greater WM capacity. To investigate the patterns of stability and/or change in cognition-emotion relations over time and their implications for problem solving, 126 14-year-olds’ algebraic WM and worry levels were assessed twice in a single day before completing an algebraic math problem solving test. We used latent transition analysis to identify stability/change in cognition-emotion relations, which yielded a six subgroup solution. Subgroups varied in WM capacity, worry, and stability/change relationships. Among the subgroups, we identified a high WM/low worry subgroup that remained stable over time and a high WM/high worry, and a moderate WM/low worry subgroup that changed to low WM subgroups over time. Patterns of stability/change in subgroup membership predicted algebraic test results. The stable high WM/low worry subgroup performed best and the low WM capacity-high worry “unstable across time” subgroup performed worst. The findings highlight the importance of assessing variations in cognition-emotion relationships over time (rather than assessing cognition or emotion states alone) to account for differences in problem solving abilities. PMID:25132830

The relation between children’s constructive play activities, spatial ability, and mathematical word problem-solving performance: a mediation analysis in sixth-grade students

PubMed Central

Oostermeijer, Meike; Boonen, Anton J. H.; Jolles, Jelle

2014-01-01

The scientific literature shows that constructive play activities are positively related to children’s spatial ability. Likewise, a close positive relation is found between spatial ability and mathematical word problem-solving performances. The relation between children’s constructive play and their performance on mathematical word problems is, however, not reported yet. The aim of the present study was to investigate whether spatial ability acted as a mediator in the relation between constructive play and mathematical word problem-solving performance in 128 sixth-grade elementary school children. This mediating role of spatial ability was tested by utilizing the current mediation approaches suggested by Preacher and Hayes (2008). Results showed that 38.16% of the variance in mathematical word problem-solving performance is explained by children’s constructive play activities and spatial ability. More specifically, spatial ability acted as a partial mediator, explaining 31.58% of the relation between constructive play and mathematical word problem-solving performance. PMID:25101038

Curricular Reforms That Improve Students' Attitudes and Problem-Solving Performance

ERIC Educational Resources Information Center

Teodorescu, Raluca E.; Bennhold, Cornelius; Feldman, Gerald; Medsker, Larry

2014-01-01

We present the most recent steps undertaken to reform the introductory algebra-based course at The George Washington University. The reform sought to help students improve their problem-solving performance. Our pedagogy relies on didactic constructs such as the" GW-ACCESS problem-solving protocol," "instructional sequences" and…

Some Little Night Problems.

ERIC Educational Resources Information Center

Cannon, Lawrence O.; Elich, Joe

In most mathematics problem solving work, students' motivation comes from trying to please their teachers or to earn a good grade. The questions students must tackle are almost never generated by their own interest. Seven open-ended college algebra-level problems are presented in which the solution of one question suggests other related questions.…

Research in nonlinear structural and solid mechanics

NASA Technical Reports Server (NTRS)

Mccomb, H. G., Jr. (Compiler); Noor, A. K. (Compiler)

1980-01-01

Nonlinear analysis of building structures and numerical solution of nonlinear algebraic equations and Newton's method are discussed. Other topics include: nonlinear interaction problems; solution procedures for nonlinear problems; crash dynamics and advanced nonlinear applications; material characterization, contact problems, and inelastic response; and formulation aspects and special software for nonlinear analysis.

From geometry to algebra and vice versa: Realistic mathematics education principles for analyzing geometry tasks

NASA Astrophysics Data System (ADS)

Jupri, Al

2017-04-01

In this article we address how Realistic Mathematics Education (RME) principles, including the intertwinement and the reality principles, are used to analyze geometry tasks. To do so, we carried out three phases of a small-scale study. First we analyzed four geometry problems - considered as tasks inviting the use of problem solving and reasoning skills - theoretically in the light of the RME principles. Second, we tested two problems to 31 undergraduate students of mathematics education program and other two problems to 16 master students of primary mathematics education program. Finally, we analyzed student written work and compared these empirical to the theoretical results. We found that there are discrepancies between what we expected theoretically and what occurred empirically in terms of mathematization and of intertwinement of mathematical concepts from geometry to algebra and vice versa. We conclude that the RME principles provide a fruitful framework for analyzing geometry tasks that, for instance, are intended for assessing student problem solving and reasoning skills.

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

NASA Astrophysics Data System (ADS)

Assanova, A. T.; Imanchiyev, A. E.; Kadirbayeva, Zh. M.

2018-04-01

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge-Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.

NASCRIN - NUMERICAL ANALYSIS OF SCRAMJET INLET

NASA Technical Reports Server (NTRS)

Kumar, A.

1994-01-01

The NASCRIN program was developed for analyzing two-dimensional flow fields in supersonic combustion ramjet (scramjet) inlets. NASCRIN solves the two-dimensional Euler or Navier-Stokes equations in conservative form by an unsplit, explicit, two-step finite-difference method. A more recent explicit-implicit, two-step scheme has also been incorporated in the code for viscous flow analysis. An algebraic, two-layer eddy-viscosity model is used for the turbulent flow calculations. NASCRIN can analyze both inviscid and viscous flows with no struts, one strut, or multiple struts embedded in the flow field. NASCRIN can be used in a quasi-three-dimensional sense for some scramjet inlets under certain simplifying assumptions. Although developed for supersonic internal flow, NASCRIN may be adapted to a variety of other flow problems. In particular, it should be readily adaptable to subsonic inflow with supersonic outflow, supersonic inflow with subsonic outflow, or fully subsonic flow. The NASCRIN program is available for batch execution on the CDC CYBER 203. The vectorized FORTRAN version was developed in 1983. NASCRIN has a central memory requirement of approximately 300K words for a grid size of about 3,000 points.

Mathematical Giftedness, Problem Solving, and the Ability To Formulate Generalizations: The Problem-Solving Experiences of Four Gifted Students.

ERIC Educational Resources Information Center

Sriraman, Bharath

2003-01-01

Nine freshmen in a ninth-grade accelerated algebra class were asked to solve five nonroutine combinatorial problems. The four mathematically gifted students were successful in discovering and verbalizing the generality that characterized the solutions to the five problems, whereas the five nongifted students were unable to discover the hidden…

Neurons and the Process Standards

ERIC Educational Resources Information Center

Zambo, Ron; Zambo, Debby

2011-01-01

The classic Chickens and Pigs problem is considered to be an algebraic problem with two equations and two unknowns. In this article, the authors describe how third-grade teacher Maria is using it to develop a problem-based lesson because she is looking to her students' future needs. As Maria plans, she considers how a series of problems with the…

Linear and Quadratic Change: A Problem from Japan

ERIC Educational Resources Information Center

Peterson, Blake E.

2006-01-01

In the fall of 2003, the author conducted research on the student teaching process in Japan. The basis for most of the lessons observed was rich mathematics problems. Upon returning to the US, the author used one such problem while teaching an algebra 2 class. This article introduces that problem, which gives rise to both linear and quadratic…

Use of a Colony of Cooperating Agents and MAPLE To Solve the Traveling Salesman Problem.

ERIC Educational Resources Information Center

Guerrieri, Bruno

This paper reviews an approach for finding optimal solutions to the traveling salesman problem, a well-known problem in combinational optimization, and describes implementing the approach using the MAPLE computer algebra system. The method employed in this approach to the problem is similar to the way ant colonies manage to establish shortest…

Algebra of Majorana doubling.

PubMed

Lee, Jaehoon; Wilczek, Frank

2013-11-27

Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.

The Adidactic Interaction with the Procedures of Peers in the Transition from Arithmetic to Algebra: A "Milieu" for the Emergence of New Questions

ERIC Educational Resources Information Center

Sadovsky, Patricia; Sessa, Carmen

2005-01-01

The purpose of the present article is to give an account of the emergence of knowledge pertaining to the transition from arithmetic to algebra in the course of a debate in a grade 7 classroom. This debate follows two other instances of work: (1) the adidactic interaction between each student and a given problem, (2) the adidactic interaction of…

The Association between Mathematical Word Problems and Reading Comprehension

ERIC Educational Resources Information Center

Vilenius-Tuohimaa, Piia Maria; Aunola, Kaisa; Nurmi, Jari-Erik

2008-01-01

This study aimed to investigate the interplay between mathematical word problem skills and reading comprehension. The participants were 225 children aged 9-10 (Grade 4). The children's text comprehension and mathematical word problem-solving performance was tested. Technical reading skills were investigated in order to categorise participants as…

Using the Relational Paradigm: Effects on Pupils' Reasoning in Solving Additive Word Problems

ERIC Educational Resources Information Center

Polotskaia, Elena; Savard, Annie

2018-01-01

Pupils' difficulties in solving word problems continue to attract attention: while researchers highlight the importance of relational reasoning and modelling, school curricula typically use short word problems to develop pupils' knowledge of arithmetic operations and calculation strategies. The Relational Paradigm attributes the leading role in…

Bilingual College Writers' Collaborative Writing of Word Problems

ERIC Educational Resources Information Center

Esquinca, Alberto

2011-01-01

Numerous researchers have studied bilingual students' performance on word problems given that reading and writing these requires that they draw on linguistic and mathematical knowledge (Barwell, 2009a, 2009b). Some researchers have studied how bilinguals write word problems in the second language, but few have considered how bilinguals use their…

Cognitive Strategy Instruction for Teaching Word Problems to Primary-Level Struggling Students

ERIC Educational Resources Information Center

Pfannenstiel, Kathleen Hughes; Bryant, Diane Pedrotty; Bryant, Brian R.; Porterfield, Jennifer A.

2015-01-01

Students with mathematics difficulties and learning disabilities (LD) typically struggle with solving word problems. These students often lack knowledge about efficient, cognitive strategies to utilize when solving word problems. Cognitive strategy instruction has been shown to be effective in teaching struggling students how to solve word…

Mathematical Word Problem Solving Ability of Children with Autism Spectrum Disorder and their Typically Developing Peers.

PubMed

Bae, Young Seh; Chiang, Hsu-Min; Hickson, Linda

2015-07-01

This study examined the difference between children with autism spectrum disorders (ASD) and children with typical development (TD) in mathematical word problem solving ability and the factors associated with these children's word problem-solving ability. A total of 20 children with ASD and 20 children with TD participated in this study. Independent sample t tests and Spearman's rho correlations were used for data analysis. This study found: (a) Children with TD had higher word problem solving ability than did children with ASD; (b) Sentence comprehension, math vocabulary, computation, and everyday mathematical knowledge were associated with word problem solving ability of children with ASD and children with TD; and (c) Children with TD had higher everyday mathematical knowledge than did children with ASD.

The effect of problem structure on problem-solving: an fMRI study of word versus number problems.

