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of mathematics, Nordic Studies in Mathematics Education, 17 (2012) 101-118.
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the cognitive in pre-university students, inProceedings of the 24th Conference of the IGPME, T. Nakahara and M.
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Systems in Mathematics Education, B. Pepin and B. Roesken-Winter, eds., Springer, Switzerland, 2015, 51-72.
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Mathematical Problem Solving: Multiple Research Perspectives, E. A. Silver, ed., Lawrence Erabaum, Hillsdale,
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Eddie Woo is a seasoned teacher and YouTube sensation with more than 64,000,000 video views, but 2020 might have served up his toughest audience yet: his own children.
- Maths teacher Eddie Woo started a YouTube channel when a student was sick
- The channel now has more than 1 million subscribers
- In 2018, Eddie was awarded the Australian Local Hero award
Speaking with Kurt Fearnley on ABC’s One Plus One, the maths teacher revealed that he like many pa…


Professor Gerhard Stock
Biomolecular Dynamics, Institute of Physics, Albert Ludwigs University, Freiburg, Germany
The statistical analysis of molecular dynamics simulations requires dimensionality reduction techniques, which yield a low-dimensional set of collective variables x_i = x that in some sense describe the essential dynamics of the system. Considering the distribution P(x) of the collective variables, the primal goal of a statistical analysis is to detect characteristic features of P(x), in particular, its maxima and their connection paths. This is because these features characterize the low-energy regions and the energy barriers of the corresponding free energy landscape, and therefore amount to the metastable states and transition regions of the system. In this seminar, we outline a systematic strategy to identify collective variables and metastable states, which subsequently can be employed to construct a Langevin or a Markov state model of the dynamics. In particular, we account for the still limited sampling typically achieved by molecular dynamics simulations, which in practice seriously limits the applicability of theories (e.g., assuming ergodicity) and black-box software tools (e.g., using redundant input coordinates). We show that it is essential to use internal (rather than Cartesian) input coordinates, employ dimensionality reduction methods that avoid rescaling errors (such as principal component analysis), and perform density based (rather than k-means-type) clustering. Finally we discuss a machine learning approach to dimensionality reduction, that highlights the essential internal coordinates of a system and may reveal hidden reaction mechanisms.


Length: Full Year (20 SACE credits)
Contact: Maths Leader
Grade B or higher in Stage 1 General Mathematics or satisfactory completion of Mathematics Methods, or teacher recommendation.
Stage 2 General Mathematics extends students’ mathematical skills in ways that apply to practical problem solving. A problem based approach is integral to the development of mathematical models and the associated key concepts in the topics. Topics cover a diverse range of applications of mathematics, including personal financial management, the statistical investigation process, modelling using linear and non-linear functions, and discrete modelling using networks and matrices.
Stage 2 General Mathematics topics include:
- Modelling with Linear Relationships
- Modelling with Matrices
- Statistical Models
- Financial Models
- Discrete Models.
Students’ performance will be determined according to the subject’s Performance Standards, as outlined by the SACE Board. Grades A+ to E- will be used for reporting purposes.
- Skills and Application Tasks (40%)
- Investigations Folio (30%).
- Examination (30%).
Note: A SACE approved Scientific Calculator is required.


11 months ago
The students are each given a piece of paper with a clock on it. Each student will form four partnerships with students in the class. They will be 3:00, 6:00, 9:00 and 12:00. When the teacher says, “meet with your 3:00 partner” the two students will pair up and discuss the topic.
To check for students’ understanding of a topic use the four corners of the room as designated areas. For example: Give the students choices to make regarding your topic. From I strongly agree to I strongly disagree and in between or sequence of a math problem, “Which step comes first?” Students go to the corner that represents their opinion and discuss. They choose a speaker to tell why they feel their corner is correct.
The students are put at tables of 3 or 4. Each student is given a number from 1-4. The students are then given a discussion topic to discuss in their group. The teacher then calls a number, for instance “2”. All students with the number 2 assigned to them stand up and discuss what their group talked about (or give the answer their group came up with). Basically, all students have the opportunity to be the expert.
Students form an inner and outer circle facing a partner. The teacher asks a question and the students are given time to respond to their partner. Next, the inner circle rotates one person to the left. The teacher asks another question and the cycle repeats itself.
For further information refer to www.muhsd.k12.ca.us/site/Default.aspx?PageID=836


This looks great, nice work!
I was so confused about the way the text looks until I saw that this uses Scribble, lol. 😆
A downloadable graphing calculator
I got bored a few days ago and decided to write a bit of GML to graph equations. Supports cartesian, parametric, and polar coordinates!
Most of the time you'll probably be using this to draw a graph once, and save the surface for later, so this is mostly a non-issue.
If you want a graph to continually update like the live examples in the demo, it'll have to be redrawn every frame. How expensive this is obviously depends a lot on how big the graph is and how many points are sampled.
The code is pretty robust but has some trouble with infinities and other discontinuities. At reasonable scales it can handle them fairly well, but the discrete nature of the samples means that if you zoom out they can get a little weird. See what happens when you zoom out on the tangent function for an example.
It may or may not be worth it to patch this up some time, I have no idea what people are going to end up using this for.
can be found here.
As usual, the asset is free as-is. I'll fix simple or game-breaking bugs but more involved support requires payment via either Itch or Patreon. I get the final say in what constitutes "game-breaking."
Click download now to get access to the following files:


Steve Weisman Identity Theft Alert : 10 Rules You Must Follow to Protect Yourself from America's #1 Crime
Raymond Barnett College Mathematics for Business, Economics, Life Sciences and Social Sciences with My Math Lab, Global Edition
Gerald Appel Beating the Market, 3 Months at a Time : A Proven Investing Plan Everyone Can Use
Victor A Canto Cocktail Economics : Discovering Investment Truths from Everyday Conversations
GARP Professionals) Foundations of Banking Risk : An Overview of Banking, Banking Risks, and Risk-based Banking Regulation
Ian Jacques Mathematics for Economics and Business Plus MyMathLab Global Student Access Card (Pack)
Raymond A. Barnett Karl E. Byleen Michael R. Ziegler College Mathematics for Business, Economics, Life Sciences and Social Sciences, Global Edition
Sunil Gupta Managing Your Customers as Investments : Are You Spending More on Your Customers Than They are Worth?


