Polygon operations: primarily intersection testing, point-contains,
polygon->edges, bounding boxes, normal vectors, linear coordinates,
bisections, box intersections, bounded-space partitioning, and so on.
Takes a region and returns two new regions, each representing one side of the axis-aligned
bisection of the original. Any edge crossing the bisecting line is split, and any gaps created by
bisection are filled by using segments from the bisecting line.
(bounded-space-partition region max-edge-length)
Returns an indexed structure that can be used to test points. The structure is repeatedly
bisected until each dimension is smaller than the given amount.
(box-contains? [[minlat minlng] [maxlat maxlng]] [lat lng])
Returns true if a bounding box contains the given point.
(boxes-intersect? [[x11 y11] [x12 y12]] [[x21 y21] [x22 y22]])
(degenerate? p1 p2)
Returns true if the edge's points are identical (i.e. the edge has no length).
(edge-intersection [p11 p12] [p21 p22])
Returns the point of intersection of two edges. Behavior is undefined if the edges do not
intersect (use edges-intersect? to test), or if the two lines are parallel.
(edges->region bisecting-line edges)
Combines edges to form a new region. The bisecting line is required here because we need to infer
edges where there are gaps.
(edges-intersect? [p11 p12 :as e1] [p21 p22 :as e2])
Returns true if two edges intersect.
(line-contains? [p1 p2] p)
Returns true if a line contains a point.
(line-coordinate [p1 p2] p)
Returns the line-coordinate of the given point. Works best when the point is on the line.
(minus [x1 y1] [x2 y2])
(normal [x1 y1] [x2 y2])
Returns a normal vector for the given line. You can then take the dot product of points against
this vector to see which side of the line they are on.
(partitioned-poly-contains? partitioned-poly p)
Returns true if the partitioned poly contains the given point.
(partitioned-poly-intersects? partitioned-poly box)
Returns true if the partitioned polygon intersects the given box. The poly must be partitioned
such that the bounding boxes end up being no larger than the one being tested for.
(plus* [x y] [dx dy] factor)
(poly-contains? lat lng poly)
Used for implicitly-closed polys. Count the number of side-intersections for a ray extending from
[lat, ∞]; if even, the point is outside.
poly should be of this form:
[[[lat1 lng1 lat2 lng2 ... latk lngk] [latk+1 lngk+1 ... ]] ; one poly with a hole
[[latj lngj latj+1 lngj+1 ...] ...] ; another poly with a hole
[[[& include] [& exclude] [& exclude] ...]
[[& include] [& exclude] ...]
The toplevel array is considered to be the union of its elements.
(positive-side? [p1 p2] p)
Returns true if the given point is on the "positive side" of the line. Positive and negative
sides are deterministic and consistent, so you can use this with a line to partition space.
Returns a bounding box for the given region. The bounding box is returned as [[minlat minlng]
(region-contains? lat lng points)
Returns a list of [[lat1 lng1] [lat2 lng2]], each of which describes an edge of the region. The
region is implicitly closed.
(single-poly-contains? lat lng [include & exclude])
Uses edge intersection testing to determine whether a single bounded shape
contains the given point.
(split-edge line [p1 p2 :as edge])
Takes a line and an edge, and returns two new edges, the first of which is on the positive side
of the line and the second of which is on the negative side. The line and the edge must


(Some of these were found by people on the mep yahoo group)
- A textbook is mentioned - but is not really necessary. You can do the program without buying it.
- When you print any pages out with measurement questions - make sure that you have Adobe set to NOT stretch/scale to fit the page, or your measurements will be wrong when you measure the items. I set it this way in general - I lose the page numbers and have to handwrite them in - but it saves the problem. (This problem is caused by the pdf's being made for the standard European A4 paper, which is a little longer than the American standard letter sized paper.)
- Don't worry about converting the pounds and pence questions, just mention that they use different money, but it works the same as dollars and cents.
- If you are from the states - don't convert all the measurements from metric. Metric is base 10, and very easy. It helps with the concepts of place value in earlier years, and is used in most of the world, and in science. If day-to-day life isn't giving enough american measurement practice, buy a cheap unit from the store on measurements to work on that one item.
One other area that can cause confusion is the numbers within a > or <> where the 1 is little and inside the > sign means ONE more than. If there was a 2 there, it would mean TWO more than. Same with the <> ? the only right answer is 4, whereas 5 > ? has answers of 4,3,2,1,0 (,.....)
- Lesson 2, Activity 2. Question asks "How many cars did you not color in?" Answer is shown as 2. Answer should be 1.
- Lesson 39, Activity 2. Question asks "How many animals are there altogether?" Answer is shown as 8. Answer should be 9
- Lesson 41, Activity 2. Question asks "What is beside the scarf?" Answer is shown as Umbrella and Pine-tree. Answer should be Umbrella and Flower
- Lesson 43, Activity 2 (Game 7) - Game is impossible to win as printed. As you can move up and down only with even numbers, you need an ODD number of rows (as you don't count the row you start on....) But there are an even number of rows in the grid, so you can never get to the winning row. ALSO, my kids found this game too frustrating, as they would roll a number that to move they had to move away from their goal... So we quit this game early.
- Lesson 48, Activity 1 - Very last line in the activity - in the Explanation - Says Three and two and five make nine altogether... the five should be four.
- Lesson 53, Activity 1 - There are 4 triangles, and 5 circles. Partway through it asks "Which are thee more of, triangles or circles?" (Circles) "How many more?" - Two more circles than triangles. Obviously should be ONE more circle than triangles.
- Some of the early practice pages have questions that are never mentioned in the lesson - ?
- Posters - they were misnumbered - but might be fixed on the website?
- Note that occasionally you do some practice pages further in the book than you are - these seem to be some writing practice pages.
- Note that on occasion the student can choose which question out of a few to do.... let them!
- Lesson 25 Q2. - This is not a typo, this question introduces the concept of negative numbers. Get out the number line with negative numbers and have a go with your child. It is fine if they don't "get it" - but many children do, and it is a nice intro.
- Lesson 34 Activity 5 - Lesson plan says to Take 12 from 3. Should be take 1 from 3.
- Lesson 73 Activity 1 - On the lesson plan, it says to write on the BB: 9-3=7.... this is obviously wrong...
- Lesson 73 Activity 2 - Says the question is "Each plate had 8 pears on it" but the question book says "Each plate has 9 pears on it" - should be 9.
- Lesson 93, Activity 2 - Says to use calendar from lesson 91 to figure out answers - but questions show year 1999, and calendar on pg 91 is from 2001. Change the 1999's to 2001 to match.
- Lesson 100 Q3 - Note that you can change the operation sign too, not just the numbers!
- Lesson 126 Activity 3 - Minor one... title is Pictures of 14, and says to look at the pictures of 14.... should be 15.
- Lesson 139 Activity 1 - Count down by 4's. Says 17, 13, 8, 5, 1. Should be 17, 13, 9, 5, 1
- Lesson 174 Q5 - We think the answer should be 1 apple = 36 bananas not 9 as in the LP.
- Lesson 10 Activity 7a - (extra activities) - The last numbers in the table are 8 with a 6 under it. Either change the 8 to a 2, or just clear out the 8 and let the student work out that it is a 2. (8 is definitely wrong!)
- Lesson 54 Activity 4 - Says 89 divided by 0 = 0. Answer should be undefined.
- Lesson 2 Activity 1 - 177 is not a prime number. Should be 277.
- Unit 1, Lesson 5, Critical Path Analysis, Problem #4 - Float time for B might be 17, not 7. Float time for G might be 27, not 20. This is unconfirmed.


New York City public schools outpaced the rest of the state by dramatically increasing the number of students who met the state’s bar for proficiency in English and math this year.
City students in grades three through eight improved their proficiency by 3.3 percent in math and 1.5 percent in English. Math proficiency improved from 54 percent last year to 57.3 percent this year while English increased from 42.4 percent to 43.9 percent. The other big urban school districts in the state—Rochester, Yonkers, Buffalo, and Syracuse—did not see any increases.
“All of our students, teachers and principals should be very proud of their progress and the fact that we continue to raise achievement levels and outpace the rest of the state,” said Mayor Michael Bloomberg. “But as much progress as we have made, we know we have much more work to do. We are fully committed to ensuring that all of our students are prepared for a successful future.”
City students’ gains have outpaced students across the state on the annual English and math exams since 2002. In fourth grade English and math tests, NYC students have improved by 11 points, and by seven points in eighth grade math. While the state has either declined or stayed the same during that time, city students have seen improvements that are even greater than those in the rest of the nation.
“This is real proof that when expectations are raised, our students can rise to the occasion,” said Schools Chancellor Dennis Walcott. “It’s a model we plan to follow across the board—with higher standards in our curriculum, graduation requirements, and accountability measures for schools—to ensure our students are on track for success in college and careers. New York City students and families should be proud of their continued progress, even with these tougher standards for success.”
For the state as a whole, there was an increase in the percentage of students who met the proficiency standard in math, from 61 percent last year to 63.3 percent this year. However, there was a decrease in the number of students who met the English proficiency standard, falling from 53.2 percent to 52.8 percent.
Test scores can be viewed within ARIS Parent Link at arisparentlink.org starting on Wednesday, August 17. If you do not know your username and password, you can visit an ARIS Parent Link Access Station at select libraries from Monday, August 22 to Saturday, August 27.