PubMed

Newman, Sharlene D; Willoughby, Gregory; Pruce, Benjamin

2011-09-02

It has long been thought that word problems are more difficult to solve than number/equation problems. However, recent findings have begun to bring this broadly believed idea into question. The current study examined the processing differences between these two types of problems. The behavioral results presented here failed to show an overwhelming advantage for number problems. In fact, there were more errors for the number problems than the word problems. The neuroimaging results reported demonstrate that there is significant overlap in the processing of what, on the surface, appears to be completely different problems that elicit different problem-solving strategies. Word and number problems rely on a general network responsible for problem-solving that includes the superior posterior parietal cortex, the horizontal segment of the intraparietal sulcus which is hypothesized to be involved in problem representation and calculation as well as the regions that have been linked to executive aspects of working memory such as the pre-SMA and basal ganglia. While overlap was observed, significant differences were also found primarily in language processing regions such as Broca's and Wernicke's areas for the word problems and the horizontal segment of the intraparietal sulcus for the number problems. Copyright © 2011 Elsevier B.V. All rights reserved.

What Mathematical Competencies Are Needed for Success in College.

ERIC Educational Resources Information Center

Garofalo, Joe

1990-01-01

Identifies requisite math skills for a microeconomics course, offering samples of supply curves, demand curves, equilibrium prices, elasticity, and complex graph problems. Recommends developmental mathematics competencies, including problem solving, reasoning, connections, communication, number and operation sense, algebra, relationships,…

Inequalities, assessment and computer algebra

NASA Astrophysics Data System (ADS)

Sangwin, Christopher J.

2015-01-01

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students' answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.

On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions

NASA Astrophysics Data System (ADS)

Talib, Imran; Belgacem, Fethi Bin Muhammad; Asif, Naseer Ahmad; Khalil, Hammad

2017-01-01

In this research article, we derive and analyze an efficient spectral method based on the operational matrices of three dimensional orthogonal Jacobi polynomials to solve numerically the mixed partial derivatives type multi-terms high dimensions generalized class of fractional order partial differential equations. We transform the considered fractional order problem to an easily solvable algebraic equations with the aid of the operational matrices. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some test problems are considered to confirm the accuracy and validity of the proposed numerical method. The convergence of the method is ensured by comparing our Matlab software simulations based obtained results with the exact solutions in the literature, yielding negligible errors. Moreover, comparative results discussed in the literature are extended and improved in this study.

A Note on Substructuring Preconditioning for Nonconforming Finite Element Approximations of Second Order Elliptic Problems

NASA Technical Reports Server (NTRS)

Maliassov, Serguei

1996-01-01

In this paper an algebraic substructuring preconditioner is considered for nonconforming finite element approximations of second order elliptic problems in 3D domains with a piecewise constant diffusion coefficient. Using a substructuring idea and a block Gauss elimination, part of the unknowns is eliminated and the Schur complement obtained is preconditioned by a spectrally equivalent very sparse matrix. In the case of quasiuniform tetrahedral mesh an appropriate algebraic multigrid solver can be used to solve the problem with this matrix. Explicit estimates of condition numbers and implementation algorithms are established for the constructed preconditioner. It is shown that the condition number of the preconditioned matrix does not depend on either the mesh step size or the jump of the coefficient. Finally, numerical experiments are presented to illustrate the theory being developed.

Relativistic Causality and Quasi-Orthomodular Algebras

NASA Astrophysics Data System (ADS)

Nobili, Renato

2006-05-01

The concept of fractionability or decomposability in parts of a physical system has its mathematical counterpart in the lattice--theoretic concept of orthomodularity. Systems with a finite number of degrees of freedom can be decomposed in different ways, corresponding to different groupings of the degrees of freedom. The orthomodular structure of these simple systems is trivially manifest. The problem then arises as to whether the same property is shared by physical systems with an infinite number of degrees of freedom, in particular by the quantum relativistic ones. The latter case was approached several years ago by Haag and Schroer (1962; Haag, 1992) who started from noting that the causally complete sets of Minkowski spacetime form an orthomodular lattice and posed the question of whether the subalgebras of local observables, with topological supports on such subsets, form themselves a corresponding orthomodular lattice. Were it so, the way would be paved to interpreting spacetime as an intrinsic property of a local quantum field algebra. Surprisingly enough, however, the hoped property does not hold for local algebras of free fields with superselection rules. The possibility seems to be instead open if the local currents that govern the superselection rules are driven by gauge fields. Thus, in the framework of local quantum physics, the request for algebraic orthomodularity seems to imply physical interactions! Despite its charm, however, such a request appears plagued by ambiguities and criticities that make of it an ill--posed problem. The proposers themselves, indeed, concluded that the orthomodular correspondence hypothesis is too strong for having a chance of being practicable. Thus, neither the idea was taken seriously by the proposers nor further investigated by others up to a reasonable degree of clarification. This paper is an attempt to re--formulate and well--pose the problem. It will be shown that the idea is viable provided that the algebra of local observables: (1) is considered all over the whole range of its irreducible representations; (2) is widened with the addition of the elements of a suitable intertwining group of automorphisms; (3) the orthomodular correspondence requirement is modified to an extent sufficient to impart a natural topological structure to the intertwined algebra of observables so obtained. A novel scenario then emerges in which local quantum physics appears to provide a general framework for non--perturbative quantum field dynamics.

A Comparison of Two Types of Bank Investments

ERIC Educational Resources Information Center

Nillsen, Rodney

2017-01-01

In this paper, an investment problem is investigated in terms of elementary algebra, recurrence relations, functions, and calculus at high school level. The problem comes down to understanding the behaviour of a function associated with the problem and, in particular, to finding the zero of the function. A wider purpose is not only to formulate…

The Multiple Pendulum Problem via Maple[R

ERIC Educational Resources Information Center

Salisbury, K. L.; Knight, D. G.

2002-01-01

The way in which computer algebra systems, such as Maple, have made the study of physical problems of some considerable complexity accessible to mathematicians and scientists with modest computational skills is illustrated by solving the multiple pendulum problem. A solution is obtained for four pendulums with no restriction on the size of the…

Learning by Understanding: The Role of Multiple Representations in Learning Algebra.

ERIC Educational Resources Information Center

Brenner, Mary E.; Mayer, Richard E.; Moseley, Bryan; Brar, Theresa; Duran, Richard; Reed, Barbara Smith; Webb, David

1997-01-01

In posttest results, 76 prealgebra students who learned about functions in a unit emphasizing multiple formats, anchoring learning in a thematic context, and problem solving in cooperative groups were more successful at problem solving and problem representation than were 56 comparison students conventionally taught. Similar results were found for…

Word Problem Strategy for Latino English Language Learners at Risk for Math Disabilities

ERIC Educational Resources Information Center

Orosco, Michael J.

2014-01-01

"English Language Learners" (ELLs) at risk for "math disabilities" (MD) are challenged in solving word problems for numerous reasons such as (a) learning English as a second language, (b) limited experience using math vocabulary, and (c) lack of strategies to improve word-problem-solving skills. As a result of these…

Helping Students with Emotional and Behavioral Disorders Solve Mathematics Word Problems

ERIC Educational Resources Information Center

Alter, Peter

2012-01-01

The author presents a strategy for helping students with emotional and behavioral disorders become more proficient at solving math word problems. Math word problems require students to go beyond simple computation in mathematics (e.g., adding, subtracting, multiplying, and dividing) and use higher level reasoning that includes recognizing relevant…

Does Calculation or Word-Problem Instruction Provide a Stronger Route to Prealgebraic Knowledge?

ERIC Educational Resources Information Center

Fuchs, Lynn S.; Powell, Sarah R.; Cirino, Paul T.; Schumacher, Robin F.; Marrin, Sarah; Hamlett, Carol L.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.

2014-01-01

The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and prealgebraic knowledge. We extended the literature, which is dominated by correlational work, by examining whether intervention conducted on calculations or word problems contributes to improved performance in the other…

Language, Arithmetic Word Problems, and Deaf Students: Linguistic Strategies Used To Solve Tasks.

ERIC Educational Resources Information Center

Zevenbergen, Robyn; Hyde, Merv; Power, Des

2001-01-01

Examines the performance of deaf and hearing-impaired students in Queensland, Australia when solving arithmetic word problems. Subjects' solutions of word problems confirmed trends for learning students but their performance was delayed in comparison. Confirms other studies in which deaf and hearing-impaired students are delayed in their language…

Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development

ERIC Educational Resources Information Center

Bae, Young Seh

2013-01-01

Mathematical Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development Young Seh Bae This study investigated mathematical word problem solving and the factors associated with the solution paths adopted by two groups of participants (N=40), students with autism spectrum disorders (ASDs) and typically…

Examining How Students with Diverse Abilities Use Diagrams to Solve Mathematics Word Problems

ERIC Educational Resources Information Center

van Garderen, Delinda; Scheuermann, Amy; Jackson, Christa

2013-01-01

This study examined students' understanding of diagrams and their use of diagrams as tools to solve mathematical word problems. Students with learning disabilities (LD), typically achieving students, and gifted students in Grades 4 through 7 ("N" = 95) participated. Students were presented with novel mathematical word problem-solving…

Why Do Disadvantaged Filipino Children Find Word Problems in English Difficult?

ERIC Educational Resources Information Center

Bautista, Debbie; Mulligan, Joanne

2010-01-01

Young Filipino students are expected to solve mathematical word problems in English, a language that many encounter only in schools. Using individual interviews of 17 Filipino children, we investigated why word problems in English are difficult and the extent to which the language interferes with performance. Results indicate that children could…

Word Problem Solving: A Schema Approach in Year 3

ERIC Educational Resources Information Center

van Klinken, Eduarda

2012-01-01

This article outlines how a Brisbane independent school, Clayfield College, improved the ability of its Year 3 students to solve addition and subtraction word problems by utilising a schematic approach. It was observed that while students could read the words in the text of a written problem, many had difficulty identifying the core information…

Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay

ERIC Educational Resources Information Center

Polotskaia, Elena; Savard, Annie; Freiman, Viktor

2015-01-01

This essay proposes a reflection on the learning difficulties and teaching approaches associated with arithmetic word problem solving. We question the development of word problem solving skills in the early grades of elementary school. We are trying to revive the discussion because first, the knowledge in question--reversibility of arithmetic…

Effects of computer-based graphic organizers to solve one-step word problems for middle school students with mild intellectual disability: A preliminary study.

PubMed

Sheriff, Kelli A; Boon, Richard T

2014-08-01

The purpose of this study was to examine the effects of computer-based graphic organizers, using Kidspiration 3© software, to solve one-step word problems. Participants included three students with mild intellectual disability enrolled in a functional academic skills curriculum in a self-contained classroom. A multiple probe single-subject research design (Horner & Baer, 1978) was used to evaluate the effectiveness of computer-based graphic organizers to solving mathematical one-step word problems. During the baseline phase, the students completed a teacher-generated worksheet that consisted of nine functional word problems in a traditional format using a pencil, paper, and a calculator. In the intervention and maintenance phases, the students were instructed to complete the word problems using a computer-based graphic organizer. Results indicated that all three of the students improved in their ability to solve the one-step word problems using computer-based graphic organizers compared to traditional instructional practices. Limitations of the study and recommendations for future research directions are discussed. Copyright © 2014 Elsevier Ltd. All rights reserved.