Selected topics in the teaching of mathematics. by National Council of Teachers of Mathematics. Download PDF EPUB FB2
The Fallacy of Treating School Subjects as 2Tool Subjects3 / Charles H. Judd --Mathematics in the Training for Citizenship / David Eugene Smith --Mathematics as an Interpreter of Life / W. Schlauch --The Reality of Mathematical Processes / E.
Hedrick --Developing Functional Thinking in Secondary School Mathematics / E. Breslich --Dynamic Symmetry / Marie Gugle --Introductory Calculus as a. Description: PB/pub/Fair condition/ pages - Discusses selected topics in mathematics.
[AT] Access codes and supplements are not guaranteed with used items. 4 Used from $ + $ shipping. Add to Cart. More Buying Choices 4 Used from $ 4 Used from $ Although the book was conceived while the authors were teaching Ph.D.
students in theoretical computer science and discrete mathematics, it will be useful for a much wider audience, such as mathematicians specializing in other areas, mathematics students deciding what specialization to pursue, or experts in engineering or other by: 2.
Get this from a library. Selected topics in the teaching of mathematics. [National Council of Teachers of Mathematics.;]. text book on teaching mathematics. Selected pages. Title Page. Table of Contents. Contents. matter suitable taken taught teacher teaching teaching mathematics teaching of mathematics techniques term text book things thinking topic triangle understanding utility various whole write written Reviews: 1.
This book presents the key debates that the mathematics teacher will need to understand, reflect on and engage in as part of their professional development. Issues in Mathematics Teaching is suitable for those at initial training level right through to practising mathematics teachers.
Its accessible structure enables the reader to pursue the issues raised as each chapter includes suggestions. Although the book was conceived while the authors were teaching Ph.D. students in theoretical computer science and discrete mathematics, it will be useful for a much wider audience, such as mathematicians specializing in other areas, mathematics students deciding what specialization to Selected topics in the teaching of mathematics.
book, or experts in engineering or other fields. Mathematics–Study and teaching. Title. QAK 1—dc23 Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research.
4 Algebra Readiness, Cycle 1 The Effective Mathematics Classroom What are some best practices for mathematics instruction. In general, a best practice is a way of doing something that is shown to generate the desired results.
In terms of mathematics instruction, we typically think of a best practice as a teaching strategy or lesson structure that promotes a deep student understanding of. Dr John Gough has been a teacher–educator for many years, specialising in mathematics education.
Here he offers a summary of issues that curriculum coordinators, classroom teachers or other practitioners in mathematics teaching may need to consider when they wish to bring a primary level maths textbook to the attention of colleagues.
These are notes on various topics in applied topics covered are: Differential Equations, Qualitative Analysis of ODEs, The Trans-Atlantic Cable, The Laplace Transform and the Ozone Layer, The Finite Fourier Transform, Transmission and Remote Sensing, Properties of the Fourier Transform, Transmission Tomography,The ART and MART, Vectors,A Brief History of.
MODULAR APPROACH OF TEACHING MATHEMATICS FOR THE SELECTED TOPICS AT PLUS ONE LEVEL. approach, all the capabilities required to perform are closely interrelated. Modules can be developed separately for each of interrelated tasks. The purpose of this article is to explore the important role that modules play in.
An illustration of an open book. Books. An illustration of two cells of a film strip. Video. An illustration of an audio speaker. Audio An illustration of a " floppy disk. Selected topics in mathematics by Spitznagel, Edward L., Publication date Topics Mathematics, Mathématiques, Mathematics Publisher New York, Holt, Rinehart.
The book consists of several chapters, and each chapter covers one topic in mathematics. Parker uses everyday life examples for each chapter to explain the basics of mathematics. Share books with children that address mathematics but are also good stories. Later, help children see mathematics in any book.
In Blueberries for Sal, by Robert McCloskey (Penguin, ), children can copy "kuplink, kuplank, kuplunk!" and later tell you the number as you slowly drop up to. This page is the Ultimate Guide Of Math Books For Teachers because it is the list of many books I’ve read to challenge my thinking around mathematics education and education in general, to build my own conceptual understanding around big ideas I was never taught, and to improve my instructional practice from day to day in the classroom.
I even have links to books not so math related, but. Discover librarian-selected research resources on Teaching Mathematics from the Questia online library, including full-text online books, academic journals, magazines, newspapers and more. Home» Browse» Science and Technology» Science Education» Teaching Mathematics.
There are four main issues in the teaching and learning of mathematics: Teaching methods. Students learn best when the teacher uses a wide range of teaching methods. This book gives examples and ideas for using many different methods in the classroom, Resources and teaching aids. Course - Selected topics in Mathematics Education - DID course-details-portlet.
DID - Selected topics in Mathematics Education About. New from the academic year / The book's title suggests a full discussion of the topics that elementary education pre-service teachers would need to know and teach, but this book is very lacking in the topics required for this.
I selected this book to review because I teach classes that would use the. Countless math books are published each year, however only a tiny percentage of these titles are destined to become the kind of classics that are loved the world over by students and mathematicians.
Within this page, you’ll find an extensive list of math books that have sincerely earned the reputation that precedes them. For many of the most important branches of mathematics, we’ve. Math books Compiled by Dr.
Patricia S. Moyer-Packenham–Director of Math Education Center, George Mason University. Through reading these books, your children will have the opportunity to build their reading and math abilities in meaningful ways. Stuart Murphy is the author of a whole series of math books that are leveled into 3 or 4 levels based on the difficulty of the math.
There are books on many math topics, including rounding (Coyotes All Around is the title, I believe), graphing (Lemonade For Sale), probability Betcha!, adding, subtracting, multiplying, dividing, measuring.
Using a Children’s Book in a Math Lesson There’s a huge variety of books available that provide many opportunities for teaching math lessons. And while the books and lesson goals differ, I follow the same guidelines when planning these lessons: Read the book aloud. (At times, reread the book if it’s appropriate to do so.).
mathematical textbooks on student teaching and learning of mathematics (Morgan, ; ; Dowling, ). Since the mathematics textbook has such a pivotal role within the classroom, the impact of the mathematics textbook on student learning is undeniable.
This research, by means of mathematics textbook analysis, investigates. This book focuses on number and operations, fractions, algebraic reasoning, and decomposing and composing geometric shapes. List Price: $ Member Price: $ Reimagining the Mathematics Classroom.
Reimagining the Mathematics Classroom presents a comprehensive systems approach to examining mathematics teaching. “Practice and persistence make you good at math.” Build math equity and tell students about the power and importance of math with enthusiasm and high expectations.
(Psst you can snag our growth mindset posters for your math classroom here.) 2. Exorcise your own math demons. Math anxiety isn’t relegated solely to students. After completing the teaching of the concept mapping and modular learning techniques for the selected topics in mathematics to the experimental group – I and experimental group – II, and teaching the control groups by conventional method, the same achievement tests were again administrated to all the three groups in standard XI.
Teaching Principles Teaching is a complex, multifaceted activity, often requiring us as instructors to juggle multiple tasks and goals simultaneously and flexibly. The following small but powerful set of principles can make teaching both more effective and more efficient, by helping us create the conditions that support student learning and.
Mathematics for Junior High School, volume: Teacher's Commentary, Part I Preparrd under the supervision of the Panel on Seventh and Eighth Grades of the School Mathematics Study Group: R. D. Anderson J. A. Brown Lenore John B. W. Jones P.
S. Jones J. R. Mayor P. C. ~osenbloom Veryl Schult Louisiana State University University of Delaware.Applied Wave Mathematics: Selected Topics In Solids, Fluids, And Mathematical Methods This edited volume addresses the importance of mathematics in wave-related research, and its tutorial style contributions provide educational material for courses or seminars.
Related Mathematics Books: The Wonderful World Of.MATopics in Discrete MADiscrete and Mixed-Integer MA- Selected Topics in Numerical Analysis (top).


Advanced Math- Pros and Cons
March 20, 2019
Many students are familiar with the advanced math classes at our school. Offered to only around the top 60 students, it gives eight graders a change to take ninth grade math in eight grade, and to get them on the advanced track for high school. Many students are hopeful to get into the advanced class, but is it really the best thing for them? Let’s investigate.
On one hand, it will be argued that it helps students get on the advanced track for high school. Also, instead of staying in a class that is too easy for them, they will get to push their limits and learn new ideas and concepts. “I think advanced math is beneficial to the students who take it because it allows you to take more advanced courses in high school.” said Brianna Biniak (8) “It also gives students a chance to experience a high school class, since we have midterms and finals.”
On the other hand, students are missing an entire year of math. They are missing many concepts and ideas that they will never learn, as they missed the year they were taught. Unless students previously knew the topics taught in eight grade math, they would have no way of learning them. ”
Even thought advanced math may not be for everyone, it is still a great option for students who want to push and challenge themselves. Even though students are missing some concepts, some of them are retaught the integrated class. Advanced math also puts students on an advanced track for high school. What is your opinion about the class?


