A Note on Mathematical Induction

1974 ◽  
Vol 67 (7) ◽  
pp. 616-618
Author(s):  
Charles Brumfiel
Keyword(s):  
Mathematical Induction ◽  
Advanced Students

Do your advanced students really understand the concept of mathematical induction? Try this phony “proof” on them.

Counting Triples, Triangles, and Acute Triangles

1999 ◽  
Vol 92 (7) ◽  
pp. 612-619
Author(s):  
Ruth McClintock
Keyword(s):  
Problem Solving ◽  
Difference Equation ◽  
Mathematical Induction ◽  
Modular Arithmetic ◽  
Advanced Students ◽  
Greatest Integer Function ◽  
First Time

Activities involving counting triples, triangles, and acute triangles enrich the curriculum with excursions into modular arithmetic, the greatest-integer function, and summation notation. In addition, more advanced students can apply difference-equation techniques to find closed forms and can use mathematical induction to prove the formulas. Students may be learning about these topics for the first time, or they may be reviewing familiar ideas in different problem-solving contexts. In either situation, personal arsenals of problem-attacking skills are strengthened.

Sourcebook in the Mathematics of Medieval Europe and North Africa

2016 ◽  
Keyword(s):  
North Africa ◽  
Cultural Influences ◽  
Mathematical Induction ◽  
Cross Cultural ◽  
General Introduction ◽  
Extensive Survey ◽  
Medieval Europe ◽  
Ceva’S Theorem ◽  
Arabic Speaking ◽  
First Time

Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This book presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; Al-Muʾtaman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī al-Dīn al-Maghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references.

Henderson Hasselbalch Relationship and Weak Acid Titration

2019 ◽  
Author(s):  
Marc Blétry
Keyword(s):  
Theoretical Framework ◽  
Weak Acid ◽  
Acid Base ◽  
Sensitive Technique ◽  
Chemistry Students ◽  
Advanced Students ◽  
Acid Titration ◽  
Acid Base Titration ◽  
Analytical Expressions

Henderson-Hasselbalch relation is generally the simplified theoretical framework used to introduce students to acid-base titration. However, it is not always valid and its limitations should be made clear to chemistry students. The appropriate parameter to evaluate its validity is K a /C 0 , in connection with Ostwald dilution law. For more advanced students, it is possible to deduce analytical expressions that always fit accurately acid-base titrations and allow an evaluation of Henderson Hasselbalch relation. Gran plot appears as a particularly sensitive technique to the breakdown of Henderson Hasselbalch relation.

T. Paul Schultz. Economics o[ Population. Reading, Mass. (USA): Addison - Wesley Publishing Co. 1981.

1981 ◽  
Vol 20 (2) ◽  
pp. 269-273
Author(s):  
Syed Nawab Haider Naqvi
Keyword(s):  
Theoretical Orientation ◽  
Chicago School ◽  
The Other ◽  
Methodological Framework ◽  
Economic Demography ◽  
Advanced Students

Unfortunately this is not the long-awaited textbook in economic demography. Indeed, it is not so much a text - a survey and introduction to the area - as it is a collection of essays on particular topics, often quite advanced and difficult for all but advanced students to follow. Also, the volume should, in all fairness, be subtitled "A Chicago Approach" since the philosophical and theoretical orientation as well as the methodological framework presented is totally that of the Becker Nerlove Chicago School. Easterlin, Leibenstein and the other non Chicago writers are mentioned only in passing. Thus, a beginner to the field would gel no feeling for the enormous, far-ranging controversies which continue to rage.

Microstructures in Elastic Media

1994 ◽  
Author(s):  
Nhan Phan-Thien ◽  
Sangtae Kim
Keyword(s):  
Computational Methods ◽  
High Performance ◽  
Chemical Engineering ◽  
Elliptic Partial Differential Equations ◽  
Parallel Computers ◽  
Computational Algorithms ◽  
Applied Physics ◽  
Advanced Students ◽  
Particulate Solids ◽  
Modeling Software

This monograph describes various methods for solving deformation problems of particulate solids, taking the reader from analytical to computational methods. The book is the first to present the topic of linear elasticity in mathematical terms that will be familiar to anyone with a grounding in fluid mechanics. It incorporates the latest advances in computational algorithms for elliptic partial differential equations, and provides the groundwork for simulations on high performance parallel computers. Numerous exercises complement the theoretical discussions, and a related set of self-documented programs is available to readers with Internet access. The work will be of interest to advanced students and practicing researchers in mechanical engineering, chemical engineering, applied physics, computational methods, and developers of numerical modeling software.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

scholarly journals Predictive processes during simultaneous interpreting from German into English

Interpreting ◽  
2017 ◽  
Vol 19 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Ena Hodzik ◽  
John N. Williams
Keyword(s):  
Language Processing ◽  
English Language ◽  
Native Speakers ◽  
Transitional Probability ◽  
Semantic Cues ◽  
Simultaneous Interpreting ◽  
Spoken Language Processing ◽  
Advanced Students ◽  
Native Speakers Of English ◽  
Predictive Processes

We report a study on prediction in shadowing and simultaneous interpreting (SI), both considered as forms of real-time, ‘online’ spoken language processing. The study comprised two experiments, focusing on: (i) shadowing of German head-final sentences by 20 advanced students of German, all native speakers of English; (ii) SI of the same sentences into English head-initial sentences by 22 advanced students of German, again native English speakers, and also by 11 trainee and practising interpreters. Latency times for input and production of the target verbs were measured. Drawing on studies of prediction in English-language reading production, we examined two cues to prediction in both experiments: contextual constraints (semantic cues in the context) and transitional probability (the statistical likelihood of words occurring together in the language concerned). While context affected prediction during both shadowing and SI, transitional probability appeared to favour prediction during shadowing but not during SI. This suggests that the two cues operate on different levels of language processing in SI.

scholarly journals THE INFLUENCE OF DESIGN BRIEF INFORMATION ON CREATIVE OUTCOMES BY NOVICE AND ADVANCED STUDENTS

2021 ◽  
Vol 1 ◽  
pp. 3041-3050
Author(s):  
Georgios Koronis ◽  
Hernan Casakin ◽  
Arlindo Silva ◽  
Jacob Kai Siang Kang
Keyword(s):  
Design Education ◽  
Design Problems ◽  
Frame Design ◽  
Design Creativity ◽  
Abstract Representations ◽  
Advanced Students ◽  
Different Types ◽  
Fishbone Diagram ◽  
Design Expertise ◽  
Novel Design

AbstractThis study centers on using different types of brief information to support creative outcomes in architectural and engineering design and its relation to design expertise. We explore the influence of design briefs characterized by abstract representations and/or instructions to frame design problems on the creativity of concept sketches produced by novice and advanced students. Abstract representations of problem requirements served as stimuli to encourage associative thinking and knowledge transfer. The Ishikawa/Fishbone Diagram was used to foster design restructuring and to modify viewpoints about the main design drives and goals. The design outcomes generated by novice and advanced engineering/architecture students were assessed for their creativity using a pairwise experimental design. Results indicated that advanced students generated more novel design solutions while also contributing the most useful solutions overall. Implications for creativity in design education and professional practice are presented. Educational programs aimed at promoting creativity in the design studio may find it helpful to consider that the way design briefs are constructed can either promote or inhibit different aspects of design creativity.

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