SD-CAS: Spin Dynamics by Computer Algebra System.

PubMed

Filip, Xenia; Filip, Claudiu

2010-11-01

A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples. Copyright © 2010 Elsevier Inc. All rights reserved.

Flowing toward Correct Contributions during Group Problem Solving: A Statistical Discourse Analysis

ERIC Educational Resources Information Center

Chiu, Ming Ming

2008-01-01

Groups that created more correct ideas (correct contributions or CCs) might be more likely to solve a problem, and students' recent actions (micro-time context) might aid CC creation. 80 high school students worked in groups of 4 on an algebra problem. Groups with higher mathematics grades or more CCs were more likely to solve the problem. Dynamic…

USSR and Eastern Europe Scientific Abstracts, Electronics and Electrical Engineering, Number 33.

DTIC Science & Technology

1977-09-27

reduces to an infinite system of linear homogeneous algebraic equations and leads to Mathieu functions of the k-th order. The solution is convergent in...cylinder walls to be infinitesimally thin ideal conductors. The problem is reduced to a system of Fredholm linear algebraic equations of the second...EXPECTED DEVELOPMENTS OF TRANSISTORIZED LOW-NOISE MICROWAVE AMPLIFIERS Prague SDELOVACI TECHNIKA in Czech Vol 25, No 2, Feb 77 pp 47-49 TALLO, ANTON

Architecture studies and system demonstrations for optical parallel processor for AI and NI

NASA Astrophysics Data System (ADS)

Lee, Sing H.

1988-03-01

In solving deterministic AI problems the data search for matching the arguments of a PROLOG expression causes serious bottleneck when implemented sequentially by electronic systems. To overcome this bottleneck we have developed the concepts for an optical expert system based on matrix-algebraic formulation, which will be suitable for parallel optical implementation. The optical AI system based on matrix-algebraic formation will offer distinct advantages for parallel search, adult learning, etc.

The Problem-Solving Nemesis: Mindless Manipulation.

ERIC Educational Resources Information Center

Hawkins, Vincent J.

1987-01-01

Indicates that only 21% of respondents (secondary school math teachers) used computer-assisted instruction for tutorial work, physical models to interpret abstract concepts, or real-life application of the arithmetic or algebraic manipulation. Recommends that creative teaching methods be applied to problem solving. (NKA)

Geometry for Pie Lovers.

ERIC Educational Resources Information Center

Fisher, William

1982-01-01

An approach to the instruction of maxima and minima problems that works with tools of geometry and algebra is presented. The focus is on a classic pie-cutting problem, which is viewed as an interesting and instructive task that is an excellent application of transformation geometry. (MP)

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions [Algebraic multigrid preconditioners for multiphase flow in porous media with phase transitions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switchingmore » technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. In conclusion, we also show that the strategy is efficient and scales optimally with problem size.« less

Algebraic multigrid preconditioners for two-phase flow in porous media with phase transitions [Algebraic multigrid preconditioners for multiphase flow in porous media with phase transitions

DOE PAGES

Bui, Quan M.; Wang, Lu; Osei-Kuffuor, Daniel

2018-02-06

Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the partial differential equations modeling the flow. In the case of multiphase multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are taken into account. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem (NCP). Unlike in the primary variable switchingmore » technique, the set of primary variables in this approach is fixed even when there is phase transition. Not only does this improve the robustness of the nonlinear solver, it opens up the possibility to use multigrid methods to solve the resulting linear system. The disadvantage of the complementarity approach, however, is that when a phase disappears, the linear system has the structure of a saddle point problem and becomes indefinite, and current algebraic multigrid (AMG) algorithms cannot be applied directly. In this study, we explore the effectiveness of a new multilevel strategy, based on the multigrid reduction technique, to deal with problems of this type. We demonstrate the effectiveness of the method through numerical results for the case of two-phase, two-component flow with phase appearance/disappearance. In conclusion, we also show that the strategy is efficient and scales optimally with problem size.« less

The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

NASA Astrophysics Data System (ADS)

Zhang, Liangyin; Chen, Michael Z. Q.; Li, Chanying

2017-07-01

In this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.

The algebraic-hyperbolic approach to the linearized gravitational constraints on a Minkowski background

NASA Astrophysics Data System (ADS)

Winicour, Jeffrey

2017-08-01

An algebraic-hyperbolic method for solving the Hamiltonian and momentum constraints has recently been shown to be well posed for general nonlinear perturbations of the initial data for a Schwarzschild black hole. This is a new approach to solving the constraints of Einstein’s equations which does not involve elliptic equations and has potential importance for the construction of binary black hole data. In order to shed light on the underpinnings of this approach, we consider its application to obtain solutions of the constraints for linearized perturbations of Minkowski space. In that case, we find the surprising result that there are no suitable Cauchy hypersurfaces in Minkowski space for which the linearized algebraic-hyperbolic constraint problem is well posed.

An algebraic interpretation of PSP composition.

PubMed

Vaucher, G

1998-01-01

The introduction of time in artificial neurons is a delicate problem on which many groups are working. Our approach combines some properties of biological models and the algebraic properties of McCulloch and Pitts artificial neuron (AN) (McCulloch and Pitts, 1943) to produce a new model which links both characteristics. In this extended artificial neuron, postsynaptic potentials (PSPs) are considered as numerical elements, having two degrees of freedom, on which the neuron computes operations. Modelled in this manner, a group of neurons can be seen as a computer with an asynchronous architecture. To formalize the functioning of this computer, we propose an algebra of impulses. This approach might also be interesting in the modelling of the passive electrical properties in some biological neurons.

An Exploratory Study Contrasting High- and Low-Achieving Students' Percent Word Problem Solving

ERIC Educational Resources Information Center

Jitendra, Asha K.; Star, Jon R.

2012-01-01

This study evaluated whether schema-based instruction (SBI), a promising method for teaching students to represent and solve mathematical word problems, impacted the learning of percent word problems. Of particular interest was the extent that SBI improved high- and low-achieving students' learning and to a lesser degree on the indirect effect of…

Tense and Aspect in Word Problems about Motion: Diagram, Gesture, and the Felt Experience of Time

ERIC Educational Resources Information Center

de Freitas, Elizabeth; Zolkower, Betina

2015-01-01

Word problems about motion contain various conjugated verb forms. As students and teachers grapple with such word problems, they jointly operationalize diagrams, gestures, and language. Drawing on findings from a 3-year research project examining the social semiotics of classroom interaction, we show how teachers and students use gesture and…

Redefining the Whole: Common Errors in Elementary Preservice Teachers' Self-Authored Word Problems for Fraction Subtraction

ERIC Educational Resources Information Center

Dixon, Juli K.; Andreasen, Janet B.; Avila, Cheryl L.; Bawatneh, Zyad; Deichert, Deana L.; Howse, Tashana D.; Turner, Mercedes Sotillo

2014-01-01

A goal of this study was to examine elementary preservice teachers' (PSTs) ability to contextualize and decontextualize fraction subtraction by asking them to write word problems to represent fraction subtraction expressions and to choose prewritten word problems to support given fraction subtraction expressions. Three themes emerged from the…

Assessing the Effect of Language Demand in Bundles of Math Word Problems

ERIC Educational Resources Information Center

Banks, Kathleen; Jeddeeni, Ahmad; Walker, Cindy M.

2016-01-01

Differential bundle functioning (DBF) analyses were conducted to determine whether seventh and eighth grade second language learners (SLLs) had lower probabilities of answering bundles of math word problems correctly that had heavy language demands, when compared to non-SLLs of equal math proficiency. Math word problems on each of four test forms…

When Best Intentions Go Awry: The Failures of Concrete Representations to Help Solve Probability Word Problems

ERIC Educational Resources Information Center

Beitzel, Brian D.; Staley, Richard K.; DuBois, Nelson F.

2011-01-01

Previous research has cast doubt on the efficacy of utilizing external representations as an aid to solving word problems. The present study replicates previous findings that concrete representations hinder college students' ability to solve probability word problems, and extends those findings to apply to a multimedia instructional context. Our…

Word Frequency, Function Words and the Second Gavagai Problem

ERIC Educational Resources Information Center

Hochmann, Jean-Remy

2013-01-01

The classic gavagai problem exemplifies the difficulty to identify the referent of a novel word uttered in a foreign language. Here, we consider the reverse problem: identifying the referential part of a label. Assuming "gavagai" indicates a rabbit in a foreign language, it may very well mean ""a" rabbit" or ""that" rabbit". How can a learner know…

The Impact of Metacognitive Strategies and Self-Regulating Processes of Solving Math Word Problems

ERIC Educational Resources Information Center

Vula, Eda; Avdyli, Rrezarta; Berisha, Valbona; Saqipi, Blerim; Elezi, Shpetim

2017-01-01

This empirical study investigates the impact of metacognitive strategies and self-regulating processes in learners' achievement on solving math word problems. It specifically analyzes the impact of the linguistic factor and the number of steps and arithmetical operations that learners need to apply during the process of solving math word problems.…

The Motivation of Secondary School Students in Mathematical Word Problem Solving

ERIC Educational Resources Information Center

Gasco, Javier; Villarroel, Jose-Domingo

2014-01-01

Introduction: Motivation is an important factor in the learning of mathematics. Within this area of education, word problem solving is central in most mathematics curricula of Secondary School. The objective of this research is to detect the differences in motivation in terms of the strategies used to solve word problems. Method: It analyzed the…

Model Drawing Strategy for Fraction Word Problem Solving of Fourth-Grade Students with Learning Disabilities

ERIC Educational Resources Information Center

Sharp, Emily; Shih Dennis, Minyi

2017-01-01

This study used a multiple probe across participants design to examine the effects of a model drawing strategy (MDS) intervention package on fraction comparing and ordering word problem-solving performance of three Grade 4 students. MDS is a form of cognitive strategy instruction for teaching word problem solving that includes explicit instruction…

Young Filipino Students Making Sense of Arithmetic Word Problems in English

ERIC Educational Resources Information Center

Bautista, Debbie; Mulligan, Joanne; Mitchelmore, Michael

2009-01-01

Young Filipino children are expected to solve mathematical word problems in English, a task which they typically encounter only in schools. In this exploratory study, task-based interviews were conducted with seven Filipino children from a public school. The children were asked to read and solve addition and subtraction word problems in English or…

Additive Relations Word Problems in the South African Curriculum and Assessment Policy Standard at Foundation Phase

ERIC Educational Resources Information Center

Roberts, Nicky

2016-01-01

Drawing on a literature review of classifications developed by each of Riley, Verschaffel and Carpenter and their respective research groups, a refined typology of additive relations word problems is proposed and then used as analytical tool to classify the additive relations word problems in South African Curriculum and Assessment Policy Standard…

Two dissimilar approaches to dynamical systems on hyper MV -algebras and their information entropy

NASA Astrophysics Data System (ADS)

Mehrpooya, Adel; Ebrahimi, Mohammad; Davvaz, Bijan

2017-09-01

Measuring the flow of information that is related to the evolution of a system which is modeled by applying a mathematical structure is of capital significance for science and usually for mathematics itself. Regarding this fact, a major issue in concern with hyperstructures is their dynamics and the complexity of the varied possible dynamics that exist over them. Notably, the dynamics and uncertainty of hyper MV -algebras which are hyperstructures and extensions of a central tool in infinite-valued Lukasiewicz propositional calculus that models many valued logics are of primary concern. Tackling this problem, in this paper we focus on the subject of dynamical systems on hyper MV -algebras and their entropy. In this respect, we adopt two varied approaches. One is the set-based approach in which hyper MV -algebra dynamical systems are developed by employing set functions and set partitions. By the other method that is based on points and point partitions, we establish the concept of hyper injective dynamical systems on hyper MV -algebras. Next, we study the notion of entropy for both kinds of systems. Furthermore, we consider essential ergodic characteristics of those systems and their entropy. In particular, we introduce the concept of isomorphic hyper injective and hyper MV -algebra dynamical systems, and we demonstrate that isomorphic systems have the same entropy. We present a couple of theorems in order to help calculate entropy. In particular, we prove a contemporary version of addition and Kolmogorov-Sinai Theorems. Furthermore, we provide a comparison between the indispensable properties of hyper injective and semi-independent dynamical systems. Specifically, we present and prove theorems that draw comparisons between the entropies of such systems. Lastly, we discuss some possible relationships between the theories of hyper MV -algebra and MV -algebra dynamical systems.

Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*

DOE PAGES

Bank, R.; Falgout, R. D.; Jones, T.; ...

2015-10-29

In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less

Linear Equations. [Student Worksheets for Vocational Agricultural Courses].

ERIC Educational Resources Information Center

Jewell, Larry R.

This learning module provides students with practice in applying algebraic operations to vocational agriculture. The module consists of unit objectives, definitions, information, problems to solve, worksheets suitable for various levels of vocational agriculture instruction, and answer keys for the problems and worksheets. This module, which…

The Shad-Fack Transom

ERIC Educational Resources Information Center

Crannell, Annalisa

2011-01-01

We provide several constructions, both algebraic and geometric, for determining the ratio of the radii of two circles in an Apollonius-like packing problem. This problem was inspired by the art deco design in the transom window above the Shadek Fackenthal Library door on the Franklin & Marshall College campus.

The International Mathematical Olympiad Training Session.

ERIC Educational Resources Information Center

Rousseau, Cecil; Patruno, Gregg

1985-01-01

The Mathematical Olympiad Training Session is designed to give United States students a problem-oriented exposure to subject areas (algebra, geometry, number theory, combinatorics, and inequalities) through an intensive three-week course. Techniques used during the session, with three sample problems and their solutions, are presented. (JN)

A high-accuracy optical linear algebra processor for finite element applications

NASA Technical Reports Server (NTRS)

Casasent, D.; Taylor, B. K.

1984-01-01

Optical linear processors are computationally efficient computers for solving matrix-matrix and matrix-vector oriented problems. Optical system errors limit their dynamic range to 30-40 dB, which limits their accuray to 9-12 bits. Large problems, such as the finite element problem in structural mechanics (with tens or hundreds of thousands of variables) which can exploit the speed of optical processors, require the 32 bit accuracy obtainable from digital machines. To obtain this required 32 bit accuracy with an optical processor, the data can be digitally encoded, thereby reducing the dynamic range requirements of the optical system (i.e., decreasing the effect of optical errors on the data) while providing increased accuracy. This report describes a new digitally encoded optical linear algebra processor architecture for solving finite element and banded matrix-vector problems. A linear static plate bending case study is described which quantities the processor requirements. Multiplication by digital convolution is explained, and the digitally encoded optical processor architecture is advanced.

Anticipating students' reasoning and planning prompts in structured problem-solving lessons

NASA Astrophysics Data System (ADS)

Vale, Colleen; Widjaja, Wanty; Doig, Brian; Groves, Susie

2018-02-01

Structured problem-solving lessons are used to explore mathematical concepts such as pattern and relationships in early algebra, and regularly used in Japanese Lesson Study research lessons. However, enactment of structured problem-solving lessons which involves detailed planning, anticipation of student solutions and orchestration of whole-class discussion of solutions is an ongoing challenge for many teachers. Moreover, primary teachers have limited experience in teaching early algebra or mathematical reasoning actions such as generalising. In this study, the critical factors of enacting the structured problem-solving lessons used in Japanese Lesson Study to elicit and develop primary students' capacity to generalise are explored. Teachers from three primary schools participated in two Japanese Lesson Study teams for this study. The lesson plans and video recordings of teaching and post-lesson discussion of the two research lessons along with students' responses and learning are compared to identify critical factors. The anticipation of students' reasoning together with preparation of supporting and challenging prompts was critical for scaffolding students' capacity to grasp and communicate generality.

Cut and join operator ring in tensor models

NASA Astrophysics Data System (ADS)

Itoyama, H.; Mironov, A.; Morozov, A.

2018-07-01

Recent advancement of rainbow tensor models based on their superintegrability (manifesting itself as the existence of an explicit expression for a generic Gaussian correlator) has allowed us to bypass the long-standing problem seen as the lack of eigenvalue/determinant representation needed to establish the KP/Toda integrability. As the mandatory next step, we discuss in this paper how to provide an adequate designation to each of the connected gauge-invariant operators that form a double coset, which is required to cleverly formulate a tree-algebra generalization of the Virasoro constraints. This problem goes beyond the enumeration problem per se tied to the permutation group, forcing us to introduce a few gauge fixing procedures to the coset. We point out that the permutation-based labeling, which has proven to be relevant for the Gaussian averages is, via interesting complexity, related to the one based on the keystone trees, whose algebra will provide the tensor counterpart of the Virasoro algebra for matrix models. Moreover, our simple analysis reveals the existence of nontrivial kernels and co-kernels for the cut operation and for the join operation respectively that prevent a straightforward construction of the non-perturbative RG-complete partition function and the identification of truly independent time variables. We demonstrate these problems by the simplest non-trivial Aristotelian RGB model with one complex rank-3 tensor, studying its ring of gauge-invariant operators, generated by the keystone triple with the help of four operations: addition, multiplication, cut and join.

- XSUMMER- Transcendental functions and symbolic summation in FORM

NASA Astrophysics Data System (ADS)

Moch, S.; Uwer, P.

2006-05-01

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums, where the harmonic sums and their generalizations appear as building blocks, originating for example, from the expansion of generalized hypergeometric functions around integer values of the parameters. In this paper we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system FORM. Program summaryTitle of program:XSUMMER Catalogue identifier:ADXQ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXQ_v1_0 Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland License:GNU Public License and FORM License Computers:all Operating system:all Program language:FORM Memory required to execute:Depending on the complexity of the problem, recommended at least 64 MB RAM No. of lines in distributed program, including test data, etc.:9854 No. of bytes in distributed program, including test data, etc.:126 551 Distribution format:tar.gz Other programs called:none External files needed:none Nature of the physical problem:Systematic expansion of higher transcendental functions in a small parameter. The expansions arise in the calculation of loop integrals in perturbative quantum field theory. Method of solution:Algebraic manipulations of nested sums. Restrictions on complexity of the problem:Usually limited only by the available disk space. Typical running time:Dependent on the complexity of the problem.

Discrete Inverse and State Estimation Problems

NASA Astrophysics Data System (ADS)

Wunsch, Carl

2006-06-01

The problems of making inferences about the natural world from noisy observations and imperfect theories occur in almost all scientific disciplines. This book addresses these problems using examples taken from geophysical fluid dynamics. It focuses on discrete formulations, both static and time-varying, known variously as inverse, state estimation or data assimilation problems. Starting with fundamental algebraic and statistical ideas, the book guides the reader through a range of inference tools including the singular value decomposition, Gauss-Markov and minimum variance estimates, Kalman filters and related smoothers, and adjoint (Lagrange multiplier) methods. The final chapters discuss a variety of practical applications to geophysical flow problems. Discrete Inverse and State Estimation Problems is an ideal introduction to the topic for graduate students and researchers in oceanography, meteorology, climate dynamics, and geophysical fluid dynamics. It is also accessible to a wider scientific audience; the only prerequisite is an understanding of linear algebra. Provides a comprehensive introduction to discrete methods of inference from incomplete information Based upon 25 years of practical experience using real data and models Develops sequential and whole-domain analysis methods from simple least-squares Contains many examples and problems, and web-based support through MIT opencourseware

Computer-Based Feedback in Linear Algebra: Effects on Transfer Performance and Motivation

ERIC Educational Resources Information Center

Corbalan, Gemma; Paas, Fred; Cuypers, Hans

2010-01-01

Two studies investigated the effects on students' perceptions (Study 1) and learning and motivation (Study 2) of different levels of feedback in mathematical problems. In these problems, an error made in one step of the problem-solving procedure will carry over to the following steps and consequently to the final solution. Providing immediate…

Assessing the Impact of Representational and Contextual Problem Features on Student Use of Right-Hand Rules

ERIC Educational Resources Information Center

Kustusch, Mary Bridget

2016-01-01

Students in introductory physics struggle with vector algebra and these challenges are often associated with contextual and representational features of the problems. Performance on problems about cross product direction is particularly poor and some research suggests that this may be primarily due to misapplied right-hand rules. However, few…

Gender Differences in Solving Mathematics Problems among Two-Year College Students in a Developmental Algebra Class and Related Factors.

ERIC Educational Resources Information Center

Schonberger, Ann K.

A study was conducted at the University of Maine at Orono (UMO) to examine gender differences with respect to mathematical problem-solving ability, visual spatial ability, abstract reasoning ability, field independence/dependence, independent learning style, and developmental problem-solving ability (i.e., formal reasoning ability). Subjects…

Layout optimization with algebraic multigrid methods

NASA Technical Reports Server (NTRS)

Regler, Hans; Ruede, Ulrich

1993-01-01

Finding the optimal position for the individual cells (also called functional modules) on the chip surface is an important and difficult step in the design of integrated circuits. This paper deals with the problem of relative placement, that is the minimization of a quadratic functional with a large, sparse, positive definite system matrix. The basic optimization problem must be augmented by constraints to inhibit solutions where cells overlap. Besides classical iterative methods, based on conjugate gradients (CG), we show that algebraic multigrid methods (AMG) provide an interesting alternative. For moderately sized examples with about 10000 cells, AMG is already competitive with CG and is expected to be superior for larger problems. Besides the classical 'multiplicative' AMG algorithm where the levels are visited sequentially, we propose an 'additive' variant of AMG where levels may be treated in parallel and that is suitable as a preconditioner in the CG algorithm.

Time-dependent interaction between a two-level atom and a su(1,1) Lie algebra quantum system

NASA Astrophysics Data System (ADS)

Abdalla, M. Sebaweh; Khalil, E. M.; Obada, A.-S. F.