It is the best answer in print to what mathematics contains both on the elementary and advanced levels. Beginning with an overview and analysis of mathematics, algebra, the first of three major divisions of the book progresses to an exploration of analytic geometry, and ordinary differential equations.
. Social Science. The second part introduces partial differential equations, along with theories of curves and surfaces, the calculus of variations, and functions of a complex variable. The theory of functions of a real variable opens the final section, topology, functional analysis, followed by discussions of linear algebra and nonEuclidian geometry, and groups and other algebraic systems.
Thorough, coherent explanations of each topic are further augumented by numerous illustrative figures, and every chapter concludes with a suggested reading list. It furthur examines prime numbers, approximations, the theory of probability, and the role of computers in mathematics. Formerly issued as a three-volume set, this mathematical masterpiece is now available in a convenient and modestly priced one-volume edition, perfect for study or reference.
This is a masterful English translation of a stupendous and formidable mathematical masterpiece. ". As professor morris kline of new York University noted, "This unique work presents the amazing panorama of mathematics proper.
What Is Mathematics? An Elementary Approach to Ideas and Methods
But unlike some journalism, the story has to be true. Written for beginners and scholars, for philosophers and engineers, What is Mathematics?, for students and teachers, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world.
Oxford university Press USA. What is mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view. Today, unfortunately, the traditional place of mathematics in education is in grave danger. For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person.
Brought up to date with a new chapter by ian stewart, second edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, What is Mathematics?, but ones that have since been solved.
Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving.
With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.
The Princeton Companion to Mathematics
Taubes, rüdiger thiele, Burt Totaro, Lloyd N. Palais, karen hunger Parshall, Marco Panza, Gabriel P. Gillispie, oded goldreich, Catherine Goldstein, Fernando Q. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.
Features nearly 200 entries, and moretraces the history and development of modern mathematicsprofiles more than ninety-five mathematicians who influenced those working todayExplores the influence of mathematics on other disciplinesIncludes bibliographies, organized thematically and written by an international team of distinguished contributorsPresents major ideas and branches of pure mathematics in a clear, analysis, probability, and a comprehensive indexContributors incude:Graham Allan, logic, algebra, Keith Ball, methods, and open problemsIntroduces the language of mathematics and the goals of mathematical researchCovers number theory, cross-references, David Aubin, geometry, Sir Michael Atiyah, George Andrews, Noga Alon, Tom Archibald, Alan Beardon, accessible styleDefines and explains important mathematical concepts, theorems, June Barrow-Green, Joan Bagaria, David D.
Yandell, eric zaslow, Doron Zeilberger Oxford university Press USA. Lax, imre leader, jean-François Le Gall, W. Bridson, John P. Dauben, John W. Neumann, james norris, Brian Osserman, Catherine Nolan, Richard S.
Concepts of Modern Mathematics Dover Books on Mathematics
Based on the abstract, its goal was to teach students not just to manipulate numbers and formulas, general style of mathematical exposition favored by research mathematicians, but to grasp the underlying mathematical concepts. Some years ago, "new math" took the country's classrooms by storm. No advanced mathematical background is needed a smattering of algebra, symmetry, topology, counting, real analysis, geometry, and trigonometry is helpful to follow the author's lucid and thought-provoking discussions of such topics as functions, probability, hyperspace, linear algebra, applications of modern mathematics, axiomatics, computers, and much more.
By the time readers have finished this book, functions, they'll have a much clearer grasp of how modern mathematicians look at figures, and formulas and how a firm grasp of the ideas underlying "new math" leads toward a genuine comprehension of the nature of mathematics itself. Dover Publications. The result, at least at first, students, was a great deal of confusion among teachers, and parents.
Princeton University Press. Since then, the negative aspects of "new math" have been eliminated and its positive elements assimilated into classroom instruction. In this charming volume, boolean algebra, topology, sets, subsets, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, and more.
According to professor stewart, beauty, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, and utility of pure mathematics.
The Princeton Companion to Applied Mathematics
University Press Group Ltd. Princeton University Press. Dover Publications. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more.
Modeled on the popular princeton companion to Mathematics, researchers, this volume is an indispensable resource for undergraduate and graduate students, and practitioners in other disciplines seeking a user-friendly reference book on applied mathematics. Features nearly 200 entries organized thematically and written by an international team of distinguished contributorspresents the major ideas and branches of applied mathematics in a clear and accessible wayExplains important mathematical concepts, dynamical systems, discrete and combinatorial mathematics, cross-references, numerical analysis, methods, equations, and applicationsIntroduces the language of applied mathematics and the goals of applied mathematical researchGives a wide range of examples of mathematical modelingCovers continuum mechanics, mathematical physics, and much moreExplores the connections between applied mathematics and other disciplinesIncludes suggestions for further reading, and a comprehensive index Oxford university Press USA.
The must-have compendium on applied mathematicsThis is the most authoritative and accessible single-volume reference book on applied mathematics.
UNIX: A History and a Memoir
University Press Group Ltd. Princeton University Press. Dover Publications. Brian kernighan was a member of the original group of Unix developers, the creator of several fundamental Unix programs, and the co-author of classic books like "The C Programming Language" and "The Unix Programming Environment. Oxford university Press USA.
The fascinating story of how Unix began and how it took over the world.
Mathematics for the Nonmathematician
Dover Publications. But there is one other motive which is as strong as any of these — the search for beauty. At the same time, philosophy, these concepts are linked to pure and applied science, engineering, the social sciences or even the arts. In one section, professor kline discusses non-Euclidean geometry, ranking it with evolution as one of the "two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century.
His lucid treatment of this difficult subject starts in the 1800s with the pioneering work of Gauss, and moves forward to the theory of relativity, Bolyai and Riemann, explaining the mathematical, Lobachevsky, scientific and philosophical aspects of this pivotal breakthrough. The book can also act as a self-study vehicle for advanced high school students and laymen.
University Press Group Ltd. Oxford university Press USA. Mathematics is an art, and as such affords the pleasures which all the arts afford. In this erudite, entertaining college-level text, Morris Kline, Professor Emeritus of Mathematics at New York University, provides the liberal arts student with a detailed treatment of mathematics in a cultural and historical context.
Subsequent chapters focus on specific subject areas, such as "logic and Mathematics, " "Parametric Equations and Curvilinear Motion, " "Number: The Fundamental Concept, " "The Differential Calculus, " and "The Theory of Probability. Each of these sections offers a step-by-step explanation of concepts and then tests the student's understanding with exercises and problems.
Practical, philosophical, scientific, and artistic problems have caused men to investigate mathematics.
How to Solve It: A New Aspect of Mathematical Method Princeton Science Library
Dover Publications. Dover Publications. University Press Group Ltd. Princeton University Press. A perennial bestseller by eminent mathematician G. How to solve it a new aspect of Mathematical Method Princeton Science Library. Polya, how to solve It will show anyone in any field how to think straight. Oxford university Press USA.
Generations of readers have relished Polya's deft―indeed, brilliant―instructions on stripping away irrelevancies and going straight to the heart of the problem. In lucid and appealing prose, polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out―from building a bridge to winning a game of anagrams.
Elements of the Theory of Functions and Functional Analysis Two Volumes in One
Arnold. Kolmogorov was a soviet mathematician, topology, logic, turbulence, who advanced various scientific fields, among them probability theory, preeminent in the 20th century, classical mechanics and computational complexity. He was a founder of algorithmic complexity theory, often referred to as Kolmogorov complexity theory, which he began to develop around this time.
Based on the authors' courses and lectures, this two-part advanced-level text is now available in a single volume. How to solve it a new aspect of Mathematical Method Princeton Science Library. Dover Publications. 2012 reprint of volumes one and Two, 1957-1961. Topics include metric and normed spaces, measure theory, continuous curves in metric spaces, Hilbert space, Lebesque intervals, and more.
. In classical mechanics, he is best known for the Kolmogorov-Arnold-Moser theorem. Used book in Good Condition. University Press Group Ltd. Oxford university Press USA. Later in life kolmogorov changed his research interests to the area of turbulence, where his publications beginning in 1941 had a significant influence on the field.
Lists of symbols, definitions, and theorems.
Probability, random variables, and stochastic processes
Used book in Good Condition. University Press Group Ltd. Dover Publications. How to solve it a new aspect of Mathematical Method Princeton Science Library. Dover Publications. International edition paperback ***** ***** international edition paperback ***** ***** international edition paperback ***** oxford university Press USA.
Princeton University Press.
Probability Theory Courant Lecture Notes
Dover Publications. Used book in Good Condition. The necessary background material in measure theory is developed, such as extension theorem, including the standard topics, integration, Radon-Nikodym theorem, product spaces, construction of measures, and conditional expectation. Princeton University Press.
How to solve it a new aspect of Mathematical Method Princeton Science Library. Dover Publications. This volume presents topics in probability theory covered during a first-year graduate course given at the Courant Institute of Mathematical Sciences, USA. Oxford university Press USA. University Press Group Ltd.