Regardless of your ethical standing on gambling, there is rich and interesting information to be had in football betting as evidenced by the extensive research in fields ranging from pure statistics to machine learning to come up with approaches to best the bookmaker.
Odds represented in decimal form are most prevalent in Europe. Decimal odds can be considered as multipliers for your money. If you bet $1 on odds of 2.5 and win, you take back $2.5. You can convert decimal odds to fractional odds by subtracting 1 from the decimal odds and converting the result into its simplest fraction.
Eg. To get the fractional odds for 2.5:
2.5 – 1 = 1.5. The simplest fraction for 1.5 is 3/2, which is the required fractional odds.
You can get probability from odds by taking the reciprocal of the odds (represented in decimal form). Theoretically, if the odds are 2.0, it means that the probability of the event occuring is 1/2 = 50%.
In slot machines, ‘theoretical payback’ refers to the percentage of the total wagered money that should be returned to the players. When used for a good amount of time, the actual payback of slot machines almost mirrors the theoretical payback promised by the machine designers. Theoretical payback exists in sports betting as well, and can be detected in a simple way.
Calculate the probabilities of all the outcomes in the market (probabilities of win, draw and loss in the 1×2 market, for example) and add them up. This should yield 1, but you will end up with a figure higher than that. This is due to the effect of theoretical payback. You can calculate the theoretical payback by multiplying the odds offered with the probability.
You can now obtain the true bookmaker probabilities considering theoretical payback: Probability = theoretical payback/odds.
A more formal presentation of this introduction can be found in this master’s thesis by Rasmus Olsen.


Welcome to Year 2|
Year 2 Information
Welcome to year 2. Here is some important information that you will need:
Our learning in Year 2|
Here are some pictures of our music, literacy and maths lessons. We enjoy learning practically where we can and enjoy a varied curriculum.
Autumn 1 Newsletter
Autumn 2 Newsletter
Here is the homework for Autumn 1. It is due in on a Thursday and pupils are rewarded dojo points for completing homework.
Here is a topic plan for this half term when our topic is 'Where do we live?'
Children are expected to know their 2, 5 and 10 times tables by the end of year 2. They should know them in and out of order and with the related division facts. We learn them with the timestable being learnt at the start e.g. 2 x 1, 2 x 2, 2 x 3, 2 x 4 etc.
Children will all be given log in details for times table rock stars. Here is the link: https://play.ttrockstars.com/auth/school
Class Dojo Reward System|
If you haven't already, please sign up to the Class Dojo Reward System being used throughout school. Your login details were sent home with your child. If these details have been misplaced please let us know and we can give you another copy.
Class Dojo is a great way to keep up to date with class news/photographs and see the reasons why your child is being awarded points by the class teacher!


Comments are the non-executable statements in a program. In Python, they are added to describe the statements. It makes the program easily readable and understandable. Actually, the interpreter ignores the comments.
Comments start with a #. All the characters following the # and up to the end of the line are part of the comment.
A program can have any number of comments.
Python Comments Example
The following program is to use a single line comment. Here sqrt() is using to return the square root of 5, and the import statement is using to import the math module.
#write a program to display the square root of number import math a=int(input("Enter the value=")) b=math.sqrt(a) print("The square root is=",b)
Enter the value=5
The square root is= 2.23606797749979
In Python, there is no syntax for multiline comments. When we want to add a multiline comment in Python, we insert a # for each line.
Now we are giving an example for using a multiline comment.
#Taking values from the user #Adding them #Printing the result a=int(input("Enter the first value=")) b=int(input("Enter the second value=")) c=a+b print(c)
Enter the first value=90
Enter the second value=80
Leave a comment


University of Virginia
Found in 3 Collections and/or Records:
This collection contains a typed script paper written by Master's student, Scott George Stadtmueller, from the University of Virginia. It is titled "The Pursuit of Land Grant Status of the University of Virginia," and is dedicated to Colonel Harry Temple at Virginia Tech.
The collection includes research files, personal notes, audio recordings, videocassettes, photographs, and publications collected by engineer and professor Earl A. Thornton on the fields of aviation, aeronautics, and aerospace.
This collection contains the papers of John Edward Williams (1867-1943), professor of mathematics (1903-1924) and Dean of the College (1924-1943) at Virginia Polytechnic Institute. Materials include class notes taken by Williams as a graduate student at the University of Virginia; notes and solutions for various mathematical problems; texts of Williams' speeches; printed articles by Williams and others; and diplomas and membership certificates.


SPRINGFIELD, MO (09/11/17) - A hands on approach to learning, Club Encore is an extended learning after school program that gives kids second and fifth grade the opportunity to explore new ways of increasing their reading and math skills.
“It’s very individualized! We are making sure that we are meeting their learning needs but the biggest thing is that they are going to be hands on, they’re going to be up, engaged, and moving, and learning without even knowing it," says Kelsey Brabo with Springfield public schools.
With it being the first day a couple of the kids said they’re ready to start learning!
“I’m most excited about learning new things and stuff," says fifth grader Jasmine.
“I’m excited to learn new things at the pace we are going at," says fourth grader Asantie.
The students are going to have an extra 60 hours of learning per semester. So each kid could see an extra 120 hours of learning all year.
With around 4,400 students invited to join Club Encore this year, only about 800 participated last year. Making Springfield schools say they’re excited that kids are more involved in reading and math.
And just this morning, Brabo says well over 3,000 seats have been filled with eager kids ready to test their skills. Brabo, also, says the number is much higher this year because the eligibility requirements were broadened.
“We identified some needs within our students and we deemed them an ineligibles student and invited them to Club Encore," says Brabo.
With two tutors per 20 kids, three assessments are given throughout the school year to test which kids might need a little more help with reading and math. Next year, Springfield schools hope to include kindergarten, first, and sixth grade.


Preparing for High School Math
A Class for advanced 8th graders and 9th graders starting High School
By Shelby Osborn
Session 1: Monday 9 - Friday 13 July 2-3.30pm
Session 2: Monday 6 - Friday 10 August 2-3.30pm
Students should enroll in this class to prepare for the advanced rigor of high school math classes, the SBA state graduation test, and the SAT/ACT. Many students feel unprepared for the complex word problems in each unit, whether it’s reading the problem carefully or the actual math involved in solving the problem. This course will help students succeed by using strategies such as inquiry, reading for comprehension, justifying, and collaborating with their peers.
Students will leave the course with the ability to:
- Solve and maximize systems of equations and inequalities
- Read word problems to fully understand the question asked
- Answer word problems in context and justify their reasoning
- Collaborate with others to gain new insights into math applications
- Reflect on new learnings to create individual story problems
Students will be given a set of word problems, each week focusing on a different type of problem. For each type of word problem, students will be able to: read carefully for comprehension, write a single equation and system of equations, write a single inequality and system of inequalities, solve the equations and inequalities, answer the question in context, and reflect on their new learnings. During the last class, students will prepare their own complex story problem and solve each other’s work. The class is not a lecture-based course, instead it will focus on learning through inquiry and collaboration.
Each class will cover the topic listed by incorporating word problems:
- Monday: linear equations and inequalities
- Tuesday: systems of linear equations
- Wednesday: systems of linear inequalities
- Thursday: maximization of systems of equations/inequalities
- Friday: students create and present own word problems
I have been a teacher for four years in Kent School District, teaching the Algebras, Geometry, and essentials classes focusing on individualized help. Throughout those years, I have been given opportunities to see how much students struggle with word problems. It is by far the main thing that makes students say: “oh no, I don’t understand!” I have created strategies and used them successfully on my students to help them understand how to approach long, complex problems. I am very personable and will make the class fun, interesting, and overall a great learning experience!
4500 9th Ave NE, Suite 300, Seattle, WA, 98105
Class may be billable through insurance in some cases; contact Hampton Tutors for further information.
Terms, conditions and refund policy:
In the event that you must cancel, your payment will be fully refundable until 5 business days prior to the class start date. Cancellations made less than 5 business days in advance of the class will be partially refunded at 50% (for example, $250 refund payment for a $500 class). Cancellations may be made by phone or email.
In the event that we must cancel the class, we will notify you via phone or email and refund you in full.