2017-06-01

The problem of the interaction between a two-level atom and a two-mode field in the parametric amplifier-type is considered. A similar problem appears in an ion trapped in a two-dimensional trap. The problem is transformed into an interaction governed by su(1,1) Lie algebraic operators with phase and coupling parameter depending on time. Under an integrability condition, that relates phase and coupling, a solution to the wavefunction is obtained using the Schrödinger equation. The effects of the functional dependence of the coupling and the initial state of the two-level atom on atomic inversion, the degree of entanglement, the fidelity and the Glauber second-order correlation function are investigated. It is shown that the acceleration term plays an important role in controlling the function behavior of the considered quantities.

An algorithm for analytical solution of basic problems featuring elastostatic bodies with cavities and surface flaws

NASA Astrophysics Data System (ADS)

Penkov, V. B.; Levina, L. V.; Novikova, O. S.; Shulmin, A. S.

2018-03-01

Herein we propose a methodology for structuring a full parametric analytical solution to problems featuring elastostatic media based on state-of-the-art computing facilities that support computerized algebra. The methodology includes: direct and reverse application of P-Theorem; methods of accounting for physical properties of media; accounting for variable geometrical parameters of bodies, parameters of boundary states, independent parameters of volume forces, and remote stress factors. An efficient tool to address the task is the sustainable method of boundary states originally designed for the purposes of computerized algebra and based on the isomorphism of Hilbertian spaces of internal states and boundary states of bodies. We performed full parametric solutions of basic problems featuring a ball with a nonconcentric spherical cavity, a ball with a near-surface flaw, and an unlimited medium with two spherical cavities.

Numerical stability in problems of linear algebra.

NASA Technical Reports Server (NTRS)

Babuska, I.

1972-01-01

Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.

Study-simulation of space station dynamics

NASA Technical Reports Server (NTRS)

Gaitens, M. J.

1971-01-01

Matrix algebra translator and executor /MATE/ takes equations describing structural control system environmental interaction problem for flexible spacecraft components and loads them into self programming computer.

Teaching materials of algebraic equation

NASA Astrophysics Data System (ADS)

Widodo, S. A.; Prahmana, R. C. I.; Purnami, A. S.; Turmudi

2017-12-01

The purpose of this paper is to know the effectiveness of teaching materials algebraic equation. This type of research used experimental method. The population in this study is all students of mathematics education who take numerical method in sarjanawiyata tamansiswa of university; the sample is taken using cluster random sampling. Instrument used in this research is test and questionnaire. The test is used to know the problem solving ability and achievement, while the questionnaire is used to know the student's response on the teaching materials. Data Analysis technique of quantitative used Wilcoxon test, while the qualitative data used grounded theory. Based on the results of the test can be concluded that the development of teaching materials can improve the ability to solve problems and achievement.

Verification of Algebra Step Problems: A Chronometric Study of Human Problem Solving. Technical Report No. 253. Psychology and Education Series.

ERIC Educational Resources Information Center

Matthews, Paul G.; Atkinson, Richard C.

This paper reports an experiment designed to test theoretical relations among fast problem solving, more complex and slower problem solving, and research concerning fundamental memory processes. Using a cathode ray tube, subjects were presented with propositions of the form "Y is in list X" which they memorized. In later testing they were asked to…

The Normals to a Parabola and the Real Roots of a Cubic

ERIC Educational Resources Information Center

Bains, Majinder S.; Thoo, J. B.

2007-01-01

The geometric problem of finding the number of normals to the parabola y = x[squared] through a given point is equivalent to the algebraic problem of finding the number of distinct real roots of a cubic equation. Apollonius solved the former problem, and Cardano gave a solution to the latter. The two problems are bridged by Neil's (semi-cubical)…

Effect of the Presence of External Representations on Accuracy and Reaction Time in Solving Mathematical Double-Choice Problems by Students of Different Levels of Instruction

ERIC Educational Resources Information Center

Leikin, Roza; Leikin, Mark; Waisman, Ilana; Shaul, Shelley

2013-01-01

This study explores the effects of the "presence of external representations of a mathematical object" (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric…

The Kadison–Singer Problem in mathematics and engineering

PubMed Central

Casazza, Peter G.; Tremain, Janet Crandell

2006-01-01

We will see that the famous intractible 1959 Kadison–Singer Problem in C*-algebras is equivalent to fundamental open problems in a dozen different areas of research in mathematics and engineering. This work gives all these areas common ground on which to interact as well as explaining why each area has volumes of literature on their respective problems without a satisfactory resolution. PMID:16461465

Security analysis of boolean algebra based on Zhang-Wang digital signature scheme

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zheng, Jinbin, E-mail: jbzheng518@163.com

2014-10-06

In 2005, Zhang and Wang proposed an improvement signature scheme without using one-way hash function and message redundancy. In this paper, we show that this scheme exits potential safety concerns through the analysis of boolean algebra, such as bitwise exclusive-or, and point out that mapping is not one to one between assembly instructions and machine code actually by means of the analysis of the result of the assembly program segment, and which possibly causes safety problems unknown to the software.

A Comparative Analysis of Word Problems in Selected United States and Russian First Grade Textbooks

ERIC Educational Resources Information Center

Grishchenko, Svetlana

2009-01-01

The purpose of this study was to explore word problems as a subject matter in mathematics textbook curricula. The motivation for the study derived from the following evidence: (a) American students find some word problems are more difficult than others (Garcia, Jimenez, & Hess, 2006; Riley & Green, 1988; Stern, 2001), and (b) one of the…

Mathematical Word Problem Solving Ability of Children with Autism Spectrum Disorder and Their Typically Developing Peers

ERIC Educational Resources Information Center

Bae, Young Seh; Chiang, Hsu-Min; Hickson, Linda

2015-01-01

This study examined the difference between children with autism spectrum disorders (ASD) and children with typical development (TD) in mathematical word problem solving ability and the factors associated with these children's word problem-solving ability. A total of 20 children with ASD and 20 children with TD participated in this study.…

The Effects of Using Diagramming as a Representational Technique on High School Students' Achievement in Solving Math Word Problems

ERIC Educational Resources Information Center

Banerjee, Banmali

2010-01-01

Methods and procedures for successfully solving math word problems have been, and continue to be a mystery to many U.S. high school students. Previous studies suggest that the contextual and mathematical understanding of a word problem, along with the development of schemas and their related external representations, positively contribute to…

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

NASA Astrophysics Data System (ADS)

Plestenjak, Bor; Gheorghiu, Călin I.; Hochstenbach, Michiel E.

2015-10-01

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

The Baker-Akhiezer Function and Factorization of the Chebotarev-Khrapkov Matrix

NASA Astrophysics Data System (ADS)

Antipov, Yuri A.

2014-10-01

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient {G(t) = α1(t)I + α2(t)Q(t)} , {α1(t), α2(t) in H(L)} , I = diag{1, 1}, Q(t) is a {2×2} zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires the finding of the {ρ} zeros of the Baker-Akhiezer function ({ρ} is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-{ρ} polynomial and solution of a certain linear algebraic system of {ρ} equations.

An Algorithm for Interactive Modeling of Space-Transportation Engine Simulations: A Constraint Satisfaction Approach

NASA Technical Reports Server (NTRS)

Mitra, Debasis; Thomas, Ajai; Hemminger, Joseph; Sakowski, Barbara

2001-01-01

In this research we have developed an algorithm for the purpose of constraint processing by utilizing relational algebraic operators. Van Beek and others have investigated in the past this type of constraint processing from within a relational algebraic framework, producing some unique results. Apart from providing new theoretical angles, this approach also gives the opportunity to use the existing efficient implementations of relational database management systems as the underlying data structures for any relevant algorithm. Our algorithm here enhances that framework. The algorithm is quite general in its current form. Weak heuristics (like forward checking) developed within the Constraint-satisfaction problem (CSP) area could be also plugged easily within this algorithm for further enhancements of efficiency. The algorithm as developed here is targeted toward a component-oriented modeling problem that we are currently working on, namely, the problem of interactive modeling for batch-simulation of engineering systems (IMBSES). However, it could be adopted for many other CSP problems as well. The research addresses the algorithm and many aspects of the problem IMBSES that we are currently handling.

The Effects of Using Drawings in Developing Young Children's Mathematical Word Problem Solving: A Design Experiment with Third-Grade Hungarian Students

ERIC Educational Resources Information Center

Csikos, Csaba; Szitanyi, Judit; Kelemen, Rita

2012-01-01

The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students' knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment…

Coping Strategies Applied to Comprehend Multistep Arithmetic Word Problems by Students with Above-Average Numeracy Skills and Below-Average Reading Skills

ERIC Educational Resources Information Center

Nortvedt, Guri A.

2011-01-01

This article discusses how 13-year-old students with above-average numeracy skills and below-average reading skills cope with comprehending word problems. Compared to other students who are proficient in numeracy and are skilled readers, these students are more disadvantaged when solving single-step and multistep arithmetic word problems. The…

A process for reaching standardization of word processing software for Sandia National Laboratories (Albuquerque) secretaries

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hudson, S.R.

1989-04-01

In the summer of 1986, a number of problems being experienced by Sandia secretaries due to multiple word processing packages being used were brought to the attention of Sandia's upper management. This report discusses how these problems evolved, how management chose to correct the problem, and how standardization of word processing for Sandia secretaries was achieved. 11 refs.

The Effects of Differing Presentations of Mathematical Word Problems Upon the Achievement of Tenth Grade Students.

ERIC Educational Resources Information Center

Sherrill, James M.

Described is a study concerned with the mode of presentation of printed mathematical word problems. Tenth grade students were given twenty word problems to solve, presented in one of three ways: (1) prose only, (2) prose with an accurate picture included, or (3) prose with a distorted picture. Experimental results showed that the group with an…

Research and Implementation of Tibetan Word Segmentation Based on Syllable Methods

NASA Astrophysics Data System (ADS)

Jiang, Jing; Li, Yachao; Jiang, Tao; Yu, Hongzhi

2018-03-01

Tibetan word segmentation (TWS) is an important problem in Tibetan information processing, while abbreviated word recognition is one of the key and most difficult problems in TWS. Most of the existing methods of Tibetan abbreviated word recognition are rule-based approaches, which need vocabulary support. In this paper, we propose a method based on sequence tagging model for abbreviated word recognition, and then implement in TWS systems with sequence labeling models. The experimental results show that our abbreviated word recognition method is fast and effective and can be combined easily with the segmentation model. This significantly increases the effect of the Tibetan word segmentation.

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Kreutz, K.; Jain, A.

1989-01-01

A spatial operator algebra for modeling the control and trajectory design of manipulation is discussed, with emphasis on its analytical formulation and implementation in the Ada programming language. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of the manipulator. Inversion is obtained using techniques of recursive filtering and smoothing. The operator alegbra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. Implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection, thus greatly simplifying the transition from an abstract problem formulation and solution to the detailed mechanization of a specific algorithm.