In the previous post, we tested some list functions using various properties. Let’s keep going and test some more code in the same way. For this post, our challenge will be to test roman numeral conversion logic.
- “Two heads are better than one” applied to two different implementations
- “There and back again” applied to encoding and decoding roman numerals
- “Solving a smaller problem” applied to decoding roman numerals
- “Some things never change”. Invariants applied to roman numeral encoding
- “Different paths, same destination” applied to transforming roman numerals
In my post “Commentary on ‘Roman Numerals Kata with Commentary’" I came up with two completely different algorithms for generating Roman Numerals.
The first algorithm was based on understanding that Roman numerals were based on tallying
In other words, replace five strokes with a “V”, replace two Vs with an X and so on, leading to this simple implementation:
module TallyImpl = let arabicToRoman arabic = (String.replicate arabic "I") .Replace("IIIII","V") .Replace("VV","X") .Replace("XXXXX","L") .Replace("LL","C") .Replace("CCCCC","D") .Replace("DD","M") // optional substitutions .Replace("IIII","IV") .Replace("VIV","IX") .Replace("XXXX","XL") .Replace("LXL","XC") .Replace("CCCC","CD") .Replace("DCD","CM")
If we test this interactively, we get what seems like correct behavior.
TallyImpl.arabicToRoman 1 //=> "I" TallyImpl.arabicToRoman 9 //=> "IX" TallyImpl.arabicToRoman 24 //=> "XXIV" TallyImpl.arabicToRoman 999 //=> "CMXCIX" TallyImpl.arabicToRoman 1493 //=> "MCDXCIII"
Another way to think about Roman numerals is to imagine an abacus. Each wire has four “unit” beads and one “five” bead.
This leads to the so-called “bi-quinary” approach:
module BiQuinaryImpl = let biQuinaryDigits place (unit,five,ten) arabic = let digit = arabic % (10*place) / place match digit with | 0 -> "" | 1 -> unit | 2 -> unit + unit | 3 -> unit + unit + unit | 4 -> unit + five // changed to be one less than five | 5 -> five | 6 -> five + unit | 7 -> five + unit + unit | 8 -> five + unit + unit + unit | 9 -> unit + ten // changed to be one less than ten | _ -> failwith "Expected 0-9 only" let arabicToRoman arabic = let units = biQuinaryDigits 1 ("I","V","X") arabic let tens = biQuinaryDigits 10 ("X","L","C") arabic let hundreds = biQuinaryDigits 100 ("C","D","M") arabic let thousands = biQuinaryDigits 1000 ("M","?","?") arabic thousands + hundreds + tens + units
Again, if we test interactively, the results look good.
BiQuinaryImpl.arabicToRoman 1 //=> "I" BiQuinaryImpl.arabicToRoman 9 //=> "IX" BiQuinaryImpl.arabicToRoman 24 //=> "XXIV" BiQuinaryImpl.arabicToRoman 999 //=> "CMXCIX" BiQuinaryImpl.arabicToRoman 1493 //=> "MCDXCIII"
But how can we be sure these implementations are correct for all numbers, not just the ones we tested?
One way to gain confidence is to use the test oracle approach – compare them to each other. It’s a great way to cross-check two different implementations when you’re not sure that either implementation is right!
let biquinary_eq_tally number = let tallyResult = TallyImpl.arabicToRoman number let biquinaryResult = BiQuinaryImpl.arabicToRoman number tallyResult = biquinaryResult
But if we try running this code, we get a
ArgumentException: The input must be non-negative due to the
Check.Quick biquinary_eq_tally // ArgumentException: The input must be non-negative.
So we need to only include inputs that are positive. We also need to exclude numbers that are greater than 4000, say, since the algorithms break down there too.
How can we implement this filter?
We saw in an earlier post that we could use preconditions. But for this example, we’ll try something different and change the generator.
First we’ll define a integer generator (an “arbitrary”) called
arabicNumber which is filtered as we want (if you recall, an “arbitrary” is a combination of a generator algorithm and a shrinker algorithm, as described earlier). We’ll only include numbers from 1 to 4000.
let arabicNumber = Arb.Default.Int32() |> Arb.filter (fun i -> i > 0 && i <= 4000)
Next, we create a new property which is constrained to only use the
arabicNumber generator by using the
let biquinary_eq_tally_withinRange = Prop.forAll arabicNumber biquinary_eq_tally
Now finally, we can do the cross-check test again:
Check.Quick biquinary_eq_tally_withinRange // Ok, passed 100 tests.
And we’re good! Both algorithms work correctly, it seems.
How many roman numbers do we have in total? 4000 we said. So why not test them all?
Let’s run all 4000 numbers through our property, and filter out the ones that succeed, leaving only the ones that fail, like this:
[1..4000] |> List.choose (fun i -> if biquinary_eq_tally i then None else Some i ) // output =>
We would expect there to be no numbers that failed, and the output list to be empty, but actually there is one number in the list: 4000!
If we check the two conversions for 4000 we can see how they differ. The biquinary implementation didn’t know how to handle it. The tally implementation is less brittle and would work for higher numbers without breaking.
TallyImpl.arabicToRoman 4000 //=> "MMMM" BiQuinaryImpl.arabicToRoman 4000 //=> "M?"
Is this something we care about? Maybe not. We might want to restrict the inputs to the implementations to be less than 4000 though. For example, we could alter the tally implementation to return
None like this:
let arabicToRoman arabic = if (arabic <= 0 || arabic >= 4000) then None else (String.replicate arabic "I") .Replace("IIIII","V") .Replace("VV","X") // etc |> Some
Tip: If your domain is small enough, why not check all the values in it?
For an example from a different domain, see the post “There are Only Four Billion Floats – So Test Them All!". It uses a “test oracle” approach to check an implementation against all four billion floats.
This little hiccup is a reminder then, that property-based checking is not a golden hammer. It generates random data, but it is not necessarily the best way of probing a domain at the boundaries. If you do have well known boundaries, it’s best to create some specific tests for them, either using a custom generator for your PBT tool, or by simply doing some explicit example-based tests for the edge cases, like this:
for i in [3999..4001] do if not (biquinary_eq_tally i) then failwithf "test failed for %i" i
Another approach to testing these roman numeral implementations would be to create an inverse function, such that applying the original and then applying the inverse gets you back to the original state.
If we think of the roman numeral conversion as an “encoding”, then we will need to write a corresponding “decoder”. Here’s a very simple one using the same “tally” based approach that we used originally:
let romanToArabic (str:string) = str .Replace("CM","DCD") .Replace("CD","CCCC") .Replace("XC","LXL") .Replace("XL","XXXX") .Replace("IX","VIV") .Replace("IV","IIII") .Replace("M","DD") .Replace("D","CCCCC") .Replace("C","LL") .Replace("L","XXXXX") .Replace("X","VV") .Replace("V","IIIII") .Length
and here’s some ad-hoc tests
TallyDecode.romanToArabic "I" //=> 1 TallyDecode.romanToArabic "IX" //=> 9 TallyDecode.romanToArabic "XXIV" //=> 24 TallyDecode.romanToArabic "CMXCIX" //=> 999 TallyDecode.romanToArabic "MCDXCIII"//=> 1493
With the inverse function now available, we can write a property based test. Note that we’re using the same
arabicNumber generator to limit the inputs:
/// encoding then decoding should return /// the original number let encodeThenDecode_eq_original = // define an inner property let innerProp arabic1 = let arabic2 = arabic1 |> TallyImpl.arabicToRoman // encode |> TallyDecode.romanToArabic // decode // should be same arabic1 = arabic2 Prop.forAll arabicNumber innerProp
And if we run the test, it passes.
Check.Quick encodeThenDecode_eq_original // Ok, passed 100 tests.
We can also check the biquinary encoding against the tally-based decoding
/// encoding then decoding should return /// the original number let encodeThenDecode_eq_original2 = // define an inner property let innerProp arabic1 = let arabic2 = arabic1 |> BiQuinaryImpl.arabicToRoman // encode |> TallyDecode.romanToArabic // decode // should be same arabic1 = arabic2 Prop.forAll arabicNumber innerProp
And if we run the test, it passes again.
Check.Quick encodeThenDecode_eq_original2 // Ok, passed 100 tests.
Yet another way of checking that the implementation is correct is to break it down into smaller components and check that the smaller components are correct.
For example, if the we break the Arabic number into 1000s, 100s, 10s, and units, and then encode them separately, the concatenation of these components should be the same as the original Arabic number encoded directly.
let recursive_prop = // define an inner property let innerProp arabic = let thousands = (arabic / 1000 % 10) * 1000 |> BiQuinaryImpl.arabicToRoman let hundreds = (arabic / 100 % 10) * 100 |> BiQuinaryImpl.arabicToRoman let tens = (arabic / 10 % 10) * 10 |> BiQuinaryImpl.arabicToRoman let units = arabic % 10 |> BiQuinaryImpl.arabicToRoman let direct = arabic |> BiQuinaryImpl.arabicToRoman // should be same direct = thousands+hundreds+tens+units Prop.forAll arabicNumber innerProp
And, it works great!
Check.Quick recursive_prop // Ok, passed 100 tests.
As we mentioned last time, invariants are often a great way to beat the EDFH. On their own, they can not prove that an implementation is correct, but in conjunction with other properties they are a very powerful tool.
So, what invariants are there for a roman numeral encoding?
Well, there are no obvious things which are preserved between the input and the output of the encoding, such as the length of a string, or the contents of a collection.
However, there are some properties which are preserved when a roman numeral is created from another roman numeral using arithmetic. For example, there are never more than three “I"s, or one “V”, etc. We can test these invariants easily.
First, we need a function to count the number of occurrences of a pattern:
let matchesFor pattern input = System.Text.RegularExpressions.Regex.Matches(input,pattern).Count (* "MMMCXCVIII" |> matchesFor "I" //=> 3 "MMMCXCVIII" |> matchesFor "XC" //=> 1 "MMMCXCVIII" |> matchesFor "C" //=> 2 "MMMCXCVIII" |> matchesFor "M" //=> 3 *)
and then we can define our property using those invariants:
let invariant_prop = let maxMatchesFor pattern n input = (matchesFor pattern input) <= n // define an inner property let innerProp arabic = let roman = arabic |> TallyImpl.arabicToRoman (roman |> maxMatchesFor "I" 3) && (roman |> maxMatchesFor "V" 1) && (roman |> maxMatchesFor "X" 4) && (roman |> maxMatchesFor "L" 1) && (roman |> maxMatchesFor "C" 4) // etc Prop.forAll arabicNumber innerProp
And if we test it, it works as expected.
Check.Quick invariant_prop // Ok, passed 100 tests.
How far can we go with these patterns? We haven’t done the commutive diagram approach yet – can we apply that to the roman numeral encoding as well?
Yes, we can. It’s not really the most appropriate way to test this, but it’s a fun game to see what you can come up with!
How about this for the two paths:
- Path 1
- given a number less than 400
- encode it first
- then replace “C” with “M”, “X” with “C”, and so on, to create a new roman numeral.
- Path 2
- given the same number
- multiply it by 10 first
- then encode it
The two paths should give the same result.
/// Encoding a number less than 400 and then replacing /// all the characters with the corresponding 10x higher one /// should be the same as encoding the 10x number directly. let commutative_prop1 = // define an inner property let innerProp arabic = // take the part < 1000 let arabic = arabic % 1000 // encode it let result1 = (TallyImpl.arabicToRoman arabic) .Replace("C","M") .Replace("L","D") .Replace("X","C") .Replace("V","L") .Replace("I","X") // encode the 10x number let result2 = TallyImpl.arabicToRoman (arabic * 10) // should be same result1 = result2 Prop.forAll arabicNumber innerProp
And, amazingly, it works!
Check.Quick commutative_prop1 // Ok, passed 100 tests.
What about the converse:
- Path 1
- given a number less than 4000
- encode it first
- then replace “C” with “X”, “M” with “C”, and so on, to create a new roman numeral.
- Path 2
- given the same number
- divide it by 10 first
- then encode it
/// Encoding a number and then replacing all the characters with /// the corresponding 10x lower one should be the same as /// encoding the 10x lower number directly. let commutative_prop2 = // define an inner property let innerProp arabic = // encode it let result1 = (TallyImpl.arabicToRoman arabic) .Replace("I","") .Replace("V","") .Replace("X","I") .Replace("L","V") .Replace("C","X") .Replace("D","L") .Replace("M","C") // encode the 10x lower number let result2 = TallyImpl.arabicToRoman (arabic / 10) // should be same result1 = result2 Prop.forAll arabicNumber innerProp
This time, it fails, with “9” as a counter-example. Why is this?
Check.Quick commutative_prop2 // Falsifiable, after 9 tests // 9
That can be an exercise for the reader!
I hope you get the idea now. By trying out the various approaches (oracles, inverses, invariants, commutative diagrams, etc.) we can almost always come up with useful properties for our design.
With that, we have come to the end of the various property categories. We’ll go over them one more time in a minute – but first, an interlude.
If you sometimes feel that trying to find properties is a mental challenge, you’re not alone. Would it help to pretend that it is a game?
As it happens, there is a game based on finding properties.
It’s called Zendo and it involves placing sets of objects (such as plastic pyramids) on a table, such that each layout conforms to a pattern – a rule – or as we would now say, a property!.
The other players then have to guess what the rule (property) is, based on what they can see.
Here’s a picture of a Zendo game in progress:
The white stones mean the property has been satisfied, while black stones mean failure. Can you guess the rule here? I’m going to guess that it’s something like “a set must have a yellow pyramid that’s not touching the ground”.
Alright, I suppose Zendo wasn’t really inspired by property-based testing, but it is a fun game, and it has even been known to make an appearance at programming conferences.
If you want to learn more about Zendo, the rules are here.
In the next post we will keep going and use the same techniques to test a classic TDD example.
Source code used in this post is available here.