Enjoy the Best times to play slots at Online Casinos
The most obvious distinction between land-based and online slot machine games is the physical one. The player pulls on the lever (usually marked by a bright green symbol) then pulls the handle and after that, pushes the button to spin the reels. Slot machine games online require that the player pulls an arm and then holds a stick or plastic tab in their mouth. The button will be activated to spin the reels. The outcome of sweet bonanza this virtual spin is decided by the computer program that controls the machine. Slots on online machines are not just the result of a random process: the random number generator (RNG) that creates the results is built on mathematical algorithms that closely simulate the way in which natural random number generators work.
While the above explanation gives the basics of how an online slot machine functions, a more detailed explanation of the way it works is required to comprehend its odds and benefits. If a player is awarded a jackpot, the player is awarded not only the cash prize, but also the entry code for subsequent draws. If a player wins a jackpot, or any other prize in the lottery, they are awarded an entry code. These codes can be used to determine which number will be the winner of the next draw is the jackpot winner. Winning combinations for future drawings are determined by probabilities which take the probability of the participant of selecting the winning numbers into account. If the odds are good then the outcome of the next draw is the same.
These seemingly random outcomes appear random on the surface. However, as mentioned previously, the random number generator (RNG) that is developed by casinos utilizes an advanced mathematical algorithm to produce the results. These algorithms take the player’s past hand history into consideration and also the amount of successes in the past when playing these types of machines. For instance, a person that has previously won a jackpot at an offline casino will benefit over a player that plays an online slot machine. The random number generators are so sophisticated that the long-term performance of these types of slots will be deeply dependent on the actions and decisions of the operators and owners on these websites.
Online slots and casinos that are located in the land both offer their players the chance to win “free” money through “vending machines” or bonus offers. These offers may be appealing to gamblers who want to experience the thrill of gambling of winning. However there are people who engage in these games for a variety of reasons, and the sites which offer bonuses might not offer them for free – some land based casino sites and online casinos use their bonus programs to try to lure people to these games by offering the chance to win additional prizes after having already spent some money on their initial entry.
Online slot machines are designed differently than traditional machines in many ways. For instance the online slot machine does not usually spin all the reels. Instead the reels are arranged randomly and then programmed into each game. Because there are not enough physical slots for a typical casino, the random pattern of the reels is carried out electronically.
In addition bonus rounds are also incorporated into a variety of slot games, giving the opportunity for players to increase the odds of their spins. Many casino games provide bonus rounds regularly. Sometimes they come at specific times during the main game, and sometimes they come randomly. You can control the timing of these bonus rounds by keeping the reels moving at a constant speed.
These random number generators minesweeper free online allow the machine to decide how many times it spins its reels. This process is governed by two aspects. The first is the random number generator (or RNG). It is a computer that is located within the slot machine and utilizes numbers generated by the software to determine which symbols to show on the reels. It also assists the machine to determine the outcome of the event.
A second factor that casinos online make use of for their slots is the bonus round. This feature lets players double their bankrolls after signing an agreement. These bonuses may be exclusive to particular games, or be offered at all times. Online casinos offer a variety of gaming bonuses that can be used to get the most out of the most optimal times to play.


Joseph wrote in to ask:
Thank you for your awesome lessons. Id like to ask if they use numbers in japan as we do or if they write it in symbols ( i forgot name 😡 ).
Usually you write numbers in regular arabic numerals i.e. 123456 etc.
Sometimes you do see kanji numbers e.g. 一 二 三 四 etc. but this is usually in things like menus on the walls in traditional restaurants etc.
So you should be fine with regular numbers.
(The reason Japan uses arabic numerals is the same reason we don’t use roman numerals anymore, there was originally no “zero” in kanji counting so it makes maths so much harder!)
P.S. Ninja Tip: There’s a great Japanese numbers poster in my Learn Japanese Pack!
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Graduate Studies Reports Access
Graduate Course Proposal Form Submission Detail - MAE6137
Tracking Number - 1986
Edit function not enabled for this course.
Approved, Permanent Archive - 2004-07-02
Course Change Information (for course changes only):
- Date & Time Submitted: 2004-02-17
- Department: Secondary Education
- College: ED
- Budget Account Number: 172400000
- Contact Person: Denisse Thompson
- Phone: 42687
- Email: firstname.lastname@example.org
- Prefix: MAE
- Number: 6137
- Full Title: Probability and Statistics for Middle Grades Teachers
- Credit Hours: 3
- Section Type: C -
Class Lecture (Primarily)
- Is the course title variable?: N
- Is a permit required for registration?: N
- Are the credit hours variable?: N
- Is this course repeatable?:
- If repeatable, how many times?: 0
- Abbreviated Title (30 characters maximum): Prob/Stat Mid Grades Tchrs
- Course Online?: -
- Percentage Online:
- Grading Option:
R - Regular
- Prerequisites: Admission to the MAT program in middle grades mathematics or CI.
- Course Description: This course examines probability and statistics topics for middle grades mathematics teachers. Topics include data collection and display, measures of central tendency and variability, probabilities, and sampling procedures.
- Please briefly explain why it is necessary and/or desirable to add this course: Data analysis and probability is one of the five strands in the Sunshine State Standards. Middle grades teachers need a strong conceptual understanding of statistics and probability in order to teach the appropriate content from this strand in the middle
- What is the need or demand for this course? (Indicate if this course is part of a required sequence in the major.) What other programs would this course service? This course is required in the MAT in Middle Grades Mathematics.
- Has this course been offered as Selected Topics/Experimental Topics course? If yes, how many times? 1 (Summer 1994)
- What qualifications for training and/or experience are necessary to teach this course? (List minimum qualifications for the instructor.) Doctorate in mathematics education
- Objectives: 1. The ability to define probability and statistics;
2. The ability to display data using a variety of graphical displays (bar, line, picture, circle, scatterplot, box and whisker, stem and leaf);
3. The ability to identify measures of central tendency and situations in which they are appropriate;
4. The ability to identify measures of variability and use such measures to interpret results;
5. The ability to find probabilities of independent and dependent events and to use tree diagrams to find probabilities;
6. The ability to find combinations or permutations as appropriate for the situation;
7. The ability to design an experiment or simulation, collect data, and interpret the results;
8. The ability to discuss different sampling procedures, including the use of random samples;
9. Knowledge of major historical developments in statistics and probability.
- Learning Outcomes: • Exams or tests will evaluate students' content knowledge on the major content topics in the course. Students will have to pass the final, comprehensive exam in order to pass the course.
• Problem sets will evaluate students' ability to explore open and extended problems.
• Historical paper will give students an opportunity to explore the historical background of a topic from statistics and/or probability.
• External project will have students engage in a statistics or probability project of the instructor's design or of their own approved design.
• A journal will provide on-going evaluation of students' facility with the content of the course and emphasize the importance of writing throughout the curriculum.
- Major Topics: 1. Definitions of probability and statistics
2. Displaying data on graphs (bar, line, picture, circle, scatterplot, box and whisker, stem and leaf)
3. Measures of central tendency
4. Measures of variability
5. Using tree diagrams to find probabilities
6. Finding probability of independent events
7. Finding probability of dependent events
8. Finding combinations and permutations
9. Using statistics to predict from samples
10. Sampling procedures – random samples as a focus
11. Using computer simulations
12. Historical connections to statistics and probability
- Textbooks: Sample Text: Probability and Statistics Connections. University of Missouri-Columbia. (Draft). To be published by Prentice Hall with 2005 copyright. This college-level text was developed as part of the Connecting Middle School and College Mathematics project funded by the National Science Foundation.
Supplemental texts: The following units from the Connected Mathematics Project are possible texts.
Data About Us
How Likely Is It?
What Do You Expect?
(Glenda Lappan, James T. Fey, William M. Fitzgerald, Susan N. Friel, and Elizabeth Difanis Phillips. Menlo Park, C
- Course Readings, Online Resources, and Other Purchases:
- Student Expectations/Requirements and Grading Policy:
- Assignments, Exams and Tests:
- Attendance Policy:
- Policy on Make-up Work:
- Program This Course Supports:
- Course Concurrence Information:


Browsing Matematik-Bilgisayar Bölümü / Department of Mathematics and Computer Science by Publisher "Hikari"
Now showing items 1-1 of 1
(Hikari, 2013)The paper deals with the Arens Multiplication which we accomplished in four steps in the order bidual ~ ~ X . It is shown that if f is an element of order dual ~ X of X with ? ( f )?0 and , + x ? X then f .x = 0 implies f (x)= .


Background:The Australian Tertiary Admissions Rank (ATAR), represented by a number between 0.00 and 99.95, is the most important metric used by Australian tertiary institutions when selecting students for entry to undergraduate degrees.
Objective: To quantify the statistical precision of an ATAR by creating a realistic database of scores for 10,000 students taking six VCE subjects in 100 different schools.
Methods: Approximately 500,000 test scores were processed into study scores, scaled study scores, aggregate scores and finally converted into an ATARfor each of the 10,000 students. The same dataset was processed twice: once using outcome scores that had been rounded to integer values, as is required by the VCAA (ATAR1), and again using students’ actual, non-rounded outcome scores (ATAR2). The rounding error (i.e. the difference between ATAR1 and ATAR2) was then analysed.
Results: In this simulation, we found that not only are the two decimal places on an ATAR statistically insignificant, but that ATAR results are only accurate for 100% of students when reported to the nearest multiple of two (i.e. to the nearest even number).
Limitations: Privacy laws prevented us from using real student data in this analysis.
Conclusion: This paper calls for ATARs to be reported only to the nearest even number, and for continued modernisation of the undergraduate admissions criteria used by Australian tertiary institutions to include a broader range of student attributes.