Matrix with Prescribed Eigenvectors

ERIC Educational Resources Information Center

Ahmad, Faiz

2011-01-01

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

Connected Mathematics Project (CMP). What Works Clearinghouse Intervention Report. Updated

ERIC Educational Resources Information Center

What Works Clearinghouse, 2017

2017-01-01

"Connected Mathematics Project" (CMP) is a math curriculum for students in grades 6-8. It uses interactive problems and everyday situations to explore mathematical ideas, with a goal of fostering a problem-centered, inquiry-based learning environment. At each grade level, the curriculum covers numbers, algebra, geometry/measurement,…

Quadratic Expressions by Means of "Summing All the Matchsticks"

ERIC Educational Resources Information Center

Gierdien, M. Faaiz

2012-01-01

This note presents demonstrations of quadratic expressions that come about when particular problems are posed with respect to matchsticks that form regular triangles, squares, pentagons and so on. Usually when such "matchstick" problems are used as ways to foster algebraic thinking, the expressions for the number of matchstick quantities are…

A Problem-Centered Approach to Canonical Matrix Forms

ERIC Educational Resources Information Center

Sylvestre, Jeremy

2014-01-01

This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…

Integrating Study Skills and Problem Solving into Remedial Mathematics

ERIC Educational Resources Information Center

Cornick, Jonathan; Guy, G. Michael; Beckford, Ian

2015-01-01

Students at a large urban community college enrolled in seven classes of an experimental remedial algebra programme, which integrated study skills instruction and collaborative problem solving. A control group of seven classes was taught in a traditional lecture format without study skills instruction. Student performance in the course was…

Reversible Reasoning and the Working Backwards Problem Solving Strategy

ERIC Educational Resources Information Center

Ramful, Ajay

2015-01-01

Making sense of mathematical concepts and solving mathematical problems may demand different forms of reasoning. These could be either domain-based, such as algebraic, geometric or statistical reasoning, while others are more general such as inductive/deductive reasoning. This article aims at giving visibility to a particular form of reasoning…

Problems for judgment and decision making.

PubMed

Hastie, R

2001-01-01

This review examines recent developments during the past 5 years in the field of judgment and decision making, written in the form of a list of 16 research problems. Many of the problems involve natural extensions of traditional, originally rational, theories of decision making. Others are derived from descriptive algebraic modeling approaches or from recent developments in cognitive psychology and cognitive neuroscience.

Non-algebraic integrability of the Chew-Low reversible dynamical system of the Cremona type and the relation with the 7th Hilbert problem (non-resonant case)

NASA Astrophysics Data System (ADS)

Rerikh, K. V.

A smooth reversible dynamical system (SRDS) and a system of nonlinear functional equations, defined by a certain rational quadratic Cremona mapping and arising from the static model of the dispersion approach in the theory of strong interactions (the Chew-Low equations for p- wave πN- scattering) are considered. This SRDS is splitted into 1- and 2-dimensional ones. An explicit Cremona transformation that completely determines the exact solution of the two-dimensional system is found. This solution depends on an odd function satisfying a nonlinear autonomous 3-point functional equation. Non-algebraic integrability of SRDS under consideration is proved using the method of Poincaré normal forms and the Siegel theorem on biholomorphic linearization of a mapping at a non-resonant fixed point. The proof is based on the classical Feldman-Baker theorem on linear forms of logarithms of algebraic numbers, which, in turn, relies upon solving the 7th Hilbert problem by A.I. Gel'fond and T. Schneider and new powerful methods of A. Baker in the theory of transcendental numbers. The general theorem, following from the Feldman-Baker theorem, on applicability of the Siegel theorem to the set of the eigenvalues λ ɛ Cn of a mapping at a non-resonant fixed point which belong to the algebraic number field A is formulated and proved. The main results are presented in Theorems 1-3, 5, 7, 8 and Remarks 3, 7.

Analysis of space telescope data collection system

NASA Technical Reports Server (NTRS)

Ingels, F. M.; Schoggen, W. O.

1982-01-01

An analysis of the expected performance for the Multiple Access (MA) system is provided. The analysis covers the expected bit error rate performance, the effects of synchronization loss, the problem of self-interference, and the problem of phase ambiguity. The problem of false acceptance of a command word due to data inversion is discussed. A mathematical determination of the probability of accepting an erroneous command word due to a data inversion is presented. The problem is examined for three cases: (1) a data inversion only, (2) a data inversion and a random error within the same command word, and a block (up to 256 48-bit words) containing both a data inversion and a random error.

Working memory components that predict word problem solving: Is it merely a function of reading, calculation, and fluid intelligence?

PubMed

Fung, Wenson; Swanson, H Lee

2017-07-01

The purpose of this study was to assess whether the differential effects of working memory (WM) components (the central executive, phonological loop, and visual-spatial sketchpad) on math word problem-solving accuracy in children (N = 413, ages 6-10) are completely mediated by reading, calculation, and fluid intelligence. The results indicated that all three WM components predicted word problem solving in the nonmediated model, but only the storage component of WM yielded a significant direct path to word problem-solving accuracy in the fully mediated model. Fluid intelligence was found to moderate the relationship between WM and word problem solving, whereas reading, calculation, and related skills (naming speed, domain-specific knowledge) completely mediated the influence of the executive system on problem-solving accuracy. Our results are consistent with findings suggesting that storage eliminates the predictive contribution of executive WM to various measures Colom, Rebollo, Abad, & Shih (Memory & Cognition, 34: 158-171, 2006). The findings suggest that the storage component of WM, rather than the executive component, has a direct path to higher-order processing in children.

Characteristics of Students at Risk for Mathematics Difficulties Predicting Arithmetic Word Problem Solving Performance: The Role of Attention, Behavior, and Reading

ERIC Educational Resources Information Center

Jitendra, Asha K.; Corroy, Kelly Cozine; Dupuis, Danielle N.

2013-01-01

The purposes of this study were (a) to evaluate differences in arithmetic word problem solving between high and low at-risk students for mathematics difficulties (MD) and (b) to assess the influence of attention, behavior, reading, and socio-economic status (SES) in predicting the word problem solving performance of third-grade students with MD.…

Pedagogical content knowledge: Knowledge of pedagogy novice teachers in mathematics learning on limit algebraic function

NASA Astrophysics Data System (ADS)

Ma'rufi, Budayasa, I. Ketut; Juniati, Dwi

2017-02-01

Teacher is one of the key aspects of student's achievement. Teachers should master content material taught, how to teach it, and can interpret the students' thinking so that students easily understand the subject matter. This research was a qualitative research that aimed at describing profile of PCK's teachers in mathematics on limit algebraic functions in terms of the differences of teaching experience. Pedagogical Content Knowledge (PCK) and understanding of teachers is defined as involving the relationship between knowledge of teaching materials, how to transfer the subject matter, and the knowledge of students in mathematics on limit algebraic functions that the subject matter may be understood by students. The PCK components in this research were knowledge of subject matter, knowledge of pedagogy, and knowledge of students. Knowledge of pedagogy defines as knowledge and understanding of teachers about the planning and organization of the learning and teaching strategy of limit algebraic function. The subjects were two mathematics high school teachers who teach in class XI IPS. Data were collected through observation of learning during five meetings and interviews before and after the lesson continued with qualitative data analysis. Focus of this article was to describe novice teacher's knowledge of student in mathematics learning on limit algebraic function. Based on the results of the analysis of qualitative data the data concluded that novice teacher's knowledge of pedagogy in mathematics on limit algebraic function showed: 1) in teaching the definitions tend to identify prior knowledge of the student experience with the material to be studied, but not in the form of a problem, 2) in posing the questions tend to be monotonous non lead and dig, 3) in response to student questions preservice teachers do not take advantage of the characteristics or the potential of other students, 4) in addressing the problem of students, tend to use the drill approach and did not give illustrations easily to understand by students, 5) in teaching application concepts, tend to explain procedurally, without explaining the reasons why these steps are carried out, 6) less varied in the use of learning strategies.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hart, William; Laird, Carl; Siirola, John

Pyomo provides a rich software environment for formulating and analyzing optimization applications. Pyomo supports the algebraic specification of complex sets of objectives and constraints, which enables optimization solvers to exploit problem structure to efficiently perform optimization.

Numerical algebraic geometry for model selection and its application to the life sciences

PubMed Central

Gross, Elizabeth; Davis, Brent; Ho, Kenneth L.; Bates, Daniel J.

2016-01-01

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology. PMID:27733697

A look at scalable dense linear algebra libraries

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dongarra, J.J.; Van de Geijn, R.A.; Walker, D.W.

1992-01-01

We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization aremore » presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 GFLOPS (double precision) for the largest problem considered.« less

A look at scalable dense linear algebra libraries

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dongarra, J.J.; Van de Geijn, R.A.; Walker, D.W.

1992-08-01

We discuss the essential design features of a library of scalable software for performing dense linear algebra computations on distributed memory concurrent computers. The square block scattered decomposition is proposed as a flexible and general-purpose way of decomposing most, if not all, dense matrix problems. An object- oriented interface to the library permits more portable applications to be written, and is easy to learn and use, since details of the parallel implementation are hidden from the user. Experiments on the Intel Touchstone Delta system with a prototype code that uses the square block scattered decomposition to perform LU factorization aremore » presented and analyzed. It was found that the code was both scalable and efficient, performing at about 14 GFLOPS (double precision) for the largest problem considered.« less

Consistency of a counterexample to Naimark's problem

PubMed Central

Akemann, Charles; Weaver, Nik

2004-01-01

We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assuming that these axioms are consistent). We prove that the statement “there exists a counterexample to Naimark's problem which is generated by \\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}{\\aleph}_{1}\\end{equation*}\\end{document} elements” is undecidable in standard set theory. PMID:15131270

Automatic Item Generation of Probability Word Problems

ERIC Educational Resources Information Center

Holling, Heinz; Bertling, Jonas P.; Zeuch, Nina

2009-01-01

Mathematical word problems represent a common item format for assessing student competencies. Automatic item generation (AIG) is an effective way of constructing many items with predictable difficulties, based on a set of predefined task parameters. The current study presents a framework for the automatic generation of probability word problems…

Structuring students’ analogical reasoning in solving algebra problem

NASA Astrophysics Data System (ADS)

Lailiyah, S.; Nusantara, T.; Sa'dijah, C.; Irawan, E. B.; Kusaeri; Asyhar, A. H.

2018-01-01

The average achievement of Indonesian students’ mathematics skills according to Benchmark International Trends in Mathematics and Science Study (TIMSS) is ranked at the 38th out of 42 countries and according to the survey result in Program for International Student Assessment (PISA) is ranked at the 64th out of 65 countries. The low mathematics skill of Indonesian student has become an important reason to research more deeply about reasoning and algebra in mathematics. Analogical reasoning is a very important component in mathematics because it is the key to creativity and it can make the learning process in the classroom become effective. The major part of the analogical reasoning is about structuring including the processes of inferencing and decision-making happens. Those processes involve base domain and target domain. Methodologically, the subjects of this research were 42 students from class XII. The sources of data were derived from the results of thinks aloud, the transcribed interviews, and the videos taken while the subject working on the instruments and interviews. The collected data were analyzed using qualitative techniques. The result of this study described the structuring characteristics of students’ analogical reasoning in solving algebra problems from all the research subjects.