“Dockside Math”, A2+B2=C2
Tying up the boat using “Dockside Math”, A2+B2=C2
When tying up along side a dock, you will more than likely hang a fender over the side at amidships up against a dock post. You will want to keep the boat in place on the dock post, yet allow for the tidal drop. If you try to do this with bow and stern lines, and add enough slack for tidal swings, the fender won’t stay in place on the dock post. It will move one way or the other resulting in scratched up topsides.
The solution is spring lines, one going forward, and one going aft. These springs can be made fairly snug if they are long enough, and still allow for a tidal drop. The lines should be fastened to the dock near the ends of the boat, and can be lead back to a cleat, or winch on the other end of the boat. Snug these up against each other. Adjust the bow and stern lines so that the ends of the boat can’t touch the dock. Leave enough slack in these to account for the tidal drop. Consider running the stern line to the side of the stern that is not up against the dock. The rule of thumb is that the longer the lines are, the less slack you’ll need in them.
Do you remember the Pythagorean Theorem from your eight grade math class, A2+b2=c2? I bet you thought you’d never have use for it, and it was a waste of brain space when there were so many other things to learn, like how to shoot toothpicks from a straw, or how to stick a pencil in the ceiling tile, or how to ask a girl out without really asking her out. Well, I finally found a use for it in my line of work, which is sailing. My math teacher, Mr. Humphreys would be so proud of me if he knew.
For those captains out there that want the math, here is a calculation for a 35 foot spring line and a 3 foot tidal drop. Where A is the length of the spring line from the dock cleat to your boat cleat, B is the tidal drop, and the resulting C will be the length the line needs to be to allow enough slack for the tidal drop. A2+b2=c2, 352 x 32 =1234, the square root of 1234 is 35.128 feet. So as you can see, for a three foot drop you’d need only two inches of slack in the spring line. This is probably a lot less slack than you would have thought. Line will stretch too, so you will even have a fudge factor built in. This will keep the boat squarely on her fender.
Using this formula for a three foot tide assumes that your dock lines are fairly horizontal. If your lines are exactly horizontal at mid tide, then the 3 foot drop is really a 1 ½ foot swing in either direction, and therefore, the lines can be even more snug, with only one inch of slack. If you are tying your dock lines up near the top of the dock pilings, which is not entirely best, then this formula will not work for you. You will have to have much more slack in the dock lines.
Now do the math for the stern line, it’s only 10 feet long. 102 x 32 = 109, the square root of 109 is 10.44, or 10 feet, 5 inches. Again, a lot less slack than you might have though. Use this thinking to keep your boat out of your neighbor’s slip too. I’ve seen a lot of boats in their slips with very loose dock lines, particularly those boats with their lines tied high on the pilings.


allows you to evaluate mathematical expressions and convert numbers
to various numerical bases. Note that all numbers are treated as floating
points numbers (doubles). The syntax and operators are very similar
to C and Java - some small derivations exist, but they are explicitly
mentioned in this manual.
2.1 Addition & Subtraction
is respresented by the plus operator + and subtraction
is represented by the minus operator -.
Example: 3.2 + 10
- 0.9 results in 12.3.
Multiplication & Division
is respresented by the * operator and division is represented
by the / operator.
Example: 2 * 2*3
/ 5 results in 2.4.
modulo operation is respresented by the % operator.
It calculates the remainder of division of one number by another.
Example: 23 % 5
results in 3.
exponentiation (or power) operation is respresented by the ^
Example: (30/3) ^
(2+1) results in 1000.
Binary Left/Right Shift
binary left shift operation is respresented by the <<
operator and the binary right shift operation is represented by the
>> operator. The bit pattern is given by the
left-hand operand, and the number of positions to shift by the right-hand
operand. In fact 'x << y' is the same as x * (2^y) and 'x >>
y' is the same as x / (2^y). Note that the operands are converted to
Example: 1 <<
4 results in 16.
Example: 34.2 >>
2 results in 8.0.
AND, OR, XOR & NOT Operators
bitwise AND operation is represented by the & operator.
The bitwise Inclusive OR operation is represented by the |
operator. The bitwise Exclusive OR (XOR) operation is represented by
the # operator. The bitwise COMPLEMENT operation is
represented by the ! operator.
Note that the operands are converted to integers except the !
operator - that's a difference to C or Java. The !
operator converts bit by bit, also floating-point numbers without converting
them to integers.
Another difference to C/Java is the XOR operator which is represented
by the # operator and not by the ^
operator which is here used for exponentiation.
Example: 29 &
23 results in 21.0.
| 8 results in 12.0.
4.33 # 5.1 results in 1.0.
results in 17.25.
available operaters, shown in order of presedence - from highest to
3.1 Square Root (sqrt)
the correctly rounded positive square root of the given value.
results in 6.081118318204308.
the trigonometric sine of an angle in radians.
results in 0.9320390859672263.
the trigonometric cosine of an angle in radians.
results in 1.0.
the trigonometric tangent of an angle in radians.
results in 0.0.
Natural Logarithm (log)
Returns the natural logarithm (base e) of the given value.
results in 1.3132616875182228.
Base 10 Logarithm (log10)
the base 10 logarithm of the given value.
results in 2.0.
Arcus Sinus (asin)
the arc sine of an angle, in the range of -pi/2 through pi/2.
results in 1.5707963267948966.
Arcus Cosinus (acos)
the arc cosine of an angle, in the range of 0.0 through pi.
results in 3.141592653589793.
Arcus Tangent (atan)
the arc tangent of an angle, in the range of -pi/2 through pi/2.
results in 1.5607966601082315.
Hyperbolic Sinus (sinh)
the hyperbolic sine of the given value. The hyperbolic sine of x is
defined to be (ex - e-x)/2 where e is Euler's
results in 27.28991719712775.
the hyperbolic cosine of the given value. The hyperbolic cosine of x
is defined to be (ex + e-x)/2 where e is Euler's
results in 1.543080634815244.
the hyperbolic tangent of the given value. The hyperbolic tangent of
x is defined to be sinh(x)/cosh(x). Note that the absolute value of
the exact tanh is always less than 1.
results in 0.9640275800758169.
Converting Radians to Degress
an angle measured in radians to an approximately equivalent angle measured
results in 180.0.
Converting Degrees to Radians
Converts an angle measured in degrees to an approximately equivalent
angle measured in radians.
results in 6.283185307179586.
Rounds the given value to the closest integer.
results in 12.0.
Returns the largest integer which is less or equal to the given value.
results in 11.0.
constants are recognized:
pi : equals 3.141592653589793.
e : Euler's Number, equals
Specify a number in a different base
You can specify a number in an arbitrary base (between 2 and 36) by
appending an '@' followed by the base.
Example: 1101@2 means
to interpret 1101 in base 2 which results in 13.0.
Of course, you can uese such expressions everywhere where you are able
to use base10 numbers:
Example: 11@2 * ff@16
+ a@17 results in 775.0.
Specify the result in a different
If you want to receive the result also in another base than base 10,
you can check the checkbox under the expression field and select your
additional desired base from the dropdown list.
About / Info
was originally coded for personal reasons cause I always forgot to take
along my hand calculator and the windows/linux calculators do not satisfy
my needs nor are they comfortable to use. After finishing coding I decided
to make it public in case someone else finds it useful.
I am gonna to release the full source code somewhere in future, but
not before all bugs are removed - therefore I need your help:
find any bugs, have suggestions or commens - please mail me:
July 2008: Version 1.0
* Initial release
5th September 2008: Version
two small bugs in the parser
* added the autocompletion feature
December 2008: Version
::Download:: (120 kb) - applet, source, manual