A question came in regarding our use of a weighted average.Â lppm05 wants to know how we calculate the weighted average when determining our figure for EPSmg.Â What we do is use a sum-of-the-years-digits method for the average.Â For example, using Microsoft (MSFT), here are the last five years of diluted EPS (note their fiscal year ended 6/30/08):
2008Â — $1.87
2007Â — $1.42
2006Â — $1.20
2005Â — $1.12
2004Â — $0.75
To use the sum-of-the-years-digits method, we simply multiply the EPS for the given year by the number of the year divided by the total sum of the years.Â Continuing with our example, the weighted amount for 2008 was $1.87 x (5/15) = $0.62333.Â For 2007, we would get $1.42 x (4/15) = $0.37867.Â Once you have completed calculating the weighted amounts for all the years, simply add up the totals to find the EPSmg for 2008.Â We found the EPSmg to be $1.44 in this case.
You can see that the EPSmg is lower than the current figure.Â This may or may not be true in all cases.Â If a company has a significantly low result in the current year, the EPSmg will be higher.Â The whole purpose is to smooth out the business cycle to determine a normalized earnings.Â We believe that completing this task is the first step in evaluating a company.
If you have any questions for us, please let us know by leaving a comment below or emailing me at firstname.lastname@example.org.Â
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- Albert Einstein once said, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?” Do you agree with Albert Einstein that mathematics was invented by humans, or do you believe it was created by a higher power? Explain.
- The author of this article refers to great mathematical theorists such as Einstein, Euclid, Newton, Fibonacci, Galileo, Riemann, and Kepler. Select a mathematical theorist and discuss what significant contributions they made in mathematics and other areas of study.
- Were there any circumstance of their life that contributed to their discoveries in mathematics?
- Make sure to cite your sources.
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Numbers & Early Math
Help return 100 chickens to their home sweet home!
Count the Wheels
Count how many wheels you see.
Count's Number Ride
Trace the numbers with Count.
Do You See My Seahorse
The baby seahorses have become separated from their fathers! Use the number clues to help find them.
Follow the Stars
See if you can follow the star pattern.
Giant Hide and Seek
Play hide and seek with giants! Find hiding places for Peg, Cat, and friends.
Help Greenelle count her toys.
Hermit Shell Game
Help Sally and Nick fit the crabs into a shell that is just the right size.
I Spy Shapes
Find all of the shapes hidden in the picutres.
Muffin is Missing
This video takes you on an adventure with a young and eager investigator to explore patterns, shapes, and counting.
The Highlight Zone
Help Peg and Cat spot ten differences in the Highlight Zone.
Toopy and Binoo's Bunny Hop
Put the bunnies in the correct pattern.
Toopy and Binoo: Secret Code Game
Match shape patterns to crack the code.
George is playing with his train set. Can you help him build trains of specific lengths?
Count, sort, and explore math games for kids.
Help Gabriela collect and count her harvest.
Welcome to Beaver City
Put objects in order from shortest to tallest; compare different lengths.
Explore spatial vocabulary, size comparison and estimation with Clifford.


Q: How important do you think it is to learn calculus for game programming? Which math fields are most useful in hobby game development?
A: I needed calculus for classwork, however I have virtually never used it for any of my dozens of freeware and small commercial projects.
That said, I’m not very involved in writing physics or 3D graphics libraries – when I need those features, I reference libraries written by other people who are more knowledgable in those areas. Someone working on rendering features or optimizations for 3D engines might well have more use for calculus than I do.
For most gameplay-related math – jumping, bumping into things, the speed of a shield recharge – simplified approximation suffices. In many cases the sloppier estimation approach is even better than trying to work out proper precision. As one example, the tried-and-true platform game mechanic of “how long the jump button is held after leaving the ground determines jump height” clearly does not in any way reflect the real physics of jumping. It’s more important for a videogame to behave as the player wants (or expects) than it is for a videogame to behave realistically.
Of course, if we were trying to design an authentic flight simulator for military training, or model a car accident for the purposes of studying safety and real-world engineering, this would be a very different story. But for games we can usually just twiddle with numbers until things look and feel right. Videogame spaceships and helicopters move because we add a velocity value to their coordinates every frame – not because of rocket propulsion or Bernoulli’s principle. For 2D game programming, spacecrafts and helicopters only need 1st grade arithmetic.
But beyond basic gameplay interactions, here are the fields of math that I run into most frequently for game programming, along with notes on what I find each most useful for:
- Basic Geometry. 2D graphics programming is a matter of nudging object coordinates around on a plane, then drawing images every frame based on those coordinates. Bounding boxes (comparing coordinate differences) and the distance formula (Pythagorean Theorem) are commonly used to test whether two objects have collided with one another.
- Trigonometry. Sine and cosine are helpful for angle-to-component translation, for example when determining what percentage of a bullet’s total velocity the x-speed and y-speed are given the angle of a firing cannon (see illustration below) – this also applies to an overhead race car. Atan2 is handy for getting an angle between two things, given their relative offset in grid locations – good for getting an enemy to face the player, pointing a simple homing missile, and so on.
- Vector Geometry. The dot product is an incredibly versatile math operation to gain mastery of, even in the simplest 2 vector 2D case. Learn how to use it, and you’ll greatly increase the number of nifty things that you can pull off in game programming. This is useful for reflection off angled surfaces, checking line-of-sight, determining which direction something is facing, and many other common spatial/angle relationships.
Addition (courtesy of fellow indie Tyler Glaiel): Normalizing vectors is another basic and useful aspect of vector geometry. Properly applied, these offer a more efficient way to accomplish some of the same things that trigonometry is commonly used for, including aggressive homing missile logic.
- Boolean Logic. It’s common in game code to want something to only happen either when multiple things are true (if player is on ground AND pressing jump button, then jump) or when at least one of several things is true (if [W Key] OR [Up Arrow] is pressed, then try to jump). At first glance this doesn’t look like math, but when these statements begin to compound there’s are math-like rules that can be used to untangle and simplify boolean logic.
- Simple Algebra. Old-fashioned line-intersection check calculations are great, and trivial to write as a function. This comes up for things like pong AI, which needs to anticipate where an elastically bouncing ball will intersect a screen edge…
… (continued in ebook)
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KeyStage 2 SATs Results 2019
Pupils in Year 6 take tests in Numeracy, Reading, a Spelling, Punctuation and Grammar (SPAG) test, and are assessed by their teachers in Writing.
72% of our pupils reached the expected standard in reading, writing and maths (combined), which is above average. 24% reached the higher standard in all three subjects, which is significantly above the national average and in the top 20% of all schools. (Ofsted’s Inspection Data Summary Report, 2019)
The figures below show the overall performance at the end of key stage 2 in 2019.
Pupils attaining the Expected Standard and above at the end of KS2 in 2019
|Grammar, Punctuation and Spelling||84%||78%|
|Reading, Writing & Maths combined||72%||65%|
Pupils attaining the Higher Standard at the end of KS2
|Grammar, Punctuation and Spelling||42%||36%|
|Reading, Writing & Maths combined||24%||11%|
Average Scaled Score
Pupil Progress Measure from KS1 to KS2
Our school’s Year 6 SATs data includes all of our eligible Year 6 pupils, including our Small Specialist Class for pupils with Special Educational Needs and Disabilities, some of whom may be working below the level of the tests, but all of whom are included in our published results.
Average Scaled Score
The average scaled score has shown a steady increase over the last three years, and is above average in all subject areas.
Pupil Progress Measure from Key Stage 1 to Key Stage 2
Ofsted’s understanding of our school’s Pupil Progress measures reflects a national picture where junior schools have higher attainment and lower progress scores than primary schools overall. This can be for a variety of reasons. These scores show how much progress pupils at this school made in reading, writing and maths between the end of key stage 1 and the end of key stage 2, compared to pupils across England who got similar results at the end of key stage 1.
“We know from national data that pupils at junior schools, on average, have higher attainment scores at the end of key stage 2 than pupils at all other primary schools. However, on average they also have lower progress scores. This may be for a variety of reasons and should be taken into account when comparing their results to schools which start educating their pupils from the beginning of Key Stage 1.” Inspection Data Summary Report, Ofsted 2019
The scores are calculated by comparing the key stage 2 test and assessment results of pupils at this school with the results of pupils in schools across England who started with similar assessment results at the end of the previous key stage – key stage 1. A score above zero means pupils made more progress, on average, than pupils across England who got similar results at the end of key stage 1.
A score below zero means pupils made less progress, on average, than pupils across England who got similar results at the end of their key stage 1.
A negative progress score does not mean pupils have made no progress, or the school has failed, rather it means pupils in the school made less progress than other pupils across England with similar results at the end of key stage 1.
The majority of schools have progress scores between -5 and +5.
See School performance tables: about the data for more information about how the progress measures are calculated.
Additional data can also be found about our School on the School Comparison website.