Dual number algebra method for Green's function derivatives in 3D magneto-electro-elasticity

NASA Astrophysics Data System (ADS)

Dziatkiewicz, Grzegorz

2018-01-01

The Green functions are the basic elements of the boundary element method. To obtain the boundary integral formulation the Green function and its derivative should be known for the considered differential operator. Today the interesting group of materials are electronic composites. The special case of the electronic composite is the magnetoelectroelastic continuum. The mentioned continuum is a model of the piezoelectric-piezomagnetic composites. The anisotropy of their physical properties makes the problem of Green's function determination very difficult. For that reason Green's functions for the magnetoelectroelastic continuum are not known in the closed form and numerical methods should be applied to determine such Green's functions. These means that the problem of the accurate and simply determination of Green's function derivatives is even harder. Therefore in the present work the dual number algebra method is applied to calculate numerically the derivatives of 3D Green's functions for the magnetoelectroelastic materials. The introduced method is independent on the step size and it can be treated as a special case of the automatic differentiation method. Therefore, the dual number algebra method can be applied as a tool for checking the accuracy of the well-known finite difference schemes.

Classification of commutator algebras leading to the new type of closed Baker-Campbell-Hausdorff formulas

NASA Astrophysics Data System (ADS)

Matone, Marco

2015-11-01

We show that there are {\\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\\exp(X)\\exp(Y)\\exp(Z)=\\exp({AX+BZ+CY+DI}) \\ , $$ derived in arXiv:1502.06589, JHEP {\\bf 1505} (2015) 113. This includes, as a particular case, $\\exp(X) \\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\\exp(X)\\exp(Y)\\exp(Z)=\\exp(X)\\exp({\\alpha Y}) \\exp({(1-\\alpha) Y}) \\exp(Z)$, with $\\alpha$ fixed in such a way that it reduces to $\\exp({\\tilde X})\\exp({\\tilde Y})$, with $\\tilde X$ and $\\tilde Y$ satisfying the Van-Brunt and Visser condition $[\\tilde X,\\tilde Y]=\\tilde u\\tilde X+\\tilde v\\tilde Y+\\tilde cI$. It turns out that $e^\\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\\it types} of commutator algebras, such an equation leads to rational solutions for $\\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.

A Process Algebra Approach to Quantum Electrodynamics

NASA Astrophysics Data System (ADS)

Sulis, William

2017-12-01

The process algebra program is directed towards developing a realist model of quantum mechanics free of paradoxes, divergences and conceptual confusions. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM) without the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting, and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. Computations are carried out using the configuration space process covering map, although the connection to second quantization has not been fully explored.

Development of abstract mathematical reasoning: the case of algebra

PubMed Central

Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

2014-01-01

Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition. PMID:25228874

Development of abstract mathematical reasoning: the case of algebra.

PubMed

Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

2014-01-01

Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

Word Problem Wizardry.

ERIC Educational Resources Information Center

Cassidy, Jack

1991-01-01

Presents suggestions for teaching math word problems to elementary students. The strategies take into consideration differences between reading in math and reading in other areas. A problem-prediction game and four self-checking activities are included along with a magic password challenge. (SM)

Math and Humane Education.

ERIC Educational Resources Information Center

DeRosa, Bill

1986-01-01

Describes an activity designed to improve students' skills at solving mathematical word problems through an awareness of the pet overpopulation problem. Uses the concept of cumulative female offspring as a focal point in assisting students to analyze and work through word problems. (ML)

Arithmetic Word-Problem-Solving in Huntington's Disease

ERIC Educational Resources Information Center

Allain, P.; Verny, C.; Aubin, G.; Pinon, K.; Bonneau, D.; Dubas, F.; Gall, D.L.

2005-01-01

The purpose of this study was to examine executive functioning in patients with Huntington's disease using an arithmetic word-problem-solving task including eight solvable problems of increasing complexity and four aberrant problems. Ten patients with Huntington's disease and 12 normal control subjects matched by age and education were tested.…

Middle School Children's Problem-Solving Behavior: A Cognitive Analysis from a Reading Comprehension Perspective

ERIC Educational Resources Information Center

Pape, Stephen J.

2004-01-01

Many children read mathematics word problems and directly translate them to arithmetic operations. More sophisticated problem solvers transform word problems into object-based or mental models. Subsequent solutions are often qualitatively different because these models differentially support cognitive processing. Based on a conception of problem…

Block iterative restoration of astronomical images with the massively parallel processor

NASA Technical Reports Server (NTRS)

Heap, Sara R.; Lindler, Don J.

1987-01-01

A method is described for algebraic image restoration capable of treating astronomical images. For a typical 500 x 500 image, direct algebraic restoration would require the solution of a 250,000 x 250,000 linear system. The block iterative approach is used to reduce the problem to solving 4900 121 x 121 linear systems. The algorithm was implemented on the Goddard Massively Parallel Processor, which can solve a 121 x 121 system in approximately 0.06 seconds. Examples are shown of the results for various astronomical images.

Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory

NASA Technical Reports Server (NTRS)

Sun, Yan; Kouri, Donald J.; Truhlar, Donald G.; Schwenke, David W.

1990-01-01

New basis sets are proposed for linear algebraic variational calculations of transition amplitudes in quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained, even for basis sets with the majority of the members independent of energy.

On the integration of a class of nonlinear systems of ordinary differential equations

NASA Astrophysics Data System (ADS)

Talyshev, Aleksandr A.

2017-11-01

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

LAPACKrc: Fast linear algebra kernels/solvers for FPGA accelerators

NASA Astrophysics Data System (ADS)

Gonzalez, Juan; Núñez, Rafael C.

2009-07-01

We present LAPACKrc, a family of FPGA-based linear algebra solvers able to achieve more than 100x speedup per commodity processor on certain problems. LAPACKrc subsumes some of the LAPACK and ScaLAPACK functionalities, and it also incorporates sparse direct and iterative matrix solvers. Current LAPACKrc prototypes demonstrate between 40x-150x speedup compared against top-of-the-line hardware/software systems. A technology roadmap is in place to validate current performance of LAPACKrc in HPC applications, and to increase the computational throughput by factors of hundreds within the next few years.

Dual-scale topology optoelectronic processor.

PubMed

Marsden, G C; Krishnamoorthy, A V; Esener, S C; Lee, S H

1991-12-15

The dual-scale topology optoelectronic processor (D-STOP) is a parallel optoelectronic architecture for matrix algebraic processing. The architecture can be used for matrix-vector multiplication and two types of vector outer product. The computations are performed electronically, which allows multiplication and summation concepts in linear algebra to be generalized to various nonlinear or symbolic operations. This generalization permits the application of D-STOP to many computational problems. The architecture uses a minimum number of optical transmitters, which thereby reduces fabrication requirements while maintaining area-efficient electronics. The necessary optical interconnections are space invariant, minimizing space-bandwidth requirements.

Methods of mathematical modeling using polynomials of algebra of sets

NASA Astrophysics Data System (ADS)

Kazanskiy, Alexandr; Kochetkov, Ivan

2018-03-01

The article deals with the construction of discrete mathematical models for solving applied problems arising from the operation of building structures. Security issues in modern high-rise buildings are extremely serious and relevant, and there is no doubt that interest in them will only increase. The territory of the building is divided into zones for which it is necessary to observe. Zones can overlap and have different priorities. Such situations can be described using formulas algebra of sets. Formulas can be programmed, which makes it possible to work with them using computer models.

Text Comprehension and Oral Language as Predictors of Word-Problem Solving: Insights into Word-Problem Solving as a Form of Text Comprehension

ERIC Educational Resources Information Center

Fuchs, Lynn S.; Gilbert, Jennifer K.; Fuchs, Douglas; Seethaler, Pamela M.; N. Martin, BrittanyLee

2018-01-01

This study was designed to deepen insights on whether word-problem (WP) solving is a form of text comprehension (TC) and on the role of language in WPs. A sample of 325 second graders, representing high, average, and low reading and math performance, was assessed on (a) start-of-year TC, WP skill, language, nonlinguistic reasoning, working memory,…

Language, arithmetic word problems, and deaf students: Linguistic strategies used to solve tasks

NASA Astrophysics Data System (ADS)

Zevenbergen, Robyn; Hyde, Merv; Power, Des

2001-12-01

There has been limited examination of the intersection between language and arithmetic in the performance of deaf students, although some previous research has shown that deaf and hearing-impaired1 students are delayed in both their language acquisition and arithmetic performance. This paper examines the performance of deaf and hearing-impaired students in South-East Queensland, Australia, in solving arithmetic word problems. It was found that the subjects' solutions of word problems confirmed trends for hearing students, but that their performance was delayed in comparison. The results confirm other studies where deaf and hearing-impaired students are delayed in their language acquisition and this impacts on their capacity to successfully undertake the resolution of word problems.

Tense and aspect in word problems about motion: diagram, gesture, and the felt experience of time

NASA Astrophysics Data System (ADS)

de Freitas, Elizabeth; Zolkower, Betina

2015-09-01

Word problems about motion contain various conjugated verb forms. As students and teachers grapple with such word problems, they jointly operationalize diagrams, gestures, and language. Drawing on findings from a 3-year research project examining the social semiotics of classroom interaction, we show how teachers and students use gesture and diagram to make sense of complex verb forms in such word problems. We focus on the grammatical category of "aspect" for how it broadens the concept of verb tense. Aspect conveys duration and completion or frequency of an event. The aspect of a verb defines its temporal flow (or lack thereof) and the location of a vantage point for making sense of this durational process.

Deriving Laws from Ordering Relations

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.

2003-01-01

It took much effort in the early days of non-Euclidean geometry to break away from the mindset that all spaces are flat and that two distinct parallel lines do not cross. Up to that point, all that was known was Euclidean geometry, and it was difficult to imagine anything else. We have suffered a similar handicap brought on by the enormous relevance of Boolean algebra to the problems of our age-logic and set theory. Previously, I demonstrated that the algebra of questions is not Boolean, but rather is described by the free distributive algebra. To get to this stage took much effort, as many obstacles-most self-placed-had to be overcome. As Boolean algebras were all I had ever known, it was almost impossible for me to imagine working with an algebra where elements do not have complements. With this realization, it became very clear that the sum and product rules of probability theory at the most basic level had absolutely nothing to do with the Boolean algebra of logical statements. Instead, a measure of degree of inclusion can be invented for many different partially ordered sets, and the sum and product rules fall out of the associativity and distributivity of the algebra. To reinforce this very important idea, this paper will go over how these constructions are made, while focusing on the underlying assumptions. I will derive the sum and product rules for a distributive lattice in general and demonstrate how this leads to probability theory on the Boolean lattice and is related to the calculus of quantum mechanical amplitudes on the partially ordered set of experimental setups. I will also discuss the rules that can be derived from modular lattices and their relevance to the cross-ratio of projective geometry.