As a demonstration of the proposed benchmarking environment, some basic experimental results have been reported in the paper Benchmarking discrete optimization heuristics with IOHprofiler. The implementation of algorithms can be found in the repository IOHalgorithm.
A detailed description of the algorithms follows. An operator frequently used in these descriptions is the mutation operator, which flips the entries of pairwise different, uniformly at random chosen bit positions, see Algorithm 1.
Since we want to avoid useless evaluations of offspring that are identical to their parents, we frequently make use of the conditional binomial distribution , which assigns probability to each positive integer , and probability zero to all other values. Sampling from is identical to sampling from until a positive value is returned (“resampling strateg”).
Greedy Hill Climber
The greedy hill climber (gHC, Algorithm 2) uses a deterministic mutation strength, and flips one bit in each iteration, going through the bit string from left to right, until being stuck in a local optimum, see Algorithm 2.
Randomized Local Search
RLS uses a deterministic mutation strength, and flips one randomly chosen bit in each iteration, see Algorithm 3.
The EA with Static Mutation Rate
The EA is defined via the pseudo-code in Algorithm 4.
Fast Genetic Algorithm
The fast Genetic Algorithm (fGA) chooses the mutation length according to a power-law distribution , which assigns to each integer a probability of , where . We use this algorithm with . The fGA was introduced in the paper Fast genetic algorithms.
The Two-Rate EA
The two-rate was introduced in the paper The Evolutionary Algorithm with Self-Adjusting Mutation Rate. It uses two mutation rates in each iteration: half of offspring are generated with mutation rate , and the other offspring are sampled with mutation rate . The parameter is updated after each iteration, from a biased random decision favoring the value from which the best of all offspring has been sampled.
The EA with normalized standard bit mutation
The , Algorithm 7, samples the mutation strength from a normal distribution with mean and variance , which is identical to the variance of the binomial distribution used in standard bit mutation. The parameter is updated after each iteration, in a similar fashion as in the 2-rate EA, Algorithm 6. The was introduced in the paper Interpolating Local and Global Search by Controlling the Variance of Standard Bit Mutation.
The EA with normalized standard bit mutation and controlled variance
The , Algorithm 8 builds on the but uses, in addition to the adaptive choice of the mutation rate, an adaptive choice of the variance.
The EA with log-Normal self-adaptation on mutation rate
The , Algorithm 9, uses a self-adaptive choice of the mutation rate.
The self-adjusting , Algorithm 10, was introduced in the paper From black-box complexity to designing new genetic algorithms. The offspring population size is updated after each iteration, depending on whether or not an improving offspring could be generated. Since both the mutation rate and the crossover bias (see Algorithm 11 for the definition of the biased crossover operator cross) depend on , these two parameters also change during the run of the . In our implementation we use update strenght .
The “Vanilla” GA
The vanilla GA (vGA, Algorithm 13) constitutes a textbook realization of the so-called Traditional GA. The algorithm holds a parental population of size . It employs the Roulette-Wheel-Selection (RWS, that is, probabilistic fitness-proportionate selection which permits an individual to appear multiple times) as the sexual selection operator to form pairs of individuals that generate the offspring population. 1-point crossover (Algorithm 12) is applied to every pair with a fixed probability of . A mutation operator is then applied to every individual, flipping every bit with a fixed probability of . This completes a single cycle.
The Univariate Marginal Distribution Algorithm
The univariate marginal distribution algorithm (UMDA, Algorithm 14) is one of the simplest representatives of the family of so-called estimation of distribution algorithms (EDAs). The algorithm maintains a population of size and uses the best of these to estimate the marginal distribution of each decision variable, by simply counting the relative frequency of ones in the corresponding position. These frequencies are capped at and , respectively. In the -th iteration, a new population is created by sampling from these marginal distributions. The UMDA was introduced in The Equation for Response to Selection and Its Use for Prediction.


Java Sudoku is a Sudoku game entirely made in Java. Thus, it is cross-platform, meaning that it can be run on any operating system that supports Java. The application doesn't come with a fixed number of puzzles for you to solve. Instead, it can generate them in different levels of difficulty. Java Sudoku can also solve the puzzles for you. It does also have the ability of giving you hints while you are solving the puzzles.
Java Sudoku allows you to input information into the Sudoku Puzzle quite easily. At the bottom of the screen there are 10 buttons. 1 for each number from 1 to 9 and one that reads "delete". Whenever you want to input a number, you simply have to click on the number at the bottom of the screen and then click on the cell that you want to put the number into.
I played a few rounds of Sudoku using Java Sudoku, and I found the experience quite pleasing. It was easy to appreciate the whole panorama and inputting numbers into the puzzle was very easy. The only feature that I'd love to see in this game is a way to easily highlight numbers on the screen. Other games allow you to double-click a number, and they show every number on every different grid. This way, it is way easier to find numbers that are repeated, for example.
All in all, this game is guaranteed entertainment for those who like Sudoku.


I decided to create the Toronto Raptors logo using graphs on Desmos. I decided to create this logo because it involved circle equations, quadratic, square root, linear, and trig functions. I used circle equations to create the outside of the ball, as well as the lower vertical curves on the basketball. I changed the radius and the multiplier on the x/y variables to match the shape. I used quadratic equations to create the horizontal and vertical-middle basketball curves because they fit the small curve shape. I compressed, vertically and horizontally translated, and reflected the graph. I used square root functions to create the vertical-upper basketball curves because it fit the medium curves. I horizontally and vertically translated the graph. I used a trig function for one of the horizontal-middle curves because the sin function matched the light curve of the ball. I also needed a new equation. Finally, I used linear equations for the claw marks on the ball. The claws required a series of straight lines, hence why I used a linear equation. The major challenge I encountered was completing the claw marks. There were so many of them and it took me a long time to complete them. Copying and pasting equations helped me speed it up. I asked other students for input on how to manipulate my graph to make it match. My strategy at the end was to use the sliders to find the values and input them into the variables. I didn’t use this strategy at the beginning, so I had to change all my equations and delete the sliders at the end. Through this project I learned how transform functions with translations, expressions/compressions, and reflections through variables in equations.


All cargo airlines and even passenger airlines base the weight of an air freight on what they call a chargeable weight, whether is a volumetric weight or a physical weight. Every gram has an obvious impact on the air shipment price they charge.
First of all, in order to calculate how much weight you are going to pay, you need to have measurement of package of goods and also weighing it. Find out how this weight is calculated and what needs to be considered to get the shipment cost.
What does it mean by a chargeable weight?
All airlines charge according, to a weight of a air shipment received. You may question, what do you mean by that? The chargeable weight is the actual gross physical weight or the volumetric weight of the commodity, whichever is greater is what they go by.
For instance, a large box filled with styrofoam its weight is light ,yet it takes up more space on an aircraft, than a small box filled with heavy items.
Calculation of a Volumetric Weight and Physical Weight
In order to find the volume weight of an item, first you need to measure the package.
By getting the length, the width and the high and then by weighing physically the same package on a scale.
For instance you have a package which dimensions are: 48 inches (length) x 40 (width) inches x 68 inches (high) and the scale physical weight is 1,200 lbs (pounds).
When you multiply 48 x 40 x 68 = 130,560, then divided by 166 = 786 lbs. This is the volume weight found on this item. Therefore as you can see the greater weight used to calculate the air shipment cost obviously is 1,200 lbs. which is the physical weight
Now let’s say we have a package which dimensions are: 160 inches x 25 inches x 10 inches = 40,000, then divided by 166 = 240 lbs (this is the volume weight) and the physical weight is only 80 lbs. The chargeable weight is obviously 240 lbs. which is greater.
What if we have 5 packages with the same measurement, what we have to do is multiply 160 x 25 x 10 and result multiply by 5 and then divided by 166 = 1,204 lbs. and the physical weight is 80 lbs each, therefore 5 x 80 lbs = 400 lbs. in this case the chargeable weight is 1,204 lbs.
The formula for calculating the dimensional weight for all kind of products is 166 cubic inches per pound. (6000 cubic centimeters per kilogram or 366 cubic inches per kilogram). Multiply the length by the width by the height to obtain the cubic inches (cm).
If you are shipping more than 1 piece, you may multiple the outcome of the formula by the number of pieces in your air shipment.
If the shipment is given in centimeter, then the formula to be used is : Length x Width x Height divided by 6000 . You will get the result in kilos.
• To obtain the dimensional weight in pounds, divide the cubic inch result by 166.
• To obtain the dimensional weight in kilograms, divide the cubic centimeter result by 6000. If the dimensional weight of an item is greater than the actual weight, you will use it to calculate the transportation charges.


Use the IFERROR function to trap and handle errors produced by other formulas or functions. IFERROR checks for the following errors: #N/A, #VALUE!, #REF!, #DIV/0!, #NUM!, #NAME?, or #NULL!.
For example, if A1 contains 10, B1 is blank, and C1 contains the formula =A1/B1, the following formula will trap the #DIV/0! error that results from dividing A1 by B1:
=IFERROR (A1/B1. "Please enter a value in B1")
In this case, C1 will display the message "Please enter a value in B1" if B1 is blank or zero.