2017-11-16 08:30:03 UTC
My comment in Nature:
Not if LIGO fraud remains unexposed:
"Merging Neutron Stars Deliver Deathblow To Dark Matter And Dark Energy Alternatives. [...] In Einstein's theory, the speed of gravity equals the speed of light, exactly, and at all times. But in many alternatives, that assumption gets tweaked. [...] But the fact that gamma rays and gravitational waves from the merging neutron star event GW170817 arrived within 1.7 seconds of one another means that the speed of gravity must equal the speed of light to better than 1 part in 10^15. As a result, a huge slew of alternatives to standard General Relativity with standard dark energy are ruled out." https://www.forbes.com/sites/startswithabang/2017/10/25/merging-neutron-stars-deliver-deathblow-to-dark-matter-and-dark-energy-alternatives/
The crucial questions are:
Does Einstein's general relativity predict that gravitational waves travel at the speed of light? Does it predict how the Shapiro delay for gravitational waves compares with the Shapiro delay for light?
The answer to both questions is 'no'. According to Arthur Eddington, Einstein's general relativity says nothing about the speed of gravitational waves, let alone their Shapiro time-delay:
Arthur Eddington: "The statement that in the relativity theory gravitational waves are propagated with the speed of light has, I believe, been based entirely upon the foregoing investigation; but it will be seen that it is only true in a very conventional sense. If coordinates are chosen so as to satisfy a certain condition which has no very clear geometrical importance, the speed is that of light; if the coordinates are slightly different the speed is altogether different from that of light. The result stands or falls by the choice of coordinates and, so far as can be judged, the coordinates here used were purposely introduced in order to obtain the simplification which results from representing the propagation as occurring with the speed of light. The argument thus follows a vicious circle." The Mathematical Theory of Relativity, pp. 130-131 https://www.amazon.com/Mathematical-Theory-Relativity-S-Eddington/dp/0521091659
So what is the a priori probability that the gravitational waves (if they exist) arrive simultaneously with the optical signal? Answer: Zero.
That is, if, in the neutron star case, LIGO's fabrication involved different times of arrival, that would at least have sounded realistic. The claim that the gravitational waves and the optical signal arrived at exactly the same time, which implies that they not only travel at the same speed but also experience the same Shapiro delay, unequivocally proves that LIGO conspirators just faked the gravitational wave signals.


Fri, 8 Mar, 2019
Maths hack goes viral and blows people’s minds: "You've changed my life"
A UK copywriter has shared an arithmetic trick that makes calculating a lot easier.
Ben Stephens took to Twitter to share his “fascinating little life hack” for doing percentage calculations.
He showed that by flipping numbers and multiplying them as per usual will result in the sum you are looking for.
“So, for example, if you needed to work out 4 per cent of 75 in your head, just flip it and and do 75 per cent of 4, which is easier,” Stephens wrote.
Fascinating little life hack, for doing percentages:
x% of y = y% of x
So, for example, if you needed to work out 4% of 75 in your head, just flip it and and do 75% of 4, which is easier.
— Ben Stephens (@stephens_ben) 3 March 2019
People on the Internet have been amazed by the simple maths hack, with Stephens’ post accumulating more than 4,000 retweets and 11,000 likes at the time of writing.
“You’ve changed my life,” a man simply replied.
Another man commented that he was “furious” for not knowing this sooner. “I'm 30 and have avoided knowing this until some hero on Twitter tweeted it.”
One wrote, “How could maths teachers let us live without this!”
The tweet also inspired some teachers who had not been aware of the switching technique.
“I teach Maths at primary level and had never realised this,” one wrote. “50 per cent blown away/50 per cent going DOH!”
A woman chimed in, “I used to teach maths for reporters as a part of journalism school and wish I’d had this explanation in my back pocket. I had other tricks for mathsphobes but this is far more elegant.”
Some complained that the trick is a simple mathematics rule rather than some little-known hack.
“Do you really think people don’t understand such a simple concept enough to know this? Good grief,” one wrote.
“Those of us who studied basic arithmetic at school are scratching our heads as to why this is a revelation,” another added. “Next you’ll be telling us that x+y = y+x and that xy = yx.”
Stephens defended his post, saying it was meant as a way to pique people’s interest in numbers. He admitted that he also just learned about the ‘switcheroo’ fact.
“I almost think stuff like this would be lost on a kid at school,” he wrote. “It has way more impact a couple of decades later when you see it and you're like OMG IT WAS SITTING RIGHT THERE THE WHOLE TIME.”
Did you know this maths trick? Let us know in the comments.


Most surveyors know, from study and training if not personal experience, that “least squares” are important. That might be the extent of what they know though. Important, yes, but why? And what exactly is least squares adjustment? What exactly does it do for land surveyors and other geospatial professionals?
Least squares adjustment can be defined, as “a model for the solution of an overdetermined system of equations based on the principle of least squares of observation residuals.” For surveyors, “overdetermined systems” are the networks of related coordinates used to establish boundaries, locate points on Earth, facilitate large construction projects, etc.
These networks are usually “overdetermined” in a mathematical sense because of varying randomness in observation technologies and measuring equipment. Usefully analyzing all this randomness to improve network accuracy and integrity requires a set of powerful mathematical techniques (including matrix algebra) that are collectively termed least squares adjustment.
This is an over-simplified explanation, of course, but perhaps establishes that least squares adjustment plays an important role in any surveying practice that uses or creates coordinate networks. That is to say, all surveying practices.
How can surveyors apply least squares adjustment efficiently? After all, even though surveyors tend to be extremely intelligent and well educated, they often aren’t comfortable using some of these types of advanced mathematics on a daily basis.
Fortunately, as computing power became widely available, practical least squares adjustment was one of the first applications to be developed specifically for land surveyors. One such program, MicroSurvey’s STAR*NET, has been around since the mid-1980s, and has been used by thousands of surveying firms in many different workflows.
Least Squares can be complicated when trying to understand how to really works, but it is however the recommended method for adjustments in the field of Land Surveying.


“Conway’s Game of Life” is a mathematical game created by John Conway, in an attempt to simplify a mathematical model for a hypothetical machine that could build copies of itself. This game opened up a whole new field of mathematical research, the field of cellular automata
I have always loved this game and thought that it was time to take a shot at recreating it, and that is what I did. Also, no java applet or any nonsense that I always see prompted when you want to play this game online. Profit!


29/03/2019 · In Canada, the United Kingdom, and several other European countries, temperature is measured in degrees Celsius °C. In the United States, Belize, the Bahamas, the Cayman Islands, and Palau, temperature is measured in degrees Fahrenheit °F. Luckily, it’s easy to convert between the two whenever you may need to. About. Celsius, or centigrade, is used to measure temperatures in most of the world. Water freezes at 0° Celsius and boils at 100° Celsius. Fahrenheit is a scale commonly used to. From these dates, it's probable that Anders Celsius invented this conversion formula. How To Convert Celsius To Fahrenheit. Using the Celsius to Fahrenheit formula: Fahrenheit °F = Celsius x 1.832, these examples show how to convert a temperature in Celsius to Fahrenheit.
21/12/2019 · Boiling water at normal pressure measures 100° in Celsius, but 212° in Fahrenheit; And as water freezes it measures 0° in Celsius, but 32° in Fahrenheit; Like this: Looking at the diagram, notice: The scales start at a different number 0 vs 32, so we will need to add or subtract 32. Come Convertire da Fahrenheit a Celsius. Per convertire i gradi Fahrenheit in Celsius e viceversa bastano una semplice addizione, una sottrazione, una moltiplicazione e una divisione. La prossima volta che ti verranno forniti dei dati. Fahrenheit was the standard unit of measure in most countries until the 1960s and 1970s when it was replaced with the Celsius scale in a widespread conversion to the more useful metric system. But in addition to the U.S. and its territories, Fahrenheit is still used in the Bahamas, Belize, and the Cayman Islands for most temperature measurements. The words Fahrenheit and Celsius are two of the hardest ones for students to spell! Here are some of the many misspellings we've seen in our visitor logs since starting this site: farenheit, faernheight, fahenheit, fahernheight, fahranheit, fahreheight, fahreheit, fahrehnheit, fahreinheigt, fahreinheit, fahren.
Fahrenheit to Celsius: Subtract 32, then multiply by 5, then divide by 9 Interesting fact: The Fahrenheit scale and the Celsius scale are the same at -40 degrees. This relation can also be used when converting from one scale to the other. Convert 36.8 °C Celsius to °F Fahrenheit. Fahrenheit proposed his temperature scale in 1724, basing it on two reference points of temperature. In his initial scale which is not the final Fahrenheit scale, the zero point was determined by placing the thermometer in a mixture "of ice, of water, and of ammonium. Fahrenheit en Celsius ºF en ºC calculatrice de conversion pour les conversions de température avec tables et formules supplémentaires. langue. conversion d'unités > Convertisseur Métrique > Convertisseur de température > Conversion de Fahrenheit > Fahrenheit en Celsius. Fahrenheit to Celsius formula. Fahrenheit °F to Celsius °C, how to convert & conversion table. How to convert Fahrenheit to Celsius. The temperature T in degrees Celsius °C is equal to the temperature T in degrees Fahrenheit °F minus 32, times 5/9.
Celsius to Fahrenheit: Multiply by 9, divide by 5, and then add 32 Fahrenheit to Celsius: Subtract 32, then multiply by 5, then divide by 9 Interesting fact: The Fahrenheit scale and the Celsius scale are the same at -40 degrees. This relation can also be used when converting from one scale to the other. Formulas can help you to convert Celsius to Fahrenheit and vice versa. You can also use various methods for making conversions in your head. Online calculator to convert Celsius to Fahrenheit °C to °F with formulas, examples, and tables. Our conversions provide a quick and easy way to convert between Temperature units. Celsius and Fahrenheit are two important temperature scales that are commonly misspelled as Celcius and Farenheit. The formula to find a Celsius temperature from Fahrenheit is: °F = °C × 9/532; The formula to find a Fahrenheit temperature from Celsius is: °F = °C × 9/532; The two temperature scales are equal at -40°.
Zero degrees Fahrenheit indicates the lowest temperature Fahrenheit could obtain by a mixture of ice and salt. The degree Celsius °C is a unit of temperature named for the Swedish astronomer Anders Celsius 1701-1744 who first proposed it. Celsius vs Fahrenheit: Key Differences. Before we explain how to convert Celsius to Fahrenheit and Fahrenheit to Celsius, let’s review the main differences between the two temperature scales. Celsius written as °C and also called Centigrade is the most common temperature scale in the world, used by all but five countries. Celsius to Fahrenheit °C to °F, how to convert & conversion table. To switch between Celsius and Fahrenheit follow these instructions: 1. Make sure the thermometer is off. If it shows anything on the display, leave the buttons untouched for 10 seconds. 2. Press and hold the Ear-button for 10 seconds until the display shows “- - -˚C” or “- - -˚F”.
Fahrenheit vs Celsius If you follow weather updates where temperature is mentioned, confusion can arise when two units are mentioned: Fahrenheit and Celsius. This article explains the two temperature scales and also focuses on the conversions from Celsius to Fahrenheit and Fahrenheit to Celsius. Before you jump into Fahrenheit to degrees switch. Instant free online tool for Celsius to Fahrenheit conversion or vice versa. The Celsius [°C] to Fahrenheit [°F] conversion table and conversion steps are also listed. Also, explore tools to convert Celsius or Fahrenheit to other temperature units or learn more about temperature conversions. To build a simple table to convert temperature from Celsius to Fahrenheit, you can use the CONVERT function. In the example shown, the formula in C5, copied down, is.
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Here are a collection of Jupyter iPython notebooks I've created to solidify my understanding of ideas during my studies of probability, stats and machine learning.
I've attempted a few Kaggle competitions, here are the relevant notebooks:
- Titanic: Machine Learning from Disaster
- Forest Cover Type Prediction
- Predicting Red Hat Business Value
- Attempt 1 quick first attempt using routine preprocessing pipeline for categorical / quantitative variables and using logistic regression and random forest models. Ignores categorical variables with thousands of unique values that can't be one-hot encoded.
- Attempt 2 exploring ways of including categorical variables that have thousands of unique values (ordinal, mix of one-hot and binary)