Dynamic Order Algebras as an Axiomatization of Modal and Tense Logics

NASA Astrophysics Data System (ADS)

Chajda, Ivan; Paseka, Jan

2015-12-01

The aim of the paper is to introduce and describe tense operators in every propositional logic which is axiomatized by means of an algebra whose underlying structure is a bounded poset or even a lattice. We introduce the operators G, H, P and F without regard what propositional connectives the logic includes. For this we use the axiomatization of universal quantifiers as a starting point and we modify these axioms for our reasons. At first, we show that the operators can be recognized as modal operators and we study the pairs ( P, G) as the so-called dynamic order pairs. Further, we get constructions of these operators in the corresponding algebra provided a time frame is given. Moreover, we solve the problem of finding a time frame in the case when the tense operators are given. In particular, any tense algebra is representable in its Dedekind-MacNeille completion. Our approach is fully general, we do not relay on the logic under consideration and hence it is applicable in all the up to now known cases.

Novel characteristics of energy spectrum for 3D Dirac oscillator analyzed via Lorentz covariant deformed algebra

PubMed Central

Betrouche, Malika; Maamache, Mustapha; Choi, Jeong Ryeol

2013-01-01

We investigate the Lorentz-covariant deformed algebra for Dirac oscillator problem, which is a generalization of Kempf deformed algebra in 3 + 1 dimension of space-time, where Lorentz symmetry are preserved. The energy spectrum of the system is analyzed by taking advantage of the corresponding wave functions with explicit spin state. We obtained entirely new results from our development based on Kempf algebra in comparison to the studies carried out with the non-Lorentz-covariant deformed one. A novel result of this research is that the quantized relativistic energy of the system in the presence of minimal length cannot grow indefinitely as quantum number n increases, but converges to a finite value, where c is the speed of light and β is a parameter that determines the scale of noncommutativity in space. If we consider the fact that the energy levels of ordinary oscillator is equally spaced, which leads to monotonic growth of quantized energy with the increment of n, this result is very interesting. The physical meaning of this consequence is discussed in detail. PMID:24225900

Novel characteristics of energy spectrum for 3D Dirac oscillator analyzed via Lorentz covariant deformed algebra.

PubMed

Betrouche, Malika; Maamache, Mustapha; Choi, Jeong Ryeol

2013-11-14

We investigate the Lorentz-covariant deformed algebra for Dirac oscillator problem, which is a generalization of Kempf deformed algebra in 3 + 1 dimension of space-time, where Lorentz symmetry are preserved. The energy spectrum of the system is analyzed by taking advantage of the corresponding wave functions with explicit spin state. We obtained entirely new results from our development based on Kempf algebra in comparison to the studies carried out with the non-Lorentz-covariant deformed one. A novel result of this research is that the quantized relativistic energy of the system in the presence of minimal length cannot grow indefinitely as quantum number n increases, but converges to a finite value, where c is the speed of light and β is a parameter that determines the scale of noncommutativity in space. If we consider the fact that the energy levels of ordinary oscillator is equally spaced, which leads to monotonic growth of quantized energy with the increment of n, this result is very interesting. The physical meaning of this consequence is discussed in detail.

Approximating smooth functions using algebraic-trigonometric polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sharapudinov, Idris I

2011-01-14

The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p{sub n}(t)+{tau}{sub m}(t), where p{sub n}(t) is an algebraic polynomial of degree n and {tau}{sub m}(t)=a{sub 0}+{Sigma}{sub k=1}{sup m}a{sub k} cos k{pi}t + b{sub k} sin k{pi}t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W{sup r}{sub {infinity}(}M) and an upper bound for similar approximations in the class W{sup r}{sub p}(M) with 4/3

Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

NASA Astrophysics Data System (ADS)

Jiang, Yunfeng; Zhang, Yang

2018-03-01

In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 super-Yang-Mills theory.

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra by quantum separation of variables

NASA Astrophysics Data System (ADS)

Pezelier, Baptiste

2018-02-01

In this proceeding, we recall the notion of quantum integrable systems on a lattice and then introduce the Sklyanin’s Separation of Variables method. We sum up the main results for the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. These results apply as well to the spectral analysis of the lattice sine-Gordon model with open boundary conditions. The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations. We state an equivalent characterization as the set of solutions to a Baxter’s like T-Q functional equation, allowing us to rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form.

Problem Solving Abilities and Perceptions in Alternative Certification Mathematics Teachers

ERIC Educational Resources Information Center

Evans, Brian R.

2012-01-01

It is important for teacher educators to understand new alternative certification middle and high school teachers' mathematical problem solving abilities and perceptions. Teachers in an alternative certification program in New York were enrolled in a proof-based algebra course. At the beginning and end of a semester participants were given a…

Recalling Prerequisite Material in a Calculus II Course to Improve Student Success

ERIC Educational Resources Information Center

Mokry, Jeanette

2016-01-01

This article discusses preparation assignments used in a Calculus II course that cover material from prerequisite courses. Prior to learning new material, students work on problems outside of class involving concepts from algebra, trigonometry, and Calculus I. These problems are directly built upon in order to answer Calculus II questions,…

Problem Solving in Calculus with Symbolic Geometry and CAS

ERIC Educational Resources Information Center

Todd, Philip; Wiechmann, James

2008-01-01

Computer algebra systems (CAS) have been around for a number of years, as has dynamic geometry. Symbolic geometry software is new. It bears a superficial similarity to dynamic geometry software, but differs in that problems may be set up involving symbolic variables and constants, and measurements are given as symbolic expressions. Mathematical…

Developing Computational Fluency with the Help of Science: A Turkish Middle and High School Grades Study

ERIC Educational Resources Information Center

Corlu, M. Sencer; Capraro, Robert M.; Corlu, M. Ali

2011-01-01

Students need to achieve automaticity in learning mathematics without sacrificing conceptual understanding of the algorithms that are essential in being successful in algebra and problem solving, as well as in science. This research investigated the relationship between science-contextualized problems and computational fluency by testing an…

Learning Algebra by Example in Real-World Classrooms

ERIC Educational Resources Information Center

Booth, Julie L.; Oyer, Melissa H.; Paré-Blagoev, E. Juliana; Elliot, Andrew J.; Barbieri, Christina; Augustine, Adam; Koedinger, Kenneth R.

2015-01-01

Math and science textbook chapters invariably supply students with sets of problems to solve, but this widely used approach is not optimal for learning; instead, more effective learning can be achieved when many problems to solve are replaced with correct and incorrect worked examples for students to study and explain. In the present study, the…

Knowledge Building and Mathematics: Shifting the Responsibility for Knowledge Advancement and Engagement

ERIC Educational Resources Information Center

Moss, Joan; Beatty, Ruth

2010-01-01

Three classrooms of Grade 4 students from different schools and diverse backgrounds collaborated in early algebra research to solve a series of linear and quadratic generalizing problems. Results revealed that high- and low-achieving students were able to solve problems of recognized difficulty. We discuss Knowledge Building principles and…

A Concise Introduction to Colombeau Generalized Functions and Their Applications in Classical Electrodynamics

ERIC Educational Resources Information Center

Gsponer, Andre

2009-01-01

The objective of this introduction to Colombeau algebras of generalized functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic nonlinear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the…

The Popcorn Box Activity and Reasoning about Optimization

ERIC Educational Resources Information Center

Whiteley, Walter J.; Mamolo, Ami

2012-01-01

A well-known optimization problem is the Popcorn Box investigation, which involves a movie theater snack container. The problem has been tailored for classroom investigations by the Ontario Association for Mathematics Education. The exploration was designed for students in grades 9 through 12. A common strategy proposed for algebra students is to…

Dogs Don't Need Calculus

ERIC Educational Resources Information Center

Bolt, Mike

2010-01-01

Many optimization problems can be solved without resorting to calculus. This article develops a new variational method for optimization that relies on inequalities. The method is illustrated by four examples, the last of which provides a completely algebraic solution to the problem of minimizing the time it takes a dog to retrieve a thrown ball,…

Investigation of learning environment for arithmetic word problems by problem posing as sentence integration in Indonesian language

NASA Astrophysics Data System (ADS)

Hasanah, N.; Hayashi, Y.; Hirashima, T.

2017-02-01

Arithmetic word problems remain one of the most difficult area of teaching mathematics. Learning by problem posing has been suggested as an effective way to improve students’ understanding. However, the practice in usual classroom is difficult due to extra time needed for assessment and giving feedback to students’ posed problems. To address this issue, we have developed a tablet PC software named Monsakun for learning by posing arithmetic word problems based on Triplet Structure Model. It uses the mechanism of sentence-integration, an efficient implementation of problem-posing that enables agent-assessment of posed problems. The learning environment has been used in actual Japanese elementary school classrooms and the effectiveness has been confirmed in previous researches. In this study, ten Indonesian elementary school students living in Japan participated in a learning session of problem posing using Monsakun in Indonesian language. We analyzed their learning activities and show that students were able to interact with the structure of simple word problem using this learning environment. The results of data analysis and questionnaire suggested that the use of Monsakun provides a way of creating an interactive and fun environment for learning by problem posing for Indonesian elementary school students.

Investigating middle school students’ difficulties in mathematical literacy problems level 1 and 2

NASA Astrophysics Data System (ADS)

Setiawati, S.; Herman, T.; Jupri, A.

2017-11-01

The background of this study is the lack of mathematical literacy skills of students. The proficiency of students’ mathematical literacy skills based on the results of the PISA 2015 study shows that Indonesian students at the proficiency level 1. This fact gave rise to this study which aims to investigate middle school students’ difficulties in mathematical literacy problems level 1 and 2. Qualitative research was used in this study. An individual written test on mathematical literacy problems was administered, followed by interviews. The subjects of the study were 61 students grade VII in Bandung and 26 of them were interviewed afterward. Data analysis revealed that students’ error in performing arithmetic most frequently observed. Other observed difficulties concerned understanding about algebra concept, applying arithmetic operation in algebraic expressions, and interpreting symbols to represent the unknown. In solving mathematical literacy problems, students use their prior knowledge, although sometimes not relevant to the questions. Based on the results, we suggest that mathematics learning in contextual learning and which invites students to participate in the processes of understanding the concepts.

Linear homotopy solution of nonlinear systems of equations in geodesy

NASA Astrophysics Data System (ADS)

Paláncz, Béla; Awange, Joseph L.; Zaletnyik, Piroska; Lewis, Robert H.

2010-01-01

A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton-Raphson.

Formulae and Flow-Diagrams

ERIC Educational Resources Information Center

Willson, William Wynne

1977-01-01

The author recommends the use of flow charting to help students understand the manipulation of algebraic formulae. He identifies some problems with flow charts and suggests an alternative method of constructing flow diagrams. (SD)