Why bother teaching maths to boys?
As a maths teacher, one of my biggest challenges is helping teenage boys find real-life connections for the maths that they learn at school. Today’s adolescent boy lives in world where we pay with plastic, not cash. Where Siri can work out percentages in seconds. If he doesn’t know the answer to a problem, he just Googles it. Why would he bother learning maths?
Many of my students have confessed that they only study maths is because it’s a core requirement for many university courses (and perhaps because their Gen X parents, and teachers, keep banging on about how important it is).
Here’s why I believe maths is important to the men – and women – of tomorrow (and it’s not just because I’m a maths teacher – honest…)
Maths builds financial savvy and independence
Maths builds on our numeracy skills – the ability to understand and work with numbers. Why is numeracy important? In its simplest terms, numeracy plays an important role in developing the life skills needed for an independent existence. As much as we love to nurture our boys, want them to grow into young men who can live an independent life. Without basic maths skills this is going to be tricky. Numeracy skills are required in order to follow a recipe, evaluate whether or not a sale item is a good deal, and manage a budget. No one wants a grown up son who runs home to mum and dad every time he’s called on to cook, make a purchase or save for the future.
“In maths, students are required to explain how they arrived at a solution to a complex problem … an extremely transferrable skill whatever your son decides he wants to do.”Katie White, Maths teacher
Maths develops logic and critical thinking
Maths teaches boys logic and order. A mathematical equation is likely to have a predictable outcome, and precise steps must be followed in order to attain that result. The ‘discipline of mind’ boys develop in math class will stand them in good stead in all areas of their life – including money, career, parenthood and relationships. Maths can also provide a vehicle through which critical-thinking skills are put into practice and refined. In maths, students are required to explain how they arrived at a solution to a complex problem or to describe the ideas behind a formula or procedure – an extremely transferrable skill whatever your son decides he wants to do for work in the future.
The jobs of today need maths
Many jobs require maths skills. There are the obvious ones: builders, architects, engineers, scientists and doctors; however some examples of other types of jobs that also require maths skills include:
- Cartoon animators use linear algebra and geometry to show the way that an object is rotated and shifted.
- Special effects directors use powerful computing equipment, numerical methods and algorithms to make most of the spectacular feats in the visual effects industry.
- Computer scientists implement computing systems in hardware and software.
- Computer games designers require game theory, a branch of applied mathematics.
- Forensic scientists use mathematics principles to figure out the location of items, when blood was shed and even the type of impact that caused injury.
The jobs of tomorrow will (likely) need maths
I don’t have a crystal ball and many of the jobs and hobbies people do today didn’t exist a generation ago, so I can only make an educated guess at the skills and knowledge that will be required for the future.
However, based on current trends we do know that to remain competitive in the workforce your son will need to be able to be creative, innovative, knowledgeable, and be able to solve complex problems. Recent studies, including a 2016 report by the University of Sydney, also suggest that around 75% of Australia’s fastest growing industries require science, technology, engineering and maths skills.
So next time your son is sitting in class wondering why he needs to learn about dividing decimals or fractions, or how y=mx+c relates to linear graphs, I hope he chooses to develop his numeracy skills for the benefit of his future, and for the not yet existent, exciting, world-changing careers that are still to come. Maths is one of the most powerful tools he can get his hands on in preparation.Katie White is a Year 7 & 8 mathematics teacher at Brighton Grammar, an all-boys school in Melbourne. This article is about Education
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Kearns Mathematics/Statistics Scholarship
1. Applicant must be a Mathematics Major. Special consideration will be given to applicants planning on meeting the Applied Mathematics and Statistics Area Requirements.
2. Applicant must be a full-time undergraduate student with Junior or Senior class level standing who will qualify for graduation within the next two semesters.
3. Applicant must have a minimum 3.0 cumulative GPA.
4. Applicant must be a U.S. citizen.
5. Applicant must be an active participant in community and/or campus student activities.
6. Applicant must require financial assistance in order to continue studies at CSUS.
No discrimination shall be made on the basis of sex or ethnicity.


CA foundation students need to do registration four months before to examination. For the May month examination, the students need to do the registration in January month. Wishing to complete the CA foundation syllabus in 30 days is a difficult one without efficient coaching. This blog is to make students know how to prepare for CA foundation in 4 months of study, a one-month crash course and six months of complete course for the CA foundation.
Uniqueness between 12th standard and CA foundation
Accounting, Business law, business mathematics, and business economics are the four papers in the CA foundation. Business law and business economics are theory papers that students can win over with sufficient time allocation. Accounts and mathematics are practical subjects that need more understanding. The study plan for the CA foundation exam 2021 is to test the individual intelligence of the students.
In accountancy, students learn about the accounting process, inventories, BRS, depreciation, Final accounts, and financial statements of not for profit organizations, partnership accounts, company accounts, and accounting for special transactions. !2th standard accountancy has partnership firms, Companies, financial statement analysis, and cash flow statements. Here the advanced level understanding of financial statements is final accounts, inventory and depreciation. The cash flow statement expresses the effect of income on the balance sheet. Anyways, BRS, and accounting for special transactions are new chapters with paramount importance in the advanced level subjects. CA foundation takes the student to the advanced level of understanding.
In mathematics, the syllabus for CA foundation goes as Ratio and proportion, logarithms, indices, linear inequalities, equations and matrices, time value of money, sequence and series – arithmetic and geometric progressions, basic concepts of permutations and combinations, sets, functions and relations, and basic concepts of differential and integral calculus. The syllabus of the 12th standard are as follows: Number series, coding, decoding and Odd Man Out, seating arrangements, direction tests, blood relations, syllogism, measures of central tendency and dispersion, statistical description of data, probability, correlation and regression, theoretical distributions, index number and time series. Relations and functions, calculus, algebra, vectors and 3-D Geometry, probability and linear programming are the topics covered in the 12th standard syllabus.
In mathematics, the 12th syllabus of statistical description of data, probability, linear programming helps for understanding ratio and proportion, linear inequalities and permutation and combinations. The 12th syllabus provides only the fundamental knowledge, but the CA foundation provides the advanced level of knowledge.
Business law in 12th standard and CA foundation are different. In 12th standard, students learn about the fundamentals of business, types of business, business services, social responsibility and business ethics, emerging modes of business, sources of business finance, international trade and internal trade. In the case of the CA foundation, it is not easy to remember different acts. The Indian contract act, 1872, the sale of goods act, 1930, companies act, 2013, the limited liability partnership act 2008, and the Indian partnership act, 1932 are explained with detailed interpretation. In economics, the chapters demand, supply, price determinants, and government budget are the same for the 12th standard and CA foundation. There is not much difference in the economics subject for CA foundation and 12th standard.
CA foundation crash course:
Students who wish to study for four months can opt for the crash course. Students can do the registration in January month and write the examination in May 2022. One month of crash course helps learn the uniqueness of the syllabus. As the course tenure is for one month, students receive recorded videos for important topics, PDF materials, and notes for the difficult subject. The mock test series for the December examination was listed on the ICAI website. The crash course provides practice on the mock test question paper. The faculty members have personal care with the students, and the doubts are cleared then and there. Two hours of coaching and two hours of study time is essential for one month of revision. Students need to complete the practical subjects first and then move on to the theoretical subjects.
CA foundation complete course:
Daily classes of four hours per day will give 4800 hours. This time dedication is enough for preparing CA foundation exam. On daily basis, two hours of coaching and two hours of study time is essential to learn the subject intensively. In the case of business law and economics, reading newspapers and magazines give the required knowledge to the students. RR academy faculties suggest the Best timetable for CA foundation students. It is helpful for fundamental knowledge.
CA course has an educational ladder that requires hard work. Students need to take initial steps to complete the foundation course on the first attempt. For this reason, the decision about the coaching through a crash course or complete course is essential for a student applying for the examination. If the student is above average and finds the subjects as easy then he can take up the one month of a crash course. After registration, the institute supplies the study material that helps for independent learning. Towards the end, the student brushes up the knowledge with the crash course. Faculty members help the students with domain-specific doubts during the crash course. A complete course of six months is beneficial to the 12th standard marks and CA foundation marks. Place enquire to RR academy to know the details of the options for the CA foundation.


Karatsuba Algorithm for Multiplying Two Integers
All of us learned how to multiply two numbers when we were kids. In case we have forgotten (😄), let’s look at how to multiply two numbers 19 and 95.
Multiplying 19 and 95 requires 4 (2²) elementary operations as shown above. Similarly multiplying 123 and 456 would require 9 (3²) elementary operations. Thus, if we generalize, given two n-digit numbers, this traditional approach requires n² elementary multiplication operations. Therefore, the time complexity of this naive approach is O(n²).
Is this the only possible way to multiply two numbers? 🤔
In 1960, a Russian mathematician named Anatoly Alexeyevich Karatsuba discovered a new algorithm for multiplying two numbers. Let us look at how this algorithm works.
The Karatsuba Algorithm
Let’s consider two 4-digit numbers x and y where x=1234 and y=5678. First of all, we should divide the n-digit numbers into n/2-digit numbers as shown below.
a and c represent the first n/2 digits of x and y. Similarly, b and d represent the last n/2 digits of x and y
The Karatsuba algorithm involves 4 main steps.
Finally, multiply the output of step 1 by 10ⁿ, the output of step 4 by 10^(n/2), and add them both with the output of step 2 (as shown below).
If you multiply 1234 and 5678, the answer would be 7006652. Seems pretty easy right!!!
But, how does this work? 🤔
Let us break this down. As abovementioned, let x = 1234 and y = 5678. We also have defined a = 12, b = 34, c = 56 and d = 78.
If we write x and y in terms of a, b, c and d:
Now, if we multiply x and y:
We have computed ac in step 1 and bd in step 2. In step 3, we computed (a+b).(c+d) = ac + ad + bc + bd. In order to get (ad + bc), we should subtract ac and bd. That is what we are doing in step 4. If we summarize,
Now, all the 4 steps that we did earlier make sense.
Python Implementation of Karatsuba Algorithm
This implementation utilizes recursion. Steps 1, 2, and 3 are performed in lines 21, 22, and 23. Finally, the output of step 4 is returned in line 25. The recursive calls should have a termination condition. Here, when the values of a, b, c, and d become single digits (0–9), recursion terminates (lines 8 and 9).
One interesting question to ask at this point would be:
We need to calculate ac, ad, bc, and bd. ac and bd are calculated in steps 1 and 2. We could have calculated ad and bc in two additional steps as well. But instead, we multiply (a+b).(c+d) and then subtract ac and bd. Why? 🤔
The reason is to reduce the number of recursive calls. The current implementation only requires three recursive calls. If we calculate ad and bc separately, the number of recursive would become 4. Keeping the number of recursive calls at a minimum would provide better performance.
How is Karatsuba Algorithm Better than the Traditional Approach?
Earlier, we saw that the traditional approach has the complexity of O(n²). Now, let us try to figure out the time complexity of the Karatsuba algorithm.
Given 2 n-digit numbers, the algorithm recurses three times on n/2-digit numbers. Hence, the complexity can be expressed as:
If we solve this, the time complexity would be:
log₂³ is approximately equal to 1.585 < 2. Therefore, for sufficiently large n, the Karatsuba algorithm will perform better (less running time) compared to the traditional naive approach.
Hope you enjoyed the article !!!


