Since we started heating with wood, I have never known how much wood we burn in a year. I am always splitting more, and never have time to count cords. So I came up with a way to estimate how much wood I need in a year. And then, lo and behold, my favorite blog "Walden Effect" had a post on their wood supply.
So I thought I would share my method. This works with any kind of wood, I burn mostly cedar and cypress (gasp!). Because all wood has the same energy per kilogram. It just takes more pieces and chimney cleaning to burn softwoods.
Step 1. About how many days will you use your woodstove? It doesn't have to be exact, but it is better to over than under estimate. Here we usually burn from mid-November to mid-April. So 5 months. 5x30 is 150 days.
Like I said, this is not rocket science.
Step 2. How much wood do you use every day on average? An easy way to figure that out is to fill your woodbox/rack in the morning, and count how many pieces of wood are in it. The next morning at about the same time, count again and subtract. We burn an average of 30 pieces of wood per day. The more days that you count it, the more accurate your estimate will become. Multiply your number by the heating days. In my case: 150 days x 30 pieces = 4500 pieces of wood. But who wants to count to 4500 when you are splitting? Not me.
Step 3. Go to your woodpile and count how many splits are in a square foot of face on your woodpile. (Yes, that is right, Mr. Metric says count in feet. The numbers are easier to imagine in this case) We average about 15 splits (I split quite small, since I procrastinate so badly- they need to dry faster than other people's wood). With 32 square feet of face on a cord, that comes to 480 per tier. If you cut to 16 inch lengths (three tiers/cord), that is 1440 pieces per cord. If you cut to 12 inches like me (four tiers/cord), that is 1920.
Step 4. Divide your needs by the pieces/cord. For my case 4500 pieces/1920 pieces = 2.34375 cords. Let's round up and say two and a half cords.
Or you can just use my handy dandy Wood Needs Calculator.
So, now I should get out there splitting, 'cause I only have 1.5 cords split and stacked!
New Zealand Bug Out Cabin
1 hour ago


Izak9 has arrived in Inch School! This new maths resource allows pupils to engage in mental maths in a way that is fun and challenging.
The resource is made up of 27 cubes. Each of the faces on the cubes has a number shape or symbol. These cubes are used by the pupils to solve the various maths problems sourced from an online website. The maths problems are based on fractions, percentages, decimals, shape and space number theory and algebra.
This maths resource develops the pupils ability to reason and problem solve in collaborative groups using concrete materials. When using this maths resource, pupils think about how they think about maths and engage in discussion and debate about the viability of the various maths solutions they come up with. When using the Izak9 pupils are encouraged to come up with more than one way to find the solution to the maths problem /challenge, thereby promoting their “thinking outside the box” skills.
We would like to extend a very sincere thank you to the PA and the parent body for providing funding for this invaluable and innovative maths resource for the pupils in Inch. Your support and generosity is very much appreciated by the staff.


I was struggling trying to figure out how and when to introduce the concept of zero to Doug. I thought that if I introduced zero first and he got a number surprise box full of nothing that it would seriously piss Doug off and put him off the number surprise box. I thought that if I delayed its introduction too long, that Doug wouldn't understand that it goes before all the other numbers he knows and not after and that it is indeed less.
A bit hard to see due to the camera flash, but nap time seems like its going to be cut short, so I don't have time to re-shoot. There are two tabs on the front side and two on the back side. They all fit very nicely together. I started with the 0-20, but we can add more, all the way up to 100, as we learn along.
Evidently the 20 is just off to the side of the frame, but you get the idea.
Doug loves playing with it. It becomes a roller coaster, a parking lot or train tracks depending on what he wants to drive over it.
I like to play a number game with it. I put out the number line and we count up and touch each of the numbers as we assemble it, taking special care to point out our number of the week. Once it's all together I ask Doug to bring me the week's number in certain objects.
For example, I asked him to bring me eight cars:
And then eight fish:
And then we ran out of Disney/Pixar toys, but you get the idea. We never put anything on the 0 obviously, but we talk about how it's a place holder. I know that Doug doesn't grasp that concept quite yet, but I've begun to plant the seed. I plan to cover the number zero after we do the number ten.


Design Science created the educational desktop application MathType 220.127.116.118 Crack. The only purpose of the graphical editor is to create mathematical equations in a fully graphical WYSIWYG (What You See Is What You Get) environment. This implies that you can directly enter TeX, LaTeX, and MathML, among other mathematical markup languages. Additionally, it is compatible with Apple Pages, Microsoft Word, and Microsoft PowerPoint, among other office and productivity programs. Working together with these desktop programs allows you to rapidly add formulas and equations to your projects. The MathType product key works with Microsoft Office 2007 and Windows 7 or later.
If math isn’t your thing lately, MathType Keygen can lend you a hand in this respect, as this software solution has been specifically developed to work with text editors to make your life a lot easier.
ChemType is a flavor designed to help you work with chemical notation. You’ll find a specific toolbar with the common chemical symbols and a user experience adapted to chemistry authors. Math Type SDK allows you to integrate professional quality math typesetting in your web, desktop, and mobile applications. Handwriting recognition SDK makes a real difference for your mobile app.
Other Informative Posts:
MathType Crack Features
Download MathType 18.104.22.1688 Crack which is the leading formula editor and equation writer used by K-12 and university STEM students and faculty worldwide. Discover why millions of them use it every day!
Easy to use
Write equations with an interface that provides a user-friendly experience, forget about having to learn LaTeX to write math on a computer. It does not matter if you are a beginner or an advanced user, MathType Patch is for everyone and adapts to your style of writing math, so you can focus on your projects at hand.
Create and edit math notation on office apps and web-based platforms. Whether you are working on Windows, Google Workspace, an LMS, or a web-based environment, MathType will meet your needs.
MathType is an online equation editor that provides perfect-quality math formulas. No matter which environment you are using it in, your documents will always look professional and of a high quality.
Share your documents and collaborate with your colleagues and students. MathType Serial Code allows you to write and share math equations in your documents and presentations.
Working on a touch device? No worries, MathType will convert your handwritten formula into a clean digital equation ready to be used in your documents.
MathType is the perfect partner for your online platforms, LMS and web editors. Its seamless integration allows anyone to use it from anywhere and on any device.
MathType is compliant with accessibility requirements to create software accessible to people with disabilities. Readable content by third-party assistive technologies or content functionality operating through a keyboard are requirements that MathType follows from WCAG (Web Content Accessibility Guidelines) 2.0 guidelines developed by W3C.
What’s New in MathType 7.5
- Fixed pasting equations from MathType Desktop on Mac not working properly in certain cases
- Fixed MathML 2 compatibility mode replacing the display attribute with the mode in math elements
- Fixed invalid MathML being output in certain cases when using phantom and script elements
- Fixed invalid MathML being output in certain cases when using multi-script and more elements
- Fixed mtable’s align attribute being able to be set and output nonstandard values
- Fixed LaTeX conversion not working sometimes when encountering empty mi, myspace, etc. elements
- Windows 2000/ XP/ Vista/ 7/ 8/ 8.1/ 10 or Server 2003/ 2008/ 2012/ 2016/ 2019
- Computer with reasonable speed
- Microsoft Office 2003 or above
How To Download MathType?
- Download MathType Crack
- Disconnect from the internet
- Extract and install the program
- Launch the program and then register offline


The 2 ½ year program will be offered in a course. Unit system in semesters. Hence the program will comprise 5 semesters, each of 15 weeks duration.
Subjects and Credits
Year 1 - Semester 1
Year 1 - Semester 2
Year 2 - Semester 1
Year 2 - Semester 2
Year 3 - Semester 1
Applicants should have one of the followings:
GCE (A/L) Examination (old syllabus) in Economics, Commerce & Finance and Accountancy or GCE (A/L) Examination in any two of the above three subjects and a pass in one of the following subjects: Political Science, Pure Mathematics, Geography, Sinhala, Tamil, English, Logic, History, Business Statistic at the same examination.
GCE (A/L) Examination (new syllabus) in Economics, Business Studies and Accountancy or GCE (A/L) Examination in any two of the above three subjects and a pass in Geography, Home Economics, Political Science, Logic & Scientific Methods, Agricultural Science, Combined Mathematics and Business Statistic at the same examination
Successfully completion of National Certificate in Accounting Technicians conducted by the Department of Technical Education
An Ordinary passes in English and Mathematics at GCE (O/L)


The LCC module allows users to define life cycle costs other than
those predicted by the RCMCost and AvSim modules. These costs may
be integrated with predicted costs in the LCC cost breakdown
structure to provide a time-dependent analysis of a system's whole
life cycle cost process.
The items may be further subdivided until the cost of each element
can be defined as a mathematical equation. At a simple level this
can be the number of man-hours multiplied by a cost rate. The
elements of cost will then be added together to give the total cost
for each item and a grand total for the system through its full
life. Life Cycle Cost analysis is used to evaluate the through-life
cost of new plant or equipment. The user builds a hierarchical Cost
Breakdown Structure (CBS) through an unlimited number of indenture
levels, starting at the conception stage and continuing through
research and development, design, operations and maintenance and
ending with system disposal.
The CBS can be directly linked to cost predictions produced by
the RCMCost or AvSim modules. Other costs may be defined as
time-dependent cost equations or simple numerical values. Global
variables may be defined and utilised in the cost equations.
High level costs are determined either by summating the cost
values for child nodes in the CBS or by applying a user-defined
cost equation. The syntax of cost equations is easy to understand
and the construction of cost equations is assisted by an
intelligent code-recognition utility that automatically reveals
global variable lists as the user types in an equation.
Phase-dependent cost equations may also be defined. Phases are
shared between the LCC and AvSim modules.


