St. Petersburg becomes the center of attraction for mathematicians from all over the world!
Only a few days left before the start of the 62nd International Mathematical Olympiad, which will be virtual this year. St. Petersburg will be the organizer of this world event again and will become a center of attraction for participants from all over the world.
St. Petersburg is the city with an amazing history, the city of drawbridges and white nights, the city that inspires the greatest minds of mankind.
It is from St. Petersburg that Russian science, Russian mathematics, Russian higher and, above all, mathematical and technical education began. In 1724, by decree of the Russian Emperor Peter I, the St. Petersburg Academy of Sciences, the first higher scientific institution in the Russian Empire, was founded in the city on the Neva River.
For three centuries, the St. Petersburg Mathematical School has been one of the leaders not only in Russia, but also in the world. We invite you to get to know the city even closer and present you a short film about the history of mathematics in St. Petersburg.


Schools are full of cutoffs: how many credits you need to graduate, the test score you need to avoid summer school or to get out of ESL class, how many checks next to your name on the board before you get sent to the principal’s office.
There is an increasingly popular evaluation design called Regression Discontinuity that makes use of cutoffs. The idea is to compare the kids right above the cutoff with those right below the cutoff, using regression to control for their difference in score. While, for example, you would expect the kids who got sent to summer school because they flunked the end-of-year math test to do worse in next year’s math course than the kids who passed, if you compare the outcomes of the kids who just barely flunked with the kids who just barely passed, and control for their test score, you can get a good picture of whether summer school helped them. Because kids can’t control if they get one point under or one point over the cutoff score very well, this design (in theory) might neutralize some of the unobserved differences in who enters a program, without the sturm und drang of making schools randomly assign some kids to summer school and some kids to summer vacation who got the exact same score on the test.
It’s a good enough study design in its way, but what if the important policy isn’t the program you are testing, but the cutoff itself? For example, one of the big controversies in math education (as I understand it) has long been how many kids can benefit from a full year of Algebra in middle school , with some zealots claiming that everyone can take it just fine, and the City of San Francisco deciding recently that if some kids can’t take Algebra in middle school, nobody can, and then taking its toys and going home.
A valuable use of federal research money, I think, would be to proffer grants to a reasonably large group of districts to agree to randomly agree to a test score cutoff for taking Algebra in middle school, each district assigned to a single cutoff across a wideish but reasonable range. Call it “random cutoff regression discontinuity.” It might make some people mad (what education policy or study doesn’t?) but it would tell us something useful about what works well for whom.
Of course, it would require an explicit acknowledgement that the appropriate course for kids varies depending on ability– a fact that literally everyone in education agrees on implicitly (thus the ubiquitous use of cutoffs) but no one likes to say.


Dr Kevin Kidney, CFA
Kevin’s role is to develop strategic ideas within the Cameron Hume investment team. He is also responsible for modelling the risks of individual investment strategies, thereby ensuring that the investment team has a clear and precise view of exposures held within portfolios.
Prior to joining Cameron Hume, Kevin worked at Standard Life Investments where he managed segregated UK fixed income mandates, contributed to economic research, and was responsible for the modelling of investment strategies, volatilities and correlations within the absolute return bonds portfolio. He began his career at JPMorgan Asset Management in 2006 as an analyst with the European Technology and Operations division.
Kevin holds a first class honours degree in mathematics from the University of Strathclyde and a PhD in the mathematics of liquid crystal theory from the same institution. He is also a CFA Charterholder.


Bringing business leaders together to recognize and empower teaching excellence in the areas of science, technology, engineering and math.
Helping students achieve success and acquire promising careers, HSU scholarship endowments impact local students year after year.
Support for programs that encourage entrepreneurship, leadership, career technical training, and cultural enrichment.
Our Science, Technology, Engineering, and Math (STEM) programs raise interest and inspire students to prepare for careers of future demand.
Game Changers Squad
The HSU Educational Foundation encourages families to explore emerging technologies together by interacting with local industry.
The AFA CyberPatriot CyberCamps are designed to excite student interest in cybersecurity and STEM career opportunities.
Drone Team Challenge
Drone Team Challenge is a student UAV competition focused on introducing students to autonomous UAV technology at an early age while fostering a community of mentorship.
INNOV8+ is a coalition of educational