NIAID renews funding of PRIME to develop mathematical models to study influenza infection
Links shared publicly online about this topic.
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View full resource at news-medical.net
Most Recently Shared on December 17, 2010 at 4:20 pm By:
Mathematical Modeling Of Influenza Infection Being Studied By Consortium - QualityPoint Technologies
qualitypointtech.net — “Mount Sinai School of Medicine has announced that the National Institute of Allergy and Infectious Diseases (NIAID), part of the National Institutes of Health (NIH), has renewed funding of the Program for Research on Immune Modeling and Experimentation (PRIME). This program seeks to develop easy-to-use, predictive mathematical models to better understand patterns of infection among individuals affected by the H1N1 and 1918 influenza viruses and other related viruses. The renewed contract provide” View full resource at qualitypointtech.net
Most Recently Shared on December 19, 2010 at 11:45 am By:


Welcome to our new monthly feature on EDNC called the NC STEM ScoreCard, written in partnership with the NC Science, Mathematics, and Technology Education Center (SMT) and the Burroughs Wellcome Fund.
North Carolina is at a critical crossroads in its economic and educational development. The proud traditional businesses and occupations that fueled our economy in the past, especially those requiring minimal education and skills, are playing a diminishing role in our state. A new more competitive economy has emerged, one that requires higher levels of knowledge and skills, many based in science, technology, engineering, and mathematics and achieved through more engaging teaching and learning strategies.
Are we ready?


In the previous sections we looked at the basic properties of idealized EM-waves and their mathematical description. Here we look at the propagation behavior of these waves in different media. The propagation behavior can be described entirely in terms of the electric permittivity () and the magnetic permeability (), which are both complex functions of the frequency and can be related to the refractive index () and the impedance (). All of these quantities can take a huge range of values for different materials and different frequency values. These differences are responsible for the effect of frequency dispersion: different propagation behavior of EM-waves for different frequencies. Dispersion gives rise to the splitting up of a ray of white light into its different frequency components in a prism and the formation of a rainbow.
Both phenomena originate from frequency dispersion in the refractive index. For each of the wavelengths in the visible spectrum, the refractive index of the prism and the collection of rain droplets is slightly different. At the interface between two media, the angle between the normal and the transmitted wave differs from the angle between the incident wave and the normal, according to Snell’s law.
This change in the transmitted angles is known as refraction and originates from the wavelength dependence of the refractive index. In order to formally derive this behavior, we consider the polarization dependent properties of electromagnetic waves that are incident on the interface between two different media. We consider two different perpendicular polarization states of a homogeneous plane wave: one with the electric field perpendicular to the plane of incidence (figure 2(a)) and one with the electric field parallel to the plane of incidence (figure 2(b)). The former is S-polarized or a transverse electric (TE) wave, while the latter is P-polarized or a transverse magnetic (TM) wave.
For both polarization states the appropriate boundary conditions need to be applied in order to derive the propagation behavior of the incident wave at the interface. Three different waves are involved in this process, which depend on the material properties of both media: the incident wave (equation 2-3), the refracted wave (equation 4-5) and the reflected wave (equation 6-7).
The magnitudes of the different wave numbers are given by
At the interface between medium 1 and medium 2 () the boundary conditions have to be satisfied for all points. The spatial and time variation of all fields must be the same there, which implies that all phase vectors should be equal, independent of the boundary conditions.
From equation 10 it is clear that all 3 wave vectors lie in the same plane. As , it follows that the angle between the incident and refracted wave should be equal, and we also can derive Snell’s law directly.
The boundary conditions that have to be satisfied can now be written in terms of the field values (equations 2-7). The normal components of and and the tangential components of and have to be continuous at the interface, which can be expressed as follows
in which is a unit vector perpendicular to the interface. From this point onwards, it is useful to split up the derivation in 2 cases for S-polarized (figure 2(a)) and P-polarized (figure 2(b)) waves, as any other polarization state can be constructed based on these two polarization states.
For the S-polarized case, equation 12 yields no result, as the electric fields are all perpendicular to the plane of incidence. From 14 and 15 we can derive equations 16-17, while equation 13 reproduces 14 when we apply Snell’s law.
From the obtained boundary conditions, the relative amplitudes of the refracted and reflected waves are obtained, which gives the following complex transmission and reflection coefficients:
in which the square roots are introduced using Snell’s law to rewrite everything as function of the properties of the incident wave.
For the P-polarized case a similar derivation can be made, for which the tangential and have to be continuous, which results in
in this case the normal component of should be continuous as well, which duplicates the second equation when we apply Snell’s law. Again, we can write the relative field amplitudes as function of the properties of the incident wave, which gives the following complex transmission and reflection coefficients:
The propagation behavior of any type of EM-wave can be fully described by means of equations 18-19 and 22-23, which are often referred to as the Fresnel equations. In general the refractive index , the electric permittivity and the magnetic permeability are all complex functions of the frequency. At optical frequencies in conventional materials it is often stated that , but as we’ll show later on, this is not valid in case of optical metamaterials which also exhibit a strong magnetic response.
The electric permittivity of a material depends primarily on the electric polarizability, which is closely related to the electron density. For this tutorial, we restrict ourselves to dielectrics and noble metals in the visible and NIR wavelength range of the EM spectrum. In case of dielectric materials (such as silica ()), the refractive index and permittivity show almost no frequency dispersion and can be treated as positive real numbers (damping can be ignored). Metals show a totally different behavior, and can be described by the Drude-Sommerfeld model:
in which is the plasma frequency, and a damping factor. The contribution of the bound electrons to the polarizability is included in , which is 1 in case only the conduction band electrons contribute. This model gives a fairly good description for noble metals in the visible and NIR spectral range, although some modifications are needed at shorter wavelengths in order to compensate for inter-band transitions. Therefore in numerical calculations we use adapted models based on experimental data to obtain a more accurate description of the investigated structures. In general, the permittivity can be written as . The real part is negative for frequencies below the plasma frequency and relates directly to the polarizability of the metal, while the imaginary part takes positive values and relates to the damping of propagating waves and the phase of the polarizability. An overview of the experimental values of the permittivity for gold (Au) is given in figure 3.
Figure 3: Electric permittivity of gold
It is clear that also the propagation behavior of EM-waves will be influenced by the effects of dispersion. So far we only considered the propagation of idealized waves at a fixed frequency. In that case the permittivity defines the amplitude, phase and damping of a propagating wave. In that respect, we introduced the concept of phase velocity , which defines the propagation velocity for a single frequency component. In practical applications even the most idealized source will contain more than one frequency component or wavelength, due to finite pulse durations or inherent broadening in the source. All of these frequency components propagate with slightly different phase velocities and as a consequence both the amplitudes and phases tend to show changes with respect to each other. This implies that in dispersive media the velocity of energy flow can differ largely from the phase velocity.
For dispersive media, the frequency depends on the wave vector (). In most spectral regions is a slowly varying function of , but in certain spectral regions the variation is much more pronounced, for example in the vicinity of the plasma frequency of a metal. If we assume for now that and are real numbers, we can write a superposition of homogeneous plane waves to construct a more general solution for a plane wave in a dispersive medium
where defines the amplitude for the different components of the superposition of plane waves. The amplitude values are obtained by the Fourier transform of the spatial amplitude at
In that way, we obtain homogeneous plane wave . If we consider a finite wave train at then will not be a delta function but a peaked function with a finite width (determined by the length of the wave train) which is centered around the wave number . If we assume a fairly sharp shape of this peaked function (i.e. the wave train is rather long), then the frequency can be expanded around .
By inserting this expansion into equation 25, we can derive
If we compare this with equation 26 then it follows that the integral in equation 28 describes with , from which we obtain
This illustrates that apart from an overall phase factor, the pulse travels undistorted in shape with a group velocity which can differ significantly from the (average) phase velocity which was already introduced in section 1.
As the group velocity of a pulse describes its propagation velocity, it also determines at which rate the pulse is transporting energy in the direction of the Poynting vector. The relationship between and can be written in its most general shape
from which we obtain the general expressions for the phase velocity and group velocity as function of the refractive index
In most conventional materials with normal dispersion ( and ) the velocity of energy flow is smaller than the phase velocity and smaller than . In regions of anomalous dispersion, huge differences between the phase and group velocity can occur.


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₿ Should You Buy Bitcoin Right Now? - An Expert Opinion Things To Know Before You Buy
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Seldom, it can be formed as a pattern reversal pattern at the end of an upward trend. when is the best time to sell bitcoin of where they take place, the pattern shows bearishness. This pattern is formed after a long downward pattern and shows trend reversal. The symmetrical triangle pattern is made up of a distinct shape created by convergence of two trend lines.
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Determined from the Fibonacci number series: 0. 618 = Main Ratio1. 618 = Primary Forecast 0. 786 = Square root of 0. 6180.886 = 4th roofing of 0. 618 or Square root of 0. 7861.130 = Fourth root of 1. 618 or Square root of 1. 271.270 = Square root of 1.
382 = (1 0. 618) or 0. 618e20.500 = 0. 770e20.707 = Square root of 0. 501.410 = Square root of 2. 02.000 = 1 + 12. 240 = Square root of 52. 618 = 1. 618e23.141 = Pi3. 618 = 1 + 2. 618 Exit based on special cost history For a range of factors, Bitcoin tends to stay in a drop or uptrend on particular months of a year.
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Nonlinear Dynamics of the Nikkei Stock Average Futures
This paper analyzes the conditional distribution of the Nikkei Stock Average Futures prices traded in the Singapore International Monetary Exchange (SIMEX). It is found that the conditional mean of the logarithmic price ratios is zero and the conditional variance is adequately described by the exponential generalized autoregressive conditional heteroscedasticity model (witht errors) suggested by Nelson (1991) and the autoregressive volatility model suggested by Hsieh (1993). The Brock, Dechert and Scheinkman (1987) statistic cannot reject the hypothesis that the standardized residuals are independently and identically distributed. The results are applied to calculate the maintenance margin and the long-term capital requirements of the contract given an assumed maximum failure rate. The margin requirements set by the SIMEX appear to be adequate compared to our estimates.
Asia-Pacific Financial Markets
TSE, Yiu Kuen.
Nonlinear Dynamics of the Nikkei Stock Average Futures. (1995). Asia-Pacific Financial Markets. 2, (3), 181-195. Research Collection School Of Economics.
Available at: https://ink.library.smu.edu.sg/soe_research/533


A strategy is strictly dominant if, regardless of what any other players do, the strategy earns a player a strictly higher payoff than any other. Hence, a strategy is strictly dominant if it is always better than any other strategy, for any profile of other players’ actions.
If a player has a strictly dominant strategy, than he or she will always play it in equilibrium. Also, if one strategy is strictly dominant, than all others are dominated. For example, in the prisoner’s dilemma, each player has a strictly dominant strategy.
Source: Shor, Mikhael, “Strictly Dominant Strategy,” Dictionary of Game Theory Terms, Game Theory .net, http://www.gametheory.net/dictionary/StrictlyDominantStrategy.html« Back to Glossary Index


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When children ask why they need to study math, the answer usually has something to do with either daily life or applications to science and technology. The problem with the first motivation is that it is an obvious and transparent lie. The second type of “motivation” tends to have the opposite of the intended effect.
The “daily life” explanation tells students that they will need math for their daily activities. For example, they will need to calculate the tip in a restaurant, or determine how much they should pay for their groceries. Most students are quick to point out that this problem can be solved by carrying around a calculator. And anyone who is worried about running out of batteries can carry around a spare set of batteries, or even two calculators. Even cell phones have built in calculators.
The arguments against the “daily life” explanation continue. In daily life, you never need to do more than add, subtract, multiply, or divide. Why learn trigonometry? Why study calculus? Why do anything beyond arithmetic? Even math-oriented jobs rarely require any really advanced math. When I worked as an actuary, the only math I used on the job was multiplication and the occasional exponent (the actuarial profession is one of the most math-oriented professions in the world.)
The other rationale for studying math focuses on science and technology. We need math to design space shuttles and satellites, to work in laboratories, and to build the newest computers. In one way this argument makes sense. Much of that work requires intensive use of advanced math. But very few people work in those areas. Those that work in those areas usually do so because of an internal passion, not because of any external motivation.
In fact, from the perspective of most students, there is very little external motivation to be a scientist. The strongest external motivators for most teenagers are money, fame, power, popularity, and attraction to the opposite sex. None of these powerfully motivate students to pursue careers in science. For every million dollars a scientist makes, the businessmen for whom he works make a billion. For every famous scientist, there are a thousand famous musicians and actors. The scientists who made the nuclear bomb were not the ones to use it; that power belonged to politicians. And in American culture, scientists have no more popularity or sex appeal than anyone else.
Thus, this argument not only fails to motivate students; it actually does the reverse. A student with no interest in being a scientist who hears the technology argument now thinks that advanced math is useful only for scientists. Thus, he does not need to learn it. If his goal is personal gain, his time is better spent doing almost anything else – studying politics, learning to play the guitar, working out, or thinking of ways to make himself rich. Math becomes just an annoying requirement.
So then why should a student learn math at all? (close)
The belief that Asians are good at math is held with good reason: even in America, students with Asian parents tend to significantly outperform every other ethnic group. For example, 2005 math proficiency testing showed that Asian students had higher math proficiency scores than White, Black, and Hispanic students at all age levels (Source: Child Trends Databank).
This section examines the techniques used by Asian parents and educators. Of course, there are variations depending on the country of origin and the individual, but there are techniques and principles that are almost universal among Asian parents and educators.
The Asian system is built on memorization. At an early age, children are taught to memorize multiplication tables and the like. As they get older, they memorize formulas, and even memorize step by step ways to solve specific problem types.
The Asian system is radically different from current American methods, which emphasize understanding over memorization. Where American parents and math teachers focus on explaining why a technique works, the Asian educators simply require that the student memorize the technique, and be ready to use it.
One might expect that such a technique would create students who simply have formulas memorized and are unable to understand what they are doing. But the reality is just the opposite. Once students have the information memorized, the understanding seems to come naturally. On the other hand, systems that drop memorization and focus on only understanding seem to have the reverse effect. Students often end up confused – unable to understand the problem, or to solve it.
This is one of the strangest paradoxes in math education, one that I wrestled with extensively at the beginning of my career as an educator. Why does memorization work in math? Why does focusing exclusively on understanding fail? Isn’t math about understanding? Shouldn’t memorization be saved for history?
To unravel this mystery, we will undertake a journey that will help us understand some of the most important cognitive principles involved in math education. (close)
You just started seventh grade, and it is the first math class of the
year. The teacher explains something, and you notice that others in
the class seem to understand the concepts faster than you do. You
are able to solve the problems, but it takes you a bit longer than the
rest. By the end of the class, you are starting to think that you just
might be bad at math.
The next day confirms it, as the other students seem to race
ahead of you. In reality, they are at best 5% faster, but from your
perspective it seems like they are at least ten times better at math
than you are. At this point (two days into the year), you are already
pretty sure that you are bad at math. By the end of the week, you are
absolutely certain. In fact, being bad at math soon becomes part of
your identity. When the teacher gives a challenging problem, you do
not really try that hard to solve it. That is just not who you are. You
are the kid who struggles with math; what chance do you have of
solving a hard problem?
You do your homework, but if you cannot get a problem, you are
not too concerned. Athletes play sports. Rockstars sing. Your role in
life is to get math problems wrong.
Every so often, you get some evidence that does not fit your
theory. You get a 9/10 on a quiz that everyone else in the class fails.
But you and the rest of the class are now so sure that you are bad at
math that you decide that the quiz is flawed. Everyone laughs about
it. How did the bad math student do better than everyone else on
the quiz? Even your parents think it is kind of funny.
From time to time you get a math teacher that tells you that you
can be an excellent math student, and that there is nothing wrong
with your abilities. Sure, he is something of an authority. But you
have years of experience, the opinions of other teachers, and the
opinions of your friends on the opposite side. Obviously, you assume
that the heretical math teacher is wrong and that every other person
on Earth is right. (close)
...For the mind to have the incentive to develop, two things are
necessary. First, it must encounter a problem that it is unable to
do; the process of figuring out how to solve this initially unsolvable
problem causes the mind to develop. If a student is only given
problems at his current ability level, what incentive does the mind
have to improve? Just as lifting a half-pound weight will not make
a person physically stronger, doing an easy math problem will not
make a person mentally stronger.
Parents and teachers of gifted students often overlook this, and
just allow them to work at a comfortable pace. The result is that
the gifted students never get the opportunity to realize their full
potential. Like natural athletes who never train hard, they end up
squandering their innate talents. (close)
If you ask the parents of an outstanding math student what kind
of rewards they use to motivate their child to do his homework, for a
split second they will look at you as if you asked what kind of reward
their child gets for not wearing diapers anymore. They will then
quickly recover and come up with some polite answer that sounds
good. But that initial look tells you everything you need to know.
Parents of outstanding math students only use incentives when
the child is very young, and then only for a short period of time. The
approach is almost like toilet training. When a child is first toilet
trained, parents may give great approval, and even some small reward,
to the child when he correctly uses the toilet. But very few parents
cheer a 15 year old of normal intelligence when he manages to use
a toilet correctly. By that age it is just taken for granted. Similarly,
no approval is given when a child does his homework every day. It is
just taken for granted.
So how do you get there? How do you make homework a
permanent part of a child’s daily life? (close)
We learned in the last chapter that motivating a child to do
his daily homework has more to do with self-perception than with
incentives. However, to really bring out the best in a child, you will
need to provide the right type of long-term incentives, in addition
to helping him develop the right self-perception.
This is especially important for brilliant but lazy children who
think outside the box. Because these students tend to look at situations
objectively, self-perception does not have the same momentum with
them as it does as with others. Thus, these students need the right
objective, long-term incentives to bring out their best.
Why might a student not do his homework? Sometimes
students look at the short-term gains from not doing their
homework (immediate fun) and ignore the long-term benefits of
strong cognitive skills. For other students, particularly for students
with learning differences such as dyslexia and students with delayed
intellectual development, the homework is such a painful process
that they avoid doing it at all costs.
But in one of the most common types of situations, neither
applies. The main details are usually the same. The child is raised
in an upper middle class family, in a wealthy neighborhood. His
parents are successful professionals, often doctors or lawyers, and
he has seen the positive results of an excellent education. The large
house in which he lives, the car in which he rides, the luxuries he
enjoys resulted from educational success. If anyone on Earth should
be convinced that education is important, it is this child.
As one looks more closely at the situation, it makes even less sense.
The child is unusually bright, and his memory and processing speed
are excellent. Even his attention span may be above average. This
is not a case in which the child is overwhelmed by the homework,
or unable to do it. Nor is the child too young to understand the
consequences; typically the child is between 13 and 15. It seems
as if the child just does not want to do any homework, so he does
not do it. The parents (and teachers) are often tearing their hair
in frustration, as increasingly elaborate threats and bribes fail to
motivate the student.
Why would an intelligent, able student who obviously knows the
value of an education not do homework? Why would he not bother
to study for a test? Is he just short sighted? Does he just not see the
Surprisingly enough, the answer is often just the opposite! The
child is not really thinking only about the short-term; in his own
way, he is thinking about the long-term. (close)
This is a section about stereotypes, and as with all stereotypes,
there will be many exceptions. However, understanding the nature
of these differences will help you teach math to both girls and boys.
You might find that “girl” techniques are appropriate for your son,
or that “boy” techniques work well for your daughter. Or you may
find that neither are completely appropriate for your child. However,
there is a good chance that the gender specific techniques in this
chapter will match exactly.
Take a boy and a girl in the same school, with the same grades,
the same level of ability, and the same level of understanding. It
would be no surprise to discover that the boy thinks of himself as
excellent at math, and the girl thinks of herself as horrendous at
There are a variety of reasons for this discrepancy. Girls tend to
attribute failures to permanent, internal qualities. For example, a girl
who fails a test might think, “I failed this test because I am not good
This is an internal attribution. The girl blames herself (instead of
some external factor, like a teacher or textbook) for the math failure.
The girl may also think of the attribute as permanent. She believes
that she is not good at math in the same way she might believe her
blood type is B: it is a permanent and unchangeable feature.
The boy who fails the same test will be more likely to think,
“I failed the test because the test was unfair/stupid/ridiculous. The
next one will be better.”
The boy attributes the failure to the test itself, rather than to his
own lack of ability. This is an external attribution. He also views the
problem as temporary: “The next test will be better.” Thus, a failing
grade does not affect his confidence as severely. (close)
Here is what you need to know about innate math ability:
1. People have varying natural math abilities. Some people
naturally find math easy to learn. Others naturally find math
harder to learn.
2. So what?
If you have a slow metabolism, you can either give up on being
fit, or you can accept the fact that you have a slow metabolism and
train harder than everyone else. If your child struggles with math,
you can make excuses for him, and hold him to a lower standard.
Or you can accept the fact that your child will have to work twice as
hard as his peers, and that if he does so, he will be at the top of his
What Does the sat test?
To most people, the SAT Math section is a mystery. It only
covers algebra and geometry; however students who have taken
calculus often get several problems wrong. Students who get A’s
in honors math classes sometimes do worse on the math section
of the SAT than do students who struggle in regular math classes.
In addition, the SAT math section does not cover any advanced
topics; a student who knows basic algebra and geometry knows
every concept and formula needed to solve any problem on the SAT.
These basic concepts include the formula to find the area of a circle,
the Pythagorean theorem, and how to factor a trinomial. There are
no logarithms, no esoteric geometry theorems, and no trigonometry.
In fact, most of the necessary formulas are actually given at the
beginning of each SAT math section!
Often, a child will do extremely well on math quizzes, but
horrendously on exams. The student’s quiz grades may be the highest
in the class, and yet he may be getting D’s on his exams. This is not
due to a lack of studying – the student studies for days for each math
exam, putting in far more effort than his peers. In fact, many of his
peers do not study at all for math exams, and still do much better
than he does. It seems that no matter how much he studies for an
exam, he does horrendously. (close)
The most significant change to American math education during
the last fifteen years has been the incorporation of calculators. As
early as seventh grade, students are using graphing calculators as
part of their math education.
You already know that for most students, math is valuable not for
its potential applications, but for the cognitive abilities and reasoning
skills it develops. For example, a student should study geometry
so that he can develop his logical and spatial reasoning skills, not
because he will one day need to decide whether or not two triangles
are congruent. A student should learn to factor polynomials because
of the cognitive skills that the task develops, not because he is likely
to use this skill in any part of his adult life.
However, many teachers and administrators do not fully
appreciate this fact. Instead, they believe that the goal of math
education is to give students the ability to somehow get the answer
to a large number of math problem types. Because calculators can
often allow students to get the answer more quickly, many teachers
teach students to solve problems using calculators. For example,
rather than teaching analytical techniques for graphing, teachers just
teach students how to graph using a calculator. In the absolute worst
cases, teachers actually teach basic math concepts, such as adding
and multiplying fractions, with calculators.
Of course, this approach completely misses the point of math.
The answers to the problems themselves do not matter; only the
cognitive skills the problems develop matter. (close)
A child cannot control the pace of a school math class. If a student
does not learn how to do a problem type, he just falls behind. He will
later have to struggle to catch up, or just end up getting low grades
in the course. The course moves at a fixed pace, and one student’s
performance will not speed up or slow down the class.
However, when a child is working with a parent, his performance
does control the pace of the training. If he moves slowly and is unable
to figure out how to solve problems, his parent may move slower and
give the child easier problems. The child will end up actually doing
less total work.
On the other hand, if the child works quickly and uses his full
ability, the parent may increase the number of problems and the
level of difficulty. By working at his peak, the student has actually
increased his total workload.
Thus, a student has a strong incentive to drag his feet and use
only a fraction of his full abilities when learning math from his
parents. At some level, he realizes that the worse he does, the less
total work he will end up doing. Thus, he might take ten minutes
to solve a problem that he could do in one minute, or even give
up on problems that he would normally be able to solve. Rather
than developing his cognitive abilities, the child ends up wasting his
Thus, to ensure that the child gains the benefits of parental math
training, the parent must give the child an incentive to push his
mind to the limit. Historically, parents have used physical and verbal
punishments to create incentives. But what if there was a more
elegant and effective system? What if any “punishment” actually
benefited the child? (close)
The Asian method can be used by all parents and teachers,
regardless of their math ability. The Ladder, the Rule of Two, and
the Hydra can be used by most parents. In this section we will
examine the Microchallenge method, which can be used by par-
ents and educators with excellent math skills. Since developing the
Microchallenge method, I have used it daily with excellent results,
even when teaching students with major deficits in their math
skills. While the Microchallenge method can be difficult to master,
those who can master it will find it extremely effective, especially
when combined with the Hydra. It builds a thorough understand-
ing of mathematical concepts, while simultaneously developing
many necessary cognitive skills. This method, combined with the
Hydra, will allow you to turn students with almost no math abil-
ity into strong math students, and to supercharge the abilities of
strong math students.
The Microchallenge method requires both patience and a strong
understanding of the subject. You should only use the Microchallenge
method with material with which you are comfortable. For example,
if you are not comfortable with trigonometry, you probably will not
be able to use the Microchallenge method to teach trigonometry;
instead you might use the Asian System to teach trigonometry.
However, you will still be able to use the Microchallenge method to
teach the other parts of math.
The Microchallenge method involves helping the student
understand how to solve a difficult problem by leading him through a
series of “microchallenges” that run parallel to the required cognitive
steps of the problem. This ensures that the student develops the
underlying understanding and cognitive abilities necessary to do the
problem, and does not instead just memorize a few steps. (close)