We may earn money or products from the companies mentioned in this post.
Have your ever wished you could pull up an example of a concept you are working on with your child and print it for them? That is exactly what I have been doing over the last month with my Individual Membership to Super Teacher Worksheets.
Last week we were talking about number families. We are working on learning our multiplication tables with the younger kids and I wanted to help them understand why both 5×9 and 9×5 had the same answer. I explained it but I wanted to be sure my kids were getting the concept in a way that would help them answer their math questions.
I pulled up this practice worksheet that showed number bonds and we got to see very clearly how each number was a part of the total answer. It helped them see multiplication problems in a new and different way. Suddenly they understood. It did not have to be written in a linear line to be a problem, and that little tweak helped them make the transition in their math drills.
We are planning a few new math games as well. I purchased some ink and card stock and that is all we need to be rolling. I might laminate my cards because I love my laminator but card stock is good enough too.
The first game is called “I have…Who has…”
The child with card one calls out, “I have 35, Who has 4×4?”
The student with the card with 16 would answer, “I have 16, Who has 10×3”
This action repeats through the entire chain of students (or cards)
The Second Game is called “SCOOT.”
In a class room setting you would lay out the cards on desks and give the students a minute to write the answer in their own answer grid before saying SCOOT. They would then move from desk to desk answering the questions on their grid to turn in. I think it would be a fun scavenger hunt to spread the cards throughout the house. The cards are already printed, all I need to do is get trimming. There are several versions of SCOOT to choose from so you could play no matter the age of your child.
We also printed the math family factor cards to reinforce what we learned from our earlier bond sheet. The kids like that we have added more hands-on learning to our math time. This is especially helpful with our family, with multiple ages there are times when a child must wait to have a question answered. The flash cards and worksheets for practice are a great way to give the kids bonus work that is still fun.
I noticed that our spelling test focuses on several word families each week. While my daughter was learning her words for the week I was looking for a way to help reinforce the letter sounds she is learning. The Super Teacher Worksheet Word Wheel is just the thing we needed. I printed the letter sound she is learning each week.
Now, there is a fun review in her notebook for the week. The best part is that I can throw them in an accordion file and keep them for later in the year. I can pull them out any time we come across that blend or diagram again in our lessons.
The site is easily searchable to find exactly the worksheet that will help your child the most. Everything is organized by subject, age and grade as well. I was able to find every sheet I needed including a review for Roman Numerals for my 6th grader. I plan on trying a couple of versions of SCOOT with our after school program as well.
Super Teachers Worksheets is a great resource for children’s groups as well. We always print worksheets for our kids and this makes it easy to pick them by grade. As a non-profit our budget is small, but $19.95 for the year is a great price to be able to pull worksheets for our students right from our phone if we need to. Any time a kiddo needs a break or some quiet space we can always have some fun learning sheets on hand for them.


Level Up Math Challenge
Ritenour students will soon be introduced to a new way to learn math facts called the Ritenour Level Up Math Challenge. The challenge is designed to bring fun and excitement to learning math.
In the fall of 2023, Ritenour received feedback from families at Community Data Meetings who were eager for additional math practice opportunities outside the classroom. The Level Up Math Challenge was created to reinforce skills taught in school and to provide a platform for students to hone their abilities in various mathematical domains.
Beginning in December, students will receive a game board each month with a set of IXL codes. IXL is an online learning program that Ritenour uses to help reinforce mathematical concepts students learn in the classroom. To successfully complete the challenge, students must achieve a score of 90 or higher on each IXL skill code on the game board. Then, they must write their score on the game board and turn it in to their school office.
The challenges should be completed at home, outside of regular school hours, allowing families to actively participate in their child's educational journey.
Students who complete the challenge will receive a certificate and a pencil every month. There will also be monthly incentives to return the game board to the school office, including awards like a stadium cup, fidget toy, pennant, brag tag, and a stress ball.
Students will receive more information and game boards in early December.
For more information about the Level Up Math Challenge, please contact your child’s teacher or school office.


Benoît Mandelbrot, inventor of fractal geometry – the branch of mathematics that describes the hidden order in nature and in seemingly chaotic systems – died this last Thursday.
For more info about how your tax dollars are allocated, see the National Priorities Project.
Coalition military deaths in Iraq since March, 2003: 4,766 (as of April 22, 2011; click here to update). At least 467 contractors have also died, based on only partial information. Total U.S. military wounded as of as of January 14, 2010: 31,882.
Coalition military deaths in Afghanistan since October, 2001: 2,416 (as of April 22, 2011; click here to update.


layout design, coding,¬¨‚Ä† photo-editing,
Textbook: Brainworks (p8) Q24(b) A mathematician proposed that
"Every even number greater than 2 can
be expressed as a sum of two prime
numbers." Do you agree? Why?
I don't agree so.4 is an even number
however it is only formed by two 2s.
In this way,it is considered as only using
one prime number hence not every
single even number can be expressed
as a sum of two prime numbers.
¬¨‚Ä†-when are you coming back? ;


In the following article, we will talk about the theory of gambling and the statistical logic implemented in it. There is information on the connections with probabilities, where estimating the odds of a precise hand will guide your gaming choices. We’ve prepared data on the three fundamental principles underlying casino games and added some of the most common queries on the topic.
Theory of Gambling – Using the Concept of Interacting Choices
The feeling you get when you gamble is caused by the released dopamine, acting like a feel-good neurotransmitter that makes you feel excited. The body responds equally strong when you lose as well. Such risk conditions resulting in decision-making form the base of any gambling theory. The process of deciding among gambling strategies implies several logical development avenues, such as the existence of an expression of preference.
Game Theory in Gambling – History
The control of the games of chance results has been an unpredictable enigma for hundreds of years. Still, around the 16th century, mathematicians started to find connections between gambling and probabilities. Follow the time stamp of the probability theory, believed to be connected to the theory of gambling and statistical logic:
Gerolamo Cardano (1501–1576) – known as the first key figure in the development of probability theory around the mid-1500s. His book ‘Games of Chance’ contained the first use of probability along with a few effective cheating techniques.
Christiaan Huygens (1629–1695) – introduced the concept of expected value, which helped in identifying the probability of likely based on average outcomes from several experiments.
Abraham de Moivre (1667–1754) – mid-1700s mathematician and analytic probabilist who put forward the central limit theorem, which serves as a cornerstone to statistics. He solved one of the most famous problems – the one where two players start with a fixed number of counters, and the essence of the problem is counting.
Jacob Bernoulli (1655–1705) – proposing the first version of the law of large numbers.
Daniel Bernoulli (1700–1782) – known for resolving the St. Petersburg Paradox, a famous paradox in decision theory. It expresses the probability that you can calculate the expected payout at each round.
Pierre-Simon Laplace (1749–1827) – worked on the blend of gambling and probability theory. He wrote the Analytic Theory of Probability, where he introduced advanced versions of many previous problems.
Nowadays, gambling probability theory is the mainspring of many scientific fields, and even quantum theory comes down to probability distribution. It helps describe the motion of particles on a tiny scale and offers a better grasp of the odds. Many great ancient mathematicians and dice players have done something very beneficial in providing such gambling probability theory.
The Link Between Gambling Theory and Maths
Quantifying luck started long ago in the 16th century with Gerolamo Cardano. He outlined the navigation of the ‘sample space’ of possible events. An example is that two dice will make 36 different combinations, but only one of those would deliver two sixes. Many say this paved the way for a new area of mathematical study. Here is where you can use it to your advantage:
🏛 Understand the Concept:
The strategic aspect of decision making
The game theory increases your advantage
♠ Poker Theory:
The theory can have a significant impact
Reflects upon the efficacy of your strategy
🎰 Eligible Games:
Look for games with other players involved
The rules of probability apply similarly to all games, and you can use maths to try to beat virtual casino systems. Online casino titles happen in an environment, making it far easier to work out the probability on the spot. Although you can use maths to calculate your risk, some versions favour maths more than others, like blackjack.
Blackjack is hugely popular in land-based and online casinos alike. You can use maths to your advantage and turn it from a game for fun to a little more serious experience. Everybody can take advantage, but MIT students will do it much faster, just like shown in 21, the movie about the MIT blackjack team. They used card-counting to predict the probability of the next card winning big.
Theory of Gambling – Main Principles
The mathematical models underlined throughout the years provide intriguing possibilities for tilting the scales in your favour. The gambling theory and other topics related to the matter allow players to try exciting approaches to their preferred titles. That is applicable in land-based betting venues and at the best online gambling sites in the UK. Below, we have gathered essential information on the three fundamental principles.
The events in gambling have absolute probabilities that depend on the total number of possible outcomes or sample spaces. For example, in dice games, the sample space is six, with the likelihood of landing on any side one in six.
Games with substantial sample spaces, like poker, have events with small probabilities. For instance, in five-card poker, the possibility of drawing four of a kind is 0.000240. The chance of drawing the rarest hand, royal flush, is a mere 0.00000154. Estimating the odds of a particular hand will guide your gambling choices.
The expected value (EV) is the sum of all possible probabilities multiplied by their associated gains or losses. Its concept is important because it tells how much money you could expect to earn or lose. All casino titles have negative EVs in the long run, which explains how casinos profit from gamblers. You can also know it as house advantage or edge.
Even if luck may be the answer for some, the mathematical answer resides in the nuanced contrast between expected and actual values. EVs rule how much a player should expect. Instead, players are more interested in each hand’s basic values and its EV fluctuation.
For example, let’s assume that a flip of a coin pays an amount every time it lands on heads. At the same time, the player takes away the same amount when it lands tails. As a result, the expected value would be zero since the heads and tails’ probability is equal.
The volatility index is the technical term for standard deviation. It tells you the chance of earning more or less than the EV. The volatility index quantifies luck by telling players their odds of making more than the expected value for a specific number of rounds. The high volatility games significantly vary between the expected and actual outcomes.
Information Theory – Formal Expression of the Theory of Gambling
Information theory is easily explained as quantifying information to make the best decision in the face of imperfect information. In other words, how to make the best decision using only the information you have. The point of betting is to assess all relevant variables rationally, then compare them using odds and place a good bet.
This type of game theory in gambling is best used in sports. The handicapping here lends itself to information theory because of the availability of statistics. For many years economists have tested various mathematical approaches using sports as their laboratory, with vastly differing results.
Gambling Theory – Combination Between Art and Science
Elements such as mathematical probability, expected value, odds of winning, volatility index, bet size, and length of play paint a numerical picture of risk. It tells you whether a bet is worth pursuing or not. Still, reading the room and the opponents involves more than simple calculations. The theory of gambling and its statistical logic helps and is good to get the hang of it.
We hope our piece has given you enough details to help you understand and use the gambling theory when gaming in the UK. Even though we tried to be as detailed as possible, some of you might have additional enquiries. Therefore, we have gathered answers to the topic’s most commonly asked questions.
1⃣ What is the gambling theory?
In time, casino games and their outcomes became an intriguing topic for mathematicians and experts. The gambling theory and other topics related to tilting the scales in your favour by statistically calculating the probabilities are an essential part of your gaming routine.
2⃣ What is theory of gambling’s idea?
The effect gaming has on you causes interesting reactions in the body resulting in excitement. The theory of gambling’s idea is aimed at connecting gaming and probabilities. It helps you predict the outcomes to some extent. Having such an approach as part of your gaming is worth it.
3⃣ Who founded the theory of gambling and statistical logic?
Gambling theory and other topics related to casinos emerged many centuries ago. Famous mathematicians wanted to see if there was any relation between probabilities and game outcomes. The first scientists studying the gambling theory were important personas in the mid-16th century.
4⃣ What are the main principles of the gambling theory?
You can try several exciting approaches when considering playing your preferred titles. However, there are three main principles established by the explorers and mathematicians through time. Each relates to a different yet connected part of the gaming process. Keep them in mind to properly use the game theory in gambling.
5⃣ Is there a link between maths and the theory of gambling?
Yes, there is a connection between maths and the gambling theory. Here the probability element plays a crucial role. Moreover, the mathematical part does not apply to all gaming types. The probability apply to enough categories. Knowing the rules can help you work out the probabilities.
6⃣ Are the gambling and the information theory connected?
The information theory basically explains data to help you make the best decision out of an imperfect situation. Overall, the information theory mimics the gambling one. Both require specific info to make the best out of what you have in terms of statistics.


All employees under DDO Salaries with Bank A/c No.
Nursery, LKG, UKG, 1st Class, 2nd Class, 3rd Class, 4th Class, 5th Class, 6th Class, 7th Class, 8th Class, 9th Class, 10th Class(SSC), Intermediate
Telugu, Hindi, English, Mathematics, Enviromental Studies, Physical Science, Biological Science, Social Studies, Computer Science
Arithmetic, Real Numbers, Polynomials, Trigonometry, Applications of Trigonometry, sets, Statistic, series, Addition, Substraction, Multiplications, Divisions, Complex Numbers, Natural Numbers, Prime Numbers, Even Numbers, Odd Numbers, Probability, Factors, Geometry, Fractions, Decimals, Logaritham
Alphabet, words, sentences, Paragraph, lessons, units, grammar, spoken english, essay, rhymes, songs, active and passive voice, direct and inderect speech, simple sentence, compound sentences, complex sentences, parts of speeches, Noun, Pronoun, verb, adverb, adjective, conjunctions
How to Learn Mathematics.
How to Learn English Grammar.
How to Learn Vedic Mathematics.
How to Speak in English
How to Solve Mathematical Problems.
How to pass 10th class.
How to Solve Puzzles.
This is Educational Channel.
My FaceBook Profile: http://www.facebook.com/PCnu79
My Facebook Fan Page : http://www.facebook.com/naabadi
My Twitter : https://www.twitter.com/naabadi
Powered by NaaBadi.
Please Subscribe My Channel
Please Like My Videos
Please Share My Videos
Please Comment in the comment box......
My Whatsapp No 9951882682
Thank You for Watching


Buy The Zakon Series on Mathematical Analysis: Basic Concepts of Mathematics on ✓ FREE SHIPPING on qualified by Elias Zakon (Author). Basic Concepts of Mathematics has 2 ratings and 0 reviews: Published February 12th by The Trillia Group, pages, ebook. Elias Zakon is the author of Basic Concepts of Mathematics ( avg rating, 2 ratings, 0 reviews, published ) and The Zakon Series On Mathematical A.
|Published (Last):||10 October 2006|
|PDF File Size:||3.17 Mb|
|ePub File Size:||12.73 Mb|
|Price:||Free* [*Free Regsitration Required]|
There are no discussion topics on this book yet.
Amazon Advertising Find, attract, and engage customers. Would you like to tell us about a lower price? Exa mples ofsets are asfollo ws: Inhejoinedthe amthematics maticsfaculty at the University of Windsor, where the first degreesin the ne wly established Honours progra m in Mathe matics were awarded in The objects belonging to a set A are calledits ele ments or me mbers. Inhe took hisfa mily to the ne wly created state of Israel and he taught at mathematucs Technionin Haifa until You may post this work to your own website or other server ftp, gopher, etc.
This may be illustrated by the following exa mples. Product details Paperback Publisher: Thanks for telling us about the problem. Throughout hislife, Zakon maintained alove of music, art, politics, history, law, and especially chess; it wasin Israel that he achieved the rank of chess master. Amazon Inspire Digital Educational Resources.
Adolfo marked it as to-read Mar 13, This book helps the student complete the transition from purely manipulative to rigorous mathematics. Learn more about Amazon Prime. Downloads This work has been downloaded times via unglue.
Get fast, free shipping with Amazon Prime.
Basic Concepts of Mathematics is a Free eBook. [pdf]
In this post- Mc Carthy era, he often had as his house-guest the prolific and eccentric mathe matician Paul Erdos, who was then banned fro m the United Statesfor his political vie ws.
A si milar distinction applies to ordered n-tuples. Amazon Rapids Fun stories for kids on the go. If you are a student using this workfor self-study, no pay mentis required.
Elias Zakon (Author of Basic Concepts of Mathematics)
The Trillia Group Published: Axio ms of an Ordered Field 52 3. Basic Concepts of Math It should be noted lf the orderin zxkon the ele ments of a setfollow each other does not affect the equality of sets as stated above.
Alexa Actionable Analytics for the Web. East Dane Designer Men’s Fashion. To ask other readers questions about Basic Concepts of Mathematicsplease sign up. The LeningradInstitute of Technology was also evacuated to Barnaul upon the siege of Leningrad, and there he met the mathe matician I.
Basic Concepts of Mathematics by Elias Zakon. Site Favorites Latest Favorites. The many exercises and optional topics isomorphism of complete ordered fields, construction of the real numbers through Dedekind cuts, introduction to normed linear spaces, etc. We are publishing here the latest co mplete version of the first of these volu mes, which was usedin a one-se mester class required of all first-year Science students at Windsor.
Jon is currently reading it Jun 16, If you are a seller for this product, would you like to suggest updates through seller support? This book was prepared by Bradley J. M, N, P, etc. Want to Read Currently Reading Read.
Theset concept itself cannot be definedinsi mplerter ms. It can be used eitherforsupple mentaryreading or as a half-year course.
Lou Animashaun marked it as to-read Dec 15, Refresh and try again. Arith metic Operationsin a Field 5 4.
Basic Concepts of Mathematics
Pay ment is made through the website http: Also, if so me ele mentis mentioned several mathemativs mes,it still counts as one ele ment only.
In particular, but not exclusively, pay mentis requiredif: There’s a problem loading this menu right now. While at Windsor, leias continued publishing his research resultsinlogic and analysis.
Published February 12th by The Trillia Group first published I’d like to read this book on Kindle Don’t have a Kindle?


- Additional Details
- Dimensions: 10.50 H x 2.00 D x 8.50 W Inches
- Format: Board book
- Number of Pages: 11 Pages
- Release Date: August 1, 2013
Where can you find basic shapes? Everywhere! This delightful adventure takes children on a trip through town, where they learn basic shape recognition—a key preschool math skill. Kids follow Brother and Sister Bear to the beach, the park, the city, and many other stops, identifying circles, squares, triangles, and rectangles. Then they get to create their own shapes and PLAY-DOH sculpts using the double-sided playmat and PLAY-DOH components inside!


发信人: stealth (顺顺当当), 信区: EE
标 题: Re: 我怎么觉得DEVICE对学CIRCUIT DESIGN的人没什么用处。
发信站: The unknown SPACE (Wed May 15 11:35:15 2002) WWW-POST
I fully agree with you, after having worked with full-custom analog circuit
design for my Ph.D. over last three years.
Intuition is important, and experience is more important. You can only play
math on very few transistors. The math would become very complicated in e.g.
five-transistor amplifier , even you use the simpliest equation for each
transistor. Of course, it never hurts to have solid math, which will help you
understand the simulation result. But from my experience, if you can only
learn one kind of math, I would recommend you learn numerical analysis,
instead of device-physics-math. Having solid knowledge in numerical analysis
would help you a lot when you design/simulate large circuit. I often have to
deal analog circuit with 10k+ transistor, and I was naive enough before to
believe everything from the simulation.
If you also fabricate and test the chip you design, experience is very
important. That's is my biggest lesson dealing with large analog circuits. You
have a good idea, you design well, you simulate well, you verify and simulate
the final layout very well, there are still good chances that the chip won't
work as you expected when you test it.
【 在 madman (NaN) 的大作中提到: 】
: Yeah. I believe if one really understands something, one can describe
: it in two or three sentences in English. One who struggles with pages of
: math does not know a thing; and those formulas can be forgotten easily.
: In real work, what is important is good intuition, simple calculation, and
: the idea of the orders of magnitude. Seldom is there time to play with
: 【 在 alanshou (maverick) 的大作中提到: 】
: : I have learned Quantum mechanics for three semester, one in China, two in
: : Now I can do the math in Quantum pretty well, but do not think I really
: : understand Quantum. With my math become much more better since my coming
: : USA, the equation in Quamtum become more and more clear to me. But how
: : the physics in Quantum? I do not think it improves a lot. If you
: : the math, this does not make sure you can understand the subject. The math
: : just a langurage to decribes a natrual subject.
: : 【 在 heading (Lee~弥乐笑生生) 的大作中提到: 】
: : : RF是偏电磁场的吧，那就是说除了DEVICE的量子力学外，还要加上电磁场这一块罗
: : 下
: : : 教
※ 来源:．The unknown SPACE bbs.mit.edu．[FROM: 184.108.40.206]


Before I made the sweater, I was a good knitter and made a gauge swatch. When knitting, the number of stitches per inch will determine your final measurements, so you do a gauge swatch of approximately four inches square (10cm x 10cm) to determine, and from a section in the middle of the swatch, count how many stitches there are in the four inches and divide by four to get an accurate stitches/inch. The bigger the item, the more important an accurate stitch count is. When making something which is 48" around (like the bust of a Fat Lady Sweater), the difference between 5 stitches/inch or 5.25 stitch/inch is a couple of extra inches.
(It mad math skillz like this that make me such a good engineer. Hah! People think that engineers must be So Good at Math, but civil engineers are not the math geniuses. C'mon, we work with geometry and concrete...just like the ancient Romans!)
I even washed the gauge swatch, and measured after it had dried. Lovely! That was a smart thing to do because after washing, my sweater actually measured the dimensions I'd calculated.
Except I didn't bother to check for the difference in length before and after washing my swatch. Most knitting patterns just say "knit until item is so many inches long". So I did.
Guess what. Not only does silk really stink when it's wet, it also stretches out in length. A lot. Even though the sweater was dried flat, it lengthened about 3 inches everywhere. The sleeves that once reached my knuckles? Now they hang way past the fingertips. And the low cut bosom? Let's just say the depth of that neckline wouldn't have looked out of place on the cover of one of those bodice ripper romance novels.


Dr. Alyson G. Wilson
Dr. Wilson’s Curriulum Vitae
Dr. Alyson Wilson is a Research Staff Member at the IDA Science and Technology Policy Institute. She is also a Collaborating Associate Professor in the Department of Statistics at Iowa State University and a Guest Scientist in the Statistical Sciences Group at Los Alamos National Laboratory. She is a Fellow of the American Statistical Association and an Elected Member of the International Statistical Institute.
Dr. Wilson’s research interests include Bayesian methods, reliability, information integration, uncertainty quantification, and the application of statistics to problems in defense and national security. Before joining STPI, she was an associate professor in the Department of Statistics at Iowa State University (2008-2011), a Technical Staff Member/Scientist 5 in the Statistical Sciences group at Los Alamos National Laboratory (1999-2008), a senior statistician and operations research analyst with Cowboy Programming Resources (1995-1999), and a mathematical statistician at the National Institutes of Health (1990-1991).
Dr. Wilson has served on numerous national panels, including the National Academy of Sciences (NAS) Committee on Mathematical Foundation of Validation, Verification, and Uncertainty Quantification (2010-2011), the NAS Committee to Review the Testing of Body Armor Materials by the U.S. Army (2009-2010), the NAS Oversight Committee for the Workshop on Industrial Methods for the Effective Test and Development of Defense Systems (2008-2009), the Sandia National Laboratories’ Predictive Engineering Science Panel (2008-2013), the NAS Panel on Methodological Improvement to the Department of Homeland Security’s Biological Agent Risk Analysis (2006-2008), and the NAS Panel on the Operational Test Design and Evaluation of the Interim Armored Vehicle (2002-2003). She was on the organizing committee for the Department of Energy Office of Science (DOE/OS) Workshop on Mathematical Issues for Petascale Data Sets (2008), and an invited participant in the Chief of Naval Operations Distinguished Fellows Workshop on Critical Infrastructure Vulnerability (2008), the DOE/OS Workshop on Mathematical Research Challenges in Optimization of Complex Systems (2006), and the DOE Simulation and Modeling for Advanced Nuclear Energy Systems Workshop (2006). In 2006, the chaired the American Statistical Association President’s Task Force on Statistics in Defense and National Security.
Dr. Wilson is the winner of the Los Alamos National Laboratory (LANL) Director’s Distinguished Performance Award (2008), the LANL Star Award (2008), the DOE Defense Programs Award of Excellence (2007), and the LANL Achievement Award (2000, 2005). She is a founder and past-chair of the American Statistical Association’s Section on Statistics in Defense and National Security. She is currently the Reviews Editor (2011-2013) for the Journal of the American Statistical Association and The American Statistician.
Dr. Wilson received her Ph.D. in Statistics from Duke University, her M.S. in Statistics from Carnegie-Mellon University, and her B.A. in Mathematical Sciences from Rice University.


Probability of winning slot machine jackpot
Probability of winning slot machine jackpot
Pontoon is a variation of Blackjack and it generally follows the same rules but Pontoon allows users to double down with more than two cards and has options to buy before taking hits, and has several other slight gameplay variations. Note: Take a look at our guide on Blackjack to discover a list of common terminology as well as more information on the game and its most common variations such as European Blackjack, Multi-hand, and Blackjack Switch. Baccarat is another card game in which the player plays against the casino or house, probability of winning slot machine jackpot. Players can also enjoy about 10% payback to their accounts, probability of winning slot machine jackpot. The top casinos in Thailand are kicking off the 2013 with exclusive promotions all month long and, unlike signup bonuses, these deals are offered to registered players too! This month, Jackpot City is offering our visitors an exclusive $10 no deposit bonus, while Royal Vegas offers our visitors 20 free spins in addition to the first 3 deposit bonuses meaning existing players can cash in too! We recommend only the best in online gambling and our top 5 list below will not disappoint check them out to get the latest promotions in 2013. List of Top 10 Thailand Online Casino Sites, probability of winning slot machine jackpot.
Slot machine probability mathematics
Similarly any two spins of a fair roulette wheel are independent. Thus, knowing that a slot machine has just paid a jackpot has no influence on the probability. Therefore, we can say that the higher the rtp, the higher the jackpot win probability. However, rtp does tell us the likelihood of wins,. Odds and return to player. Many gamblers believe that video poker machines are a waste of time and money and instead, they head straight to the slots to try. Now, if you are wondering if the probability of getting a jackpot is 1 in 10. Progressive slots lure players with the promise of life-changing prizes, but slot fans should bear in mind the probability of actually winning the pot is rather. The law of averages ultimately will let the casino win. 09 by 100 percent, we can see that the probability of winning is 9 percent. As you might know, each spin on a slot game is made with. This means that the slot will have: 4x4x4 = 64 winning combinations. If the jackpot pays out for three mango symbols, with a mango on each reel,. The odds of winning the top prize at maximum coin play on the slot machines ranges from 1 in 4,096 to 1 in 33,554,000. Another way to look at the odds of. Winning money by playing the slots can certainly happen, but as with all other forms of gambling, the odds are usually in favor of the house. The odds are very low, but you can literally win millions if you hit the jackpot on some progressive payout slot machines Understand the payouts As a baccarat player, you choose to bet on either hand, with the payout being 1:1 on a winning player bet and 19:20 on a winning banker hand, probability of winning slot machine jackpot. Gambling at online casinos in Pennsylvania can be a lot of fun. We pride ourselves in the knowledge and information we bring to PA players. We also know that gambling is something that requires a level of respect and care. By ensuring you practice responsible gambling measures, you can feel confident in knowing you will have a great time, slot machine probability mathematics. Your chances of winning any major jackpot start at 50 million to more than 100 million to one. But the odds of hitting that jackpot are at least 1 in 1000. It would take 2 hours of playing before the increase in the jackpot amounted to a. When you play one of these progressive jackpot slots like major millions, which is a game where your probability of hitting the jackpot. For example, on a three-reel slot machine with 64 stops on each reel, only one of which is mapped to a jackpot-paying symbol, the probability of. Pick 3 lottery spreadsheet. Load the lottery payment calculator into your browser and scroll to the bottom of the page. The winning numbers and jackpot. Poker machines vary in the odds they offer of winning a big prize. Using probability when hitting the slot machines bally fitness locations nj scratch off game free win big jackpot app. Worldwide bonus odds machine a of are. A slot machine consists of 3 reels, and each real consists of 32 stops. To win the jackpot of $10,000, a player must get a wild symbol on the center line of. Hit frequency is defined as how often a slot machine will have a winning. Odds and return to player. Many gamblers believe that video poker machines are a waste of time and money and instead, they head straight to the slots to try. The odds of winning the lottery stand at 1 in 300 million, while your chances of hitting the jackpot in a progressive slot game is around 50. Casino slot machine strategy - odds on slot machines. Visit the igt home page which indicates the progressive jackpots on a variety of slot machines Die Starken von Platinum Play sind: Riesige Auswahl an Lizenzspielen Eine offizielle Partnerschaft mit einem der fuhrenden Unternehmen auf dem Glucksspielmarkt Microgaming Das Vorhandensein aller popularen Genres des Glucksspiels von Roulette und Slots bis hin zu Video Poker Das Vorhandensein von beeindruckenden progressiven Jackpots Bequeme und stabile mobile Anwendung Tolles Bonusprogramm Vertraulichkeit und vollstandige Sicherheit personenbezogener Daten. Stabile und schnelle Auszahlung aller Gewinne, vegas casino online no deposit code. It offers a safe gambling environment for players, casino puerto madero banda xxi. In Canada, posh casino while the slot H-plane pattern is the same as the dipole Starcity casino darwin i am extremely pleased with their service and am. Residents from Thai are allowed to participate in lotteries owned by the government, which are normally drawn on the fifth and 30th every month or place bets at Turf Club races based in Bangkok, quick hits slot free games red. On the other hand, casinos are banned in Thailand and players who would like to play have to travel to other parts of Asia where casinos are permitted like Singapore and Macau. The Payout percentage of an Online Casino combines the payout rate of each game and game category. The return to players, also called (RTP) is one of the most important figures in casinos, horario casino flotante puerto madero. While choosing a casino where to play games at, the player must watch out for several things. Two of them were already mentioned earlier the selection of available games and offerings in terms of promotions and bonuses, especially no deposit ones, wheel of fortune board game australia. While you will not be banished for landing lower than 6, you will lose the game, vegas casino online no deposit code. In Italy, the original game of Baccarat was played slightly differently than its modern counterpart. Looking for casinos with No Deposit Bonuses that allow you to Keep Your Winn. We have the Latest Real Money No Deposit Bonus offers for Leading Online Casinos, bbb sugharhouse online casino. PLAY WITH FRIENDS - Recruit friends to play multiplayer poker in the HD Poker app and earn an exclusive chest as a reward! Invite friends to join your table - play poker, grind chips, gems & cards and win prizes & rewards, riddler trophy under gotham casino. However, do note that there are different withdrawal limits and duration for the various payment methods, online casino med free spins. Therefore, the onus is on you to check on which payment method is best suited to your convenience. It may require a really long time then you will see it difficult to explain all, casino poker table dimensions. However, there are bunch ideas, procedures realities notwithstanding concerning playing betting games.
Slot machine probability mathematics, slot machine odds calculator
You are not facing a machine, and you can watch the dealer in real-time at a live casino, which is also a strategic advantage. A real casino, everywhere, all the time, probability of winning slot machine jackpot. This seems obvious, but the major advantage of live casino is to be able to play as in a real casino, in the atmosphere of a real casino, but everywhere. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices, probability of winning slot machine jackpot. This could sway customers away from Mount Airy potentially, slot machine probability mathematics. Slot machines of yesteryear were mechanical, while today's slots are computers. All slot machines contain a microchip called a random number. What are the probabilities of winning on a slot machine? we previously defined that randomness means the lack of pattern or predictability in. The number of possible combinations is fairly easy to calculate. You multiply the number symbols each slot has together. For example for a three. This eighth book of the author on gambling math presents in accessible terms the cold mathematics behind the sparkling slot machines, either physical or. Assuming we have three identical reels as listed above, then the odds of getting the jackpot symbol on any reel is 2/128. The probability of hitting the jackpot. Barboianu's eighth book on gambling math presents in accessible terms the cold mathematics behind the sparkling slot machines, either physical or virtual. Slot machine probability: a mathematical approach to slots. Free spins raging bull it is integrated to risk as player may lose his money or. One example that came up during the input was the link of probability to gambling. Gambling is a widespread, popular and recreational activity (. Machine – slot machines are a good example here. Some slots have better odds than others. Denomination – the. In accessible terms the cold mathematics behind the sparkling slot machines, either physical or virtual. Slot machines are very simple to play and don't require any specific gambling knowledge. This is especially true when we consider the chances of joint events (e. , three win symbols showing on a slot machine) or the chance of one event compared to Numbers in the drawing. “the probability of slot machines” answers. The backbone of casino game mathematics is “probability. ” informally, we understand probability as a number that describes the chance that something will occur. I came to know that the odds of slot machines are officially described in what is called a par sheet. I set up a par sheet in google sheets and added several. Gaming regulation and mathematics elements or events that have definite probabilities of occurrence. For example, a slot machine is honest if the outcome. Quite simply, to calculate the total number of possible combinations of symbols on a machine like this, you multiply the number of symbols on each reel by the. The number of possible combinations on any slot machine is easy to calculate. You need to multiply the total number of symbols (stops) each slot. The fundamentals of slot math. The confidence interval – the probability that the. Slot machine, markov chain, strong law of large numbers, central limit. Probability assignment - slot machine. Math 10+ probability assignment - fruit machine 2011. With one coin bet: what it pays, combinations, probability (combinations divided by 643,. Knowing how much a slot frequently pays out and the amount of each win allows you to find a betting strategy to turn the odds slightly in your. In slot machine games the odds paid on the game are based on the Here you will find that the amount of slot machines at each casino has an average of 1,400. Although these figures do include video poker, the majority account for slot games, slot machine probability mathematics. This should be more than enough to cater to the large amount of players that visit. You will find a large variety of different games, ranging from penny slots to hundred dollar slots, as well as classic to video slots. Bitcoin casino winners: Eagle Bucks - 1.9 dog Jason’s Quest - 403.8 eth Dolphin Quest - 462.1 eth Fairytale Legends Red Riding Hood - 126.1 ltc Fruit Shop Christmas Edition - 662.2 bch Eagles Wings - 571.2 bch Couch Potato - 25.4 btc Kicka$$ - 317.7 eth Cleopatra’s Coins - 208.8 usdt True Illusions - 726.9 ltc Wunderfest - 70.5 btc Crazy Bot - 461.2 usdt La Dolce Vita Golden Nights - 289.1 eth Magician Dreaming - 657.7 bch Good Girl Bad Girl - 319.9 ltc Videoslots, card and board games: Vegas Crest Casino Enchanted Mermaid OneHash Wild Turkey mBit Casino Lucky Coin CryptoGames La Cucaracha Betchan Casino Super 7 Stars Bitcoin Penguin Casino Super 10 Stars Bitcasino.io Pampa Treasures Sportsbet.io Wunderfest FortuneJack Casino A Night in Paris FortuneJack Casino KTV 1xBit Casino Wild Carnival Diamond Reels Casino Treasure Island 22Bet Casino Dragon Born 1xSlots Casino Royal Unicorn Oshi Casino Arabian Nights


Stefania obtained a Bachelor’s Degree in Statistics and Actuarial Science in September 2010 followed by an MSc degree in the same field on September 10 2012 at the University of Sannio in Benevento, Italy. The main subjects of her degree course were: statistics, mathematics, financial mathematics, econometrics, risk theory, statistical modeling, time series, statistical inference, microeconomics, macroeconomics, marketing, samples theory and data analysis. During her Master’s she has had an Erasmus experience in Lithuania where she also started to write her Master’s thesis.
After her Master’s she started to give some private lessons in Statistics and Mathematics of Finance, and since March 2013 she is a PhD Student at SERAF (Norwegian Center for Addiction Research, Institute of Clinical Medicine, Faculty of Medicine, University of Oslo).
Can variations in drug presence in sewage indicate prescription drug abuse?
The primary goal of this project is to look at weekly temporal variation in concentration of drugs detected in waste water analysis. The goal is to develop a statistical model for weekly patterns in drugs of abuse vs. not abused drugs in Norway and compare these results with those obtained in different European cities.


The analysis group is broadly interested in the areas partial differential equations, differential geometry and geometric analysis.
Follow the tabs above to find more information about the members of the analysis group, and the PDE Seminar we are running.
Students who are interested in taking courses in analysis, geometry, and PDE, are encouraged to look at the information page for MSc students, and should contact us directly for MSc thesis topics.
Senior Lecturer in Pure Mathematics
Office: Peter Hall 202
Research interests: Microlocal analysis, differential geometry, index theory, spectral asymptotics, singular spaces, scattering theory.
Lecturer in Pure Mathematics
Office: Peter Hall G28
Research interests: Minimal surfaces, isoperimetry, elliptic and parabolic differential equations, geometric measure theory, geometric analysis
Lecturer in Pure Mathematics
Office: Peter Hall 204
Research interests: General relativity, global evolution problems for hyperbolic partial differential equations, geometric analysis.
We are very excited that the MATRIX workshop on Hyperbolic Differential Equations in Geometry and Physics begins today!3 April, 2022
Informal lectures on general relativity
We are offering a few informal lectures on general relativity, Thursdays, Evan Williams Theatre (Peter Hall G03), 4:15-5:15pm, starting this Thursday, March 3. This will not be a formal lecture course at all, instead I am hoping to give a loose introduction to a few research topics in mathematical general relativity, which are accessible to Master's students with interests in analysis, differential geometry, …1 March, 2022
We're launching an Analysis Seminar at the University of Melbourne this year. We are hoping for many talks to happen in person, maybe even en plein air, like the seminar given by our first speaker, Zoe Wyatt. Others will be online, like by our second speaker this Friday, Allen Fang. Either follow these pages, or contact us directly, to be sure …28 February, 2022
Jan Sbierski (University of Edinburgh)
Wednesday, March 30, Peter Hall 213 2:15PM
Title: On holonomy singularities and inextendibility results for Lorentzian manifolds
Given a solution of the Einstein equations a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution. The latter question, as will be explained in the talk, is intimately tied to the strong cosmic censorship conjecture in general relativity which states in the language of partial differential equations that global uniqueness holds generically for the initial value problem for the Einstein equations. This talk will give a basic introduction to the problem of inextendibility of Lorentzian manifolds, beginning with classical methods exploiting a blow-up of curvature to show the inextendibility with a twice continuously differentiable Lorentzian metric and concluding with the presentation of a recent methodology exploiting a blow-up in holonomy to show inextendibility with a locally Lipschitz regular Lorentzian metric.
Brian Krummel (Melbourne University)
Wednesday, March 23, Peter Hall 213 2:15PM
Title: Fine structure of the free boundary for a penalized thin obstacle problem
We consider a two-penalty elliptic boundary obstacle problem, which is motivated by applications to fluid dynamics and thermics. Using monotonicity formulas of Almgren, Weiss, and Monneau, we establish rectifiability of the free boundary and uniqueness of blow-ups at free boundary points. We briefly discuss analogous parabolic problem, which represents a physical system evolving in time. Joint work with Donatella Danielli.
Allen Fang (Sorbonne University)
Friday, March 4, on zoom, 9AM
Title: A new proof for the nonlinear stability of slowly-rotating
Abstract: The stability of black hole spacetimes is a critical question in mathematical relativity. The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter
spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof utilizes spectral methods to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.
Zoe Wyatt (University of Cambridge)
Thursday, February 10, Evan Williams Theatre, 2PM
Title: Stabilising relativistic fluids on slowly expanding cosmological spacetimes
Abstract: On a background Minkowski spacetime, the relativistic Euler equations are known, for a relatively general equation of state, to admit unstable homogeneous solutions with finite-time shock formation. By contrast, such shock formation can be suppressed on background cosmological spacetimes whose spatial slices expand at an accelerated rate. The critical case of linear, i.e. zero-accelerated, spatial expansion, is not as well understood. In this talk, I will present recent work concerning the relativistic Euler and the Einstein-Dust equations for geometries expanding at a linear rate. This is based on joint works with David Fajman, Todd Oliynyk and Max Ofner.
A MSc thesis is a project over 3 semesters typically on a topic close to the current research interests of your supervisor. Talk to us directly to hear what they are!
A list of topics is also maintained on MS Prime
Students who are interested in writing an MSc thesis in the analysis group are encouraged to take the following courses:
- Measure Theory
Measure Theory (MAST90012) is a core course offered every two years in Semester 1.
- Functional Analysis
Functional Analysis (MAST90020) is offered every two years in Semester 1.
- Partial Differential Equations
Partial Differential Equations (MAST90133) is offered every two years in Semester 2.
- Differential Geometry
Differential Geometry (MAST90143) is offered every two years in Semester 2.


Way back in G-Engine #4, we rendered a triangle, and there was much rejoicing. Since then, the posts in this series have primarily focused on implementing mathematical constructs in code. This point is the final math pre-requisite before we can move on to more interesting things.
Think of a game engine as its own little universe: at first, there is nothing - the void. Concepts such as time, position, and orientation don’t exist yet. Suddenly, the game loop and delta time introduces the concept of time. With vectors, we can convey positions, directions, and distances. Slowly, our universe takes shape and we can convey important concepts in code.
But what about rotations? Vectors can store position, direction, and scale data, but they are not effective structures for 3D rotations. We need some other option to convey rotations in code.
One structure that is very effective for 3D rotations is the Quaternion. Much-maligned for their apparant complexity, Quaternions allow us to efficiently store and use 3D rotation data. In this post, we’ll explore why we use quaternions, what they are, how to perform common operations with them, and finally I’ll provide some tips for writing your own
A 3D rotation can be defined in terms of two pieces of information: the axis of rotation and the angle of rotation.
For example, if you stand up and turn your head to the left or right, the axis of rotation points straight up in the air and the angle of rotation will be either a positive or negative number depending on which direction and how far you turned your head.
Rotations are commonly described like so: “a rotation of A units about the X axis.” In that description, the axis of rotation is the X axis, and the angle of rotation is A.
Usually, a rotation angle’s units will be either degrees or radians. A full rotation is 360 degrees or 2π radians. A half rotation is 180 degrees or π radians. The unit circle can be helpful to visualize the full range of angles.
The coordinate axes X, Y, and Z are commonly used as axes of rotation. However, any unit-length vector can be used as an axis of rotation. For example,
(0.707, 0.707, 0) is an axis halfway between the X and Y axes.
What Does a Rotation Do?
Conceptually, a rotation changes an object’s orientation. If you are facing north, and you “rotate 90 degrees about the up axis”, you’ll be facing west (assuming that positive angles are counter-clockwise). A rotation is applied to an object to change its orientation.
In a virtual world, changing an object’s orientation means changing its facing direction or its position. When the object is a 3D model, this means that each of the model’s vertices are rotated. Generally, we can say that rotations in a 3D world primarily function by modifying a vector in some way (since we usually respresent positions and directions using vectors).
When a rotation is applied to a vector, the portion of the vector that is parallel to the axis of rotation does not change, but the portion that is perpendicular to the axis does change. Though we describe rotations as being “about” an axis, the rotation actually affects everything on the 2D plane perpendicular to the axis.
In the earlier example about facing north and rotating 90 degrees, it was unclear whether you’d end up facing east or west after rotation unless you specify whether positive angles are clockwise or counter-clockwise.
This decision speaks to the handedness of the rotation. It is so-called because you can choose either a “left-handed” system or a “right-handed” system. In a left-handed system, positive angles are clockwise. In a right-handed system, positive angles are counter-clockwise.
The concept of handedness comes from a handy (hah) method of remembering what convention is being used: point your thumb in the direction of the axis of rotation; your fingers curl in the direction of a positive rotation. If you point your thumb/axis towards your face, your fingers curl clockwise on your left hand and counter-clockwise on your right hand.
Handedness is a topic we briefly touched on when discussing the cross product, and we will discuss it again in more detail when we dig into transforms and coordinate systems in a future post.
In 2D, a rotation can be represented by a single
float value (the angle of rotation). A 2D game only has one 2D plane to rotation, so there’s only one possible axis of rotation!
In 3D, a rotation can occur about any of the three coordinate axes or an arbitrary axis. The possible rotations are more vast than in 2D, so the method for representing them also becomes more complex.
There are a few different methods we could use to represent 3D rotations. But what makes a good representation? The primary criteria are:
- Doesn’t take up too much memory.
- Concatenating multiple rotations into a single rotation should be efficient.
- Rotating a vector should be efficient.
- Converting to and from matrix form should be efficient.
Some secondary criteria worth considering:
- When animating objects, rotations need to be interpolated. So, interpolating ought to be correct and efficient.
- A physics simulation can affect an object’s rotation. So, a representation that can be easily used in numeric integration would be nice.
Based on these criteria, let’s briefly discuss a few possible representation options: Euler angles, axis/angles, matrices, and finally quaternions.
Euler Angle Representation
Perhaps the most intuitive way to represent a 3D rotation is with three
float values, each representing an angle of rotation about the x, y, and z coordinate axes. This seems like a natural extension of a 2D rotation.
This representation has two benefits: it takes up little memory, and it is fairly intuitive. But otherwise, this representation has some downsides.
Concatenating rotations and rotating vectors are not straightforward or efficient operations. Interpolating between rotations also doesn’t work correctly. Converting to and from matrix representation is not efficient and has some limitations. This representation also suffers from a phenomonon called gimbal lock.
Because Euler angles are so intuitive and easy to visualize, they are often used in 3D modeling package and commercial game engine interfaces to specify rotations. However, they are usually converted to/from another internal format to avoid the problems outlined above.
Another way to represent a 3D rotation is by specifying an axis of rotation and a rotation about that axis. Using this method, it is possible to represent any 3D rotation using just four
float values (three for the axis vector, one for the angle).
This representation doesn’t suffer from gimbal lock and it only uses four
float values. However, it still has a lot of the problems of Euler angles: concatenating rotations and rotating vectors are both inefficient. Converting to matrix form is doable, but converting back has some limitations.
The idea of axis/angle is powerful, easy to understand, and easy to visualize. If only we could use the axis/angle concept with a more efficient structure… (mysterious music and foreshadowing)
3D graphics APIs require that scale, rotation, and translation data be provided in a matrix structure, so it seems reasonable to just store rotations in a matrix directly and be done with it.
Actually, this is not a bad idea! A matrix allows us to concatenate rotations and rotate vectors efficiently using matrix multiplication. Matrices also interpolate pretty well, and they can be used in physics simulations. And obviously, there’s no need to convert to/from matrix form in this case!
Matrices do take up more space than other options. A 3D rotation requires a 3x3 matrix, or nine
Matrices tick a lot of our boxes, but quaternions usually edge them out by being somewhat more performance and memory efficient. Position and scale are represented as vectors before being converting to matrix form for the rendering system - rotations can also benefit from an intermediate representation.
Quaternions are fairly complex mathematical objects, but when used to represent rotations, you can think of them as axis/angle rotations. Quaternions allow us to use the idea of axis/angle representation, but take advantage of a more efficient structure.
The benefits of quaternions include:
- Memory efficient, at only four
- Concatenating rotations is efficient.
- Rotating a vector is (fairly) efficient.
- Converting to and from matrix form is efficient.
- Interpolation works well and is more efficient than matrix interpolation.
- They work well in physics simulations.
There are two downsides to quaternions. First, the math is somewhat obtuse and difficult to understand. Second, vector rotation is slightly less efficient than in matrix form - if you need to rotate a lot of vectors, consider converting to matrix form first.
Because of the benefits quaternions provide and the minimal downsides, they are commonly used in game engines, and they’re what I’ll spend the rest of this post discussing.
What is a Quaternion?
Quaternions were originally divised as a way to work with complex numbers. If, like me, you are rusty with complex numbers, that’s OK! Quaternions used as rotations do not require a deep knowledge of complex numbers.
A quaternion consists of four values: x, y, z, w. The
(x, y, z) portion is called the vector part and
w is called the scalar part.
The vector part roughly correlates to an axis of rotation, while the scalar part roughly correlates to an angle of rotation. The axis and angle data are encoded in these values, but some math is required to extract them.
A quaternion may seem similar to a 4D vector - both consist of four
float values called x, y, z, w. Though not interchangeable, it’s true that some quaternion operations behave just like vector operations, and they even share some of the same terminology.
Fortunately, there are only a few quaternion operations that we need to define to use them for 3D rotations.
Addition and Subtraction
Let’s start with something simple: to add and subtract quaternions, simply add or subtract corresponding components.
(4, 3, 2, 1) + (-3, 5, 2, 6) = (1, 8, 4, 7)
You might expect that adding two rotation quaternions together would concatenate the two rotations, but that isn’t the case.
Multiplying two rotation quaternions has the effect of concatenating the rotations. So, it’s a pretty important operation!
Quaternions have unique multiplication rules that also act as their defining characteristic. This wikipedia section explains the multiplication rules in a pretty digestible way (keeping in mind the imaginary number equality i 2 = -1).
Given two quaternions
q2, consider the vector parts and scalar parts separately. Say
q1.v refers to the vector part and
q1.s refers to the scalar part.
Multiplication can then be performed like so:
result.v = (q1.v ⨯ q2.v) + (q1.s * q2.v) + (q2.s * q1.v); result.s = (q1.s * q2.s) - (q1.v • q2.v);
In other words, the result’s vector part is calculated by summing three things:
- Cross product of the two vector parts.
q1’s scalar part multiplied by
q2’s vector part
q2’s scalar part multiplied by
q1’s vector part
And the result’s scalar part is the product of the two scalar parts minus the dot product of the two vector parts.
The order of multiplication affects the result.
q1 * q2 has the effect of rotating by
q2 and then by
q2 * q1 has the effect of rotation by
q1 and then
q2. In other words, concatenated rotations occur in right-to-left order.
Length and Normalization
In this regard, vectors and quaternions share behavior and terminology. You can calculate the length of a quaternion using the Pythagorean theorem:
length = sqrt(x * x + y * y + z * z + w * w);
A unit quaternion has a length of
1. To normalize, divide each component of the quaternion by the quaternion’s length.
Quaternions used for 3D rotations should always be unit length. This usually makes the math simpler and more efficient. But it also means that accidentally using a non-unit quaternion will give incorrect results.
Multiplying two unit-length quaternions always results in another unit-length quaternion. But be careful of other operations that might de-normalize the quaternion (many floating-point operations or interpolation). If a quaternion has recently been subjected to a lot of operations, it may be a good idea to normalize it before using it to rotate something.
As with vectors, you can multiply a quaternion by a scalar value, which multiplies each component by the scalar value and affects the length of the quaternion.
5 * (3, 2, 8, 0.5) = (5 * 3, 5 * 2, 5 * 8, 5 * 0.5) = (15, 10, 40, 2.5)
You might expect that negating a quaternion (multiplying by
-1) would negate the rotation it represents. But actually, a negated quaternion represents the exact same rotation as the original quaternion!
Negation will negate both the axis of rotation and the angle of rotation. Imagine a counter-clockwise rotation about an axis. Negating the axis causes the angle of rotation to flip to clockwise. But negating the angle causes the angle to flip again back to counter-clockwise!
If we want to actually negate a rotation, we have to negate the axis or the angle, but not both. To do this, we can use quaternion inversion.
As with matrices, multiplying a quaternion by its inverse gives us the Identity quaternion, which has a vector part of
(0, 0, 0) and a scalar part of
If a quaternion represents a rotation about an axis, the inverse represents the opposite rotation about that axis. The inverse undoes the rotation performed by the original quaternion.
Fortunately, the inverse is simple to calculate for quaternions.
Quaternions have a concept borrowed from complex numbers called the conjugate. A quaternion’s conjugate has the same scalar part, but the vector part is negated.
conjugate.v = -q.v; conjugate.s = q.s;
A quaternion’s inverse can then be calculated as
conjugate / lengthSquared. Since the quaternions we use for rotations are unit length, this equation simplifies: a unit-length quaternion’s inverse is equal to its conjugate!
Negating only the axis via inversion causes the angle to negate, which is exactly what we want in order to calculate an opposite rotation.
Applying a quaternion to a vector causes the vector to rotate about the axis. As described previously, the portion of the vector that is perpendicular to the axis of rotation is modified, while the parallel part does not change.
To rotate a 3D vector using a quaternion, we first treat the vector as though it was a quaternion with a
w value of zero. We can then use quaternion multiplication to rotate the vector using this equation:
rotatedVector = quat * vector * quatInverse;
In other words:
- Multiply the vector by the quaternion.
- Multiply the result by the quaternion’s inverse.
This somewhat unusual structure is called a sandwich product because the vector is “sandwiched” between the quaternion and the quaternion’s inverse.
The reason this causes the vector to be rotated correctly involves some fairly complex mathematical manipulation - I recommend one of the resources mentioned at the end of this post if you’re interested in learning more.
Quaternions are efficient, but the problem remains that other representations like Euler angles and axis/angle format are easier to manipulate and debug. We must also be able to convert to matrix form to send rotation data to the rendering system.
So, let’s discuss how we can convert between quaternions and other representations.
If the vector rotation equation is expanded and some substitutions are performed, you end up with this equality:
quaternion = sin(angle/2) * axis + cos(angle/2)
This tells us a few interesting things:
- The scalar part
wof a quaternion is equal to
- The vector part
(x, y, z)of a quaternion is equal to
axis * sin(angle/2).
- The length of the vector part is
Using this, we have a fairly simple way to create a quaternion from an axis and an angle:
// Assuming the axis is normalized... q.s = cos(angle * 0.5f); q.v = sin(angle * 0.5f) * axis;
To do the opposite (quaternion to axis/angle), we just need to reverse this math:
angle = acos(q.s) * 2.0f; axis = q.v / sin(angle * 0.5f);
Euler Angles Conversion
Each Euler angle is a rotation about the X/Y/Z coordinate axes - represent each one as an axis/angle. From there, generate one quaternion per axis. Finally, concatenate the three quaternions to get the final quaternion.
Converting from a quaternion to Euler angles is not inefficient, but it is complex. I’m not sure I could fully explain these equations if I tried! See this wikipedia page for full details, and of course the G-Engine source code for source code.
Converting a matrix to a quaternion exploits some fortunate observations about the matrix format:
- The trace of a rotation matrix (the sum of diagonal elements) equals
2 * cos(θ) + 1. We’ll use this as the scalar part of the quaternion.
- Subtracting certain symmetric elements of a rotation matrix for each component of a vector gives us
2 * sin(θ) * axis. We’ll use this as the vector part of the quaternion.
These two pieces of data are very close, but not quite, the data required to calculate a quaternion from an axis/angle. To get it in the right form, it’s actually possible to simply:
1to the scalar part (resulting in
2 * cos(θ) + 2).
- Normalize the quaternion.
And that’s it! Surprisingly, these operations cause the quaternion to fit the equation mentioned in the axis/angle section above, and we’re good to go.
Converting a quaternion to a matrix simply requires populating each entry in the matrix using specific values from the quaternion. The values required for each entry are outlined here.
Writing a Quaternion Class
Unlike vectors and matrices, we only need one quaternion class. The class should support at least the following operations:
- Construction, Copy Construction, and Assignment
- Equality checks
- Addition and subtraction
- Quaternion multiplication
- Scalar multiplication
- Length calculation and normalization
- Inverse calculation
- Vector rotation
- Extraction of axis/angle and euler angles
Next, I’ll highlight some considerations when writing a quaternion class.
I decided to name my class
Quaternion (shocking, I know). If you don’t like typing, I’ve also seen
Quat used. As always, we want the name to be clear!
Like a 4D vector, the x/y/z/w elements should all be
float values, and they should be laid out sequentially in memory. There should be no other data members in the class.
Constructors & Accessors
You want this class to hide the internal complexity of quaternion math and expose an easy to use public interface. This allows developers to use quaternions to perform rotations without having to understand everything happening under the hood.
One way you can do this is by providing a set of constructors that allow you to generate quaternions from various friendlier representations. In addition to the default constructor and a constructor that lets you set the x/y/z/w component directly, I’d recommend having constructors that take in axis/angle, matrix, and euler angle representations.
Likewise, you will want accessors that allow users to convert a quaternion back to axis/angle or Euler angle representations. You could also have a function to convert to a matrix, but this logic commonly exists inside separate functions related to transformation matrices (something I’ll cover in a future post).
As with vectors, I think it’s OK to leave the data members public on a quaternion. The structure of the internal data is well-defined and unlikely to ever change.
There are times when it may be valuable to directly manipulate or access the internal values. For example, direct access results in more readable code when writing out equations involving quaternions.
It can be helpful to output a quaternion to the log for debugging or data recording. You can overload
operator<< as a convenient way to do this.
When debugging rotation issues, the x/y/z/w values of the quaternion are not very helpful - it’s hard to tell whether the values are correct or not. For this reason, I usually output the quaternion in axis/angle format. This makes it more obvious when there’s a bug in rotation code (maybe the axis is pointing the wrong way or the angle is obviously wrong).
For similar reasons, I usually output angles in degrees instead of radians. I think most people find degrees a little more intuitive than radians, but it’s a matter of personal preference.
Defining some static quaternions for
Zero are useful. The Identity quaternion has a vector part of
(0, 0, 0) and a scalar part of
1. The Zero quaternion has all zeros.
The vector, matrix, and quaternion classes form a core set of math classes from which we can start to build a 3D transform system and rendering pipeline. The next post in this series will finally start to dive into this more exciting material!
If you plan to write your own game engine, creating these low-level mathematical constructs can be a huge hurdle. Some aspects are difficult to understand, it’s easy to make mistakes, and it must be done before you can see 3D objects moving around a 3D world. Hopefully, these last few posts act as a helpful guide!
As with previous math topics, I’ve tried to provide enough info to enable effective use of quaternions in a game engine. If you want more in-depth explanations for some of the math and derivations that I glossed over, check out these resources:


Why does the casino have better odds in blackjack?
In blackjack the house edge is the statistical advantage the casino has over the player. At any blackjack table the dealer is the casino representative. By acting last in a hand, the dealer gains the advantage over the players by seeing their cards, actions and potential mistakes.
What are the odds of winning at blackjack in a casino?
What Are the Odds of Winning at Blackjack? Blackjack has one of the best odds of winning, among many other casino games. In any round of a game of blackjack, the probability of an overall win is about 42.22%.
What is the casino advantage in blackjack?
The house edge is the name given to the percentage a casino will win over the long-term in a particular game. In blackjack, the house edge over an inexperienced player is around 2%. What does that mean? Well, in simple terms, you can say that for every $100 you bet in blackjack, you’ll lose $2.
Is blackjack a skill or luck?
Simply put, blackjack is both a game of skill and chance. But, many believe that the former is more critical to winning. Blackjack is a card game that uses a standard 52 deck of cards, but some casinos may use more than one pack.


Yes indeed. It is designed specifically for audio application. Read a comprehensive description below. It is more info. than you may need but informative.
Wire gage is an index which shows, indirectly (inversely and logarithmically), the cross-sectional area of a round wire. In the case of solid conductors, measurement of this area is pretty straightforward: the area is the radius of the wire squared, times pi, and for the sake of ease of expression, a measure called "Circular MIL area" is instead often used instead; one circular mil is the area of a circle with a diameter of one mil (1/1000 inch), and the circular mil area of a solid wire, consequently, is always the diameter of the wire, in mils, squared.
Stranded wire is another matter. For any given AWG size, a stranded wire will occupy more space than a solid wire, because the wire gage is measured by summing the cross-sectional area of the strands. Because there are air pockets between the strands, any given cross-sectional area of wire will take up more overall space in a stranded configuration than it will in a solid wire. Consequently, when we talk about "diameter" relative to wire gage, it's well to remember that diameter will vary not only with gage but also with stranding. In this article, when we talk about relative diameters, our examples are based on solid wire for the sake of simplicity.
The relationship of gage to wire size is, for a lot of people, counterintuitive. The larger the gage number, the smaller the wire. What's more, the relationship isn't linear, but logarithmic. Two 16 AWG wires, combined, amount to a 13 AWG conductor. If you're familiar with decibels (dB), this will make good sense. If we go up or down 10 gage sizes, we increase or decrease the area of the conductor by a factor of 10. If we go up or down 3 gage sizes, we increase or decrease the area by a factor of about 2. For some reason (we're not really sure why) the relationship isn't precise, but it's close enough, for most purposes, to a straight logarithmic formula. For example, a 40 AWG solid wire has a circular mil area, as specified by the National Bureau of Standards, of 9.61; a 30 AWG wire has a circular mil area of 100.5, a 20 AWG wire comes in at 1020, and a 10 AWG at 10380.
Incidentally, it's important to remember that it is the size of the WIRE, not the size of the wire with its insulation, that is measured in AWG.
[ Gary Jan 10, 2018 ]


TSI Math Boot Camp
Date: Mon, October 16, 2017 12:00 AM
The Adult Education and Literacy (AEL) department at Grayson College will host a TSI Math Boot Camp Oct. 30 through Nov. 16. Classes will be held Monday through Thursday from 2 p.m. until 4 p.m.
TSI Math Boot Camp will be taught through a combination of online and face-to-face classes that will be held on the college's Denison campus in the Advanced Technology and Adult Eduction building.
Classes are $99 or FREE if attendees do not miss two or more classes. The course will also provide a free TSI math test to attendees.
"Students who plan to take the TSI before the spring semester or students who took the test previously but failed the math portion are strongly encouraged to attend," said Chris Voss, AEL Data Analyst at Grayson.
The TSI Assessment (TSIA) is part of the Texas Success Initiative program designed to help colleges determine if students are ready for college-level course work in the areas of reading, writing, and mathematics. Incoming college students in Texas are required to prove readiness for college-level work, and that often involves taking the TSI Assessment. Based on performance, students may either be enrolled in a college-level course and/or be placed in the appropriate developmental course or intervention to improve skills.
"If students score below a certain level on the math section of the TSIA, they have to take and pass development math at Grayson prior to taking any other math classes for their major." said Voss. "By enrolling in the TSI Boot Camp, they can hopefully pass the TSIA and skip developmental math, which saves students time and money."
To learn more about the TSIA, see the information provided here http://www.collegeforalltexans.com/index.cfm?objectid=63176344-FFFA-217B-60C9A0E86629B3CA on the College for All Texans site. It will fully explain the intent, test and exceptions.
For more information about Grayson's TSI Math Boot Camp, contact AEL at email@example.com or 903-463-8784. Space is limited, so call today.


Office: Room 1027 SSW
Program(s): Full-Time Faculty, Senior Faculty, Statistics Department
Research Interests: Inequalities in Probability and Statistics, Sequential Analysis, Boundary Crossing and Change Point Problems, U-Statistics and Processes, Self-Normalized Processes, General Dependence Structures Including Martingales, Decoupling, Copulas. Managing Risk in Complex Systems, Banking, Actuarial Sciences, Earth Sciences, Climatology Global Warming.
Add to address book
Permanent link to this profile


PR1ME Problem Solving Pedagogy
—what it is and how it works
Webinars are conducted by:
Christine Vale, Scholastic Education Manager
Fiona Robertson-Neil, PR1ME Maths Consultant.
These practical sessions will provide you with:
- PR1ME pedagogy for problem solving
- Practical strategies, including the Bar Model and Concrete-Pictorial-Abstract
- Embedding concepts with mathematical thinking and metacognition.
12 March 2018:


Prof. Francis Allotey dies
Prof. Francis Kofi Ampenyin Allotey, one of Ghana’s and Africa’s foremost mathematicians has died. According to the Facebook page of the Africa Institute for Mathematical Sciences – Ghana, (AIMS-Ghana), he died last night November 2, 2017 in Accra, Ghana. He was 85 years old.
Prof Allotey who was the President of AIMS-Ghana was educated at the Roman Catholic Elementary School, Saltpond; the Ghana National College, Cape Coast; the Tutorial College, London; Borough Polytechnic (now University of South Bank), London; Imperial College of Science and Technology, London and Princeton University in the United States.
He taught at the Kwame Nkrumah University of Science and Technology (KNUST) and rose through the ranks to become the first Ghanaian Full Professor in Mathematics and Head of the Department of Mathematics in 1973.
He held a number of positions such as Dean of the Faculty of Science, Founding Director of the Computer Science Centre in Ghana, Pro-Vice Chancellor and member of the KNUST University Council.
Prof Allotey has also held several other local and international appointments such as being appointed by the UN Secretary-General to be among a group of 12 international experts commissioned to advise the UN on Nuclear Weapon.
By Emmanuel K. Dogbevi