In the previous article we learned about spaces and how to position and orient objects in world space by applying transformation matrices on them. We also learned about camera space, that is simply another coordinate system within the world space.
Recall that during rendering, a vertex first gets mapped from world space into camera space and then projected onto the 2D view plane using a projection matrix (roughly speaking). So in this post, we deal with the question how to set up the camera space by positioning and orienting the camera and how to derive a matrix from it so that we can map a vertex from world space into camera space. In OpenGL, this matrix (called the "view matrix") plays a big role and requires to be specified in every OpenGL program.
The coordinate system of the camera, as discussed in the previous section, is spaned by three orthonormal vectors . The position of the camera is defined by its focal point (or eye) named in world space. Note that is also the origin of the camera coordinate system. By convention the camera looks into a direction , which is most often calculated by subtracting a "look-at" point from the focal point
Now we compute and with help of an "up-vector", (0, 0, 1), that basically points upwards. With the help of the cross-product we can now compute the vectors
that spans our new orthonormal camera coordinate system.
Now that we set up the camera space, we need to construct a matrix that maps from world space into camera space. More concretely, to map a given vertex from world space to camera space, we apply the following two steps:
- translate with respect to the camera position, and then
- map the translated point into the coordinate system .
These two steps will later be combined into one matrix which are then together called camera transformation.
The translation part is fairly easy. With our given camera position , we use a translation matrix to move relative to the camera position
Ok. Now for the mapping part, we have two options how to proceed: Either we try to set up a rotation matrix that rotates the vertex into place in camera space. This would require to determine the angles between the axis coordinates so that we can use them to rotate the dimensions of the point. Or, we make use of a wonderful trick that is applicable when dealing with orthonormal coordinate systems.
We are going to do the latter.
Let us quickly review definition of the dot product, which says that for two given vectors , , where the dot product yields
and is the angle between the two vectors. The dot product basically computes the scaling factor of both and to the point where is orthogonally projected on (and vice versa). This may be a little confusing which is why I tried to elaborate on the dot product a bit deeper in the math appendix.
Now, , and are orthogonal to each other, meaning that
holds true, so that they span a so-called orthonormal basis in world space. This allows to set up a mapping matrix that "rotates" a point from world space into camera space by simply computing the dot product between the point and each coordinate axis vector
Now that we have both transformations ready, we are able to multiply them into one matrix . This is called the view matrix.
The result of applying the view matrix on in world space is a new set of coordinates in camera space. It is important to understand that the coordinates are just scaling coefficients for the axes which itself lay in world space. The linear combination describes the position of the point in camera space. But the point has not been "moved", its position is just now been described relative to the origin and orientation of the camera space.
Let us apply the above example in code. I make use of my own vector and matrix implementation which are very similar to those other countless implementations that can be found on the web.
Note that I start using heterogenious coordinates right from the beginning.
// the position of the camera, called 'eye' Vector3f c = new Vector3f(5, -5, 8); Vector3f u = new Vector3f(); Vector3f v = new Vector3f(); Vector3f w = new Vector3f(); // compute "negative" look direction by substracting // c from the look-at point (3,4,0) w.subAndAssign(c, new Vector3f(3, 4, 0)); // w = c - (3,4,0) w.normalize(); // compute cross product u.crossAndAssign(new Vector3f(0, 0, 1), w); // side = (0,0,1) x w u.normalize(); v.crossAndAssign(w, u); // up = side x look v.normalize(); Matrix4f rotation = new Matrix4f(); // identity rotation.setIdentity(); // note the format: set(COLUMN, ROW, value) // it may be different for your matrix implementation rotation.set(0, 0, u.x); rotation.set(1, 0, u.y); rotation.set(2, 0, u.z); rotation.set(0, 1, v.x); rotation.set(1, 1, v.y); rotation.set(2, 1, v.z); rotation.set(0, 2, w.x); rotation.set(1, 2, w.y); rotation.set(2, 2, w.z); Matrix4f translation = new Matrix4f(); // identity translation.set(3, 0, -c.x); translation.set(3, 1, -c.y); translation.set(3, 2, -c.z); // view matrix Matrix4f view = new Matrix4f(); view.multAndAssign(rotation, translation); // view = rotation * translation // print matrix on console view.print();
At the end of the code snipped we print out the matrix on the console. This is what it says.
0.9761871 0.21693046 0.0 -3.7962832 -0.1421731 0.6397789 0.75529456 -2.1325965 0.16384639 -0.73730874 0.65538555 -9.748859 0.0 0.0 0.0 1.0
If you are building your own Matrix class, I recommend to incorporate a lookAt method that sets up the matrix. Also, I recommend creating a camera object that caches the , and vectors and provides general heper methods to deal with positioning, pitch, yaw, roll and other stuff that you may need (eg. a matrix stack).


Pi Day: Why are we transfixed by the delicious mystery of pi? Pi is more than just a number. So it's only appropriate that Pi Day -- today, March 14 -- celebrates more than just 3.14: Pi is like a beguiling lover who can never be truly known. It's a mathematical constant. And yet, it's irrational. Pi transcends reason as it skips off into infinity. Pi Day sliced 3.14 ways Today marks the mathematically quirky Pi Day, celebrated because the month and day of today's date — 3/14 — correspond to the first three digits of π, the unique number that has infinite decimal places but begins with 3.14. How the Internet Is Paying Tribute to Pi Day Today is Pi day and since nobody brought us a pie, we'd thought we'd make our own, in chart form. The nerdy place the Internet is, it's all about celebrating March 14, or 3/14, the date that happens to correspond with the mathematical number π, or (roughly) 3.1415926535897932384626433832795028841971. But, given that the day is just about a numerical coincidence and also happens to involve a very ... Can You Determine Pi From the Digits of Pi? Happy Pi Day! Isn't Pi Day one of the best holidays all year? For last year's Pi Day, I determined a value of Pi without even using a circle . How can you top that? Pi Day 2012: Geeks' siren song sung in a round Get ready for a lot of meta-flavored pie tomorrow. Pi Day is on its way and it's time to let that geek flag fly. If you don't spend a lot of time in the nerd-o-sphere (pun totally intended), you may not realize that March 14 (3/14) is a beloved day to the mathematically aware. I shouldn't have to tell CNET readers, but pi is a mathematical constant--the ratio of a circle's circumference to its ... Pi Day: 3.14 ways you can celebrate It's a holiday for math teachers: March 14 = 3.14 = Pi Day, named after the mathematical constant that is the ratio of a Euclidean circle's circumference to its diameter. On Pi Day, we celebrate mathematics and things that are round — but instead of rounding up for pi, students relish every digit. Here's how you can celebrate, even after you've graduated. Read full article >> Writing Pi Text by Accident In honor of Pi Day , we have to address everyone's favorite irrational number. One of the areas where pi coincides with society is in the written word. Well, not really. While pi is found in many places, it is not found in many societal situations. Pi Day: five fun facts about 3.14 March 14, or 3.14, is Pi Day . Get it? Pi Day celebrates all things related to the ever-important mathematical constant that measures the ratio of a circle's circumference to its diameter. Throughout history, much has been done to understand more about this unconventional number; determining its value definitely tops the list. And while we can't tell you what pi is worth, we think there are a ... Pi Day: How 3.14 Helps Find Other Planets, and More Happy Pi Day! A favorite holiday among geeks, March 14 commemorates one of the most fundamental numbers in math. Celebrating Pi Day with a Raspberry Pi: element14 announces winners of the Pi Day "Raspberry Road Test" Challenge element14, the first collaborative community and electronics store for design engineers and electronics enthusiasts, powered by Premier Farnell [LSE:PFL], has today announced the three winners of its global "Raspberry Road Test" challenge.
Key Words: pi


Trapezoid.Trepresents a set of points lying in a quadrilateral whose north and south edges are horizontal and whose west and east edges have arbitrary non-horizontal slopes. For example, a diagonal line can be represented as a tall skinny trapezoid.
INTERFACEFor a trapezoid
Trapezoid; IMPORT Point; TYPE T = RECORD vlo, vhi: INTEGER; m1, m2: Rational; p1, p2: Point.T; END; Rational = RECORD n, d: INTEGER END;
tr.vhi are the
v coordinates of its north and south edges, respectively;
tr.m2 are the slopes of its west and east edges, respectively, as
(delta v) / (delta h). A denominator
of zero represents an infinite slope; i.e., a vertical
edge. A numerator of zero is illegal.
tr.p2 are points on the infinite lines
that extend the west and east edges, respectively.
\medskip Trapezoids are closed on the north and west edges, open on the south and east edges, closed on the northwest corner, and open on the other corners.
q represents the rational number
PROCEDURE FromEdges (y1, p1, q1: INTEGER; y2, p2, q2: INTEGER): T;
Return a trapezoid whose vertices are
(p2, y2), and
(q2, y2). The altitude of the trapezoid must be non-zero.
PROCEDURE FromVertices (READONLY p1, p2, q1, q2: Point.T): T;
Return a trapezoid from four vertices. The
p2vertices must have the same y-coordinates, as must the
q2vertices. Furthermore, the altitude of the trapezoid must be non-zero.
PROCEDURE FromTriangle (READONLY a, b, c: Point.T): T;
Return a trapezoid from the vertices of a triangle. One of the sides of the triangle must be parallel to the x-axis, and the triangle's altitude must be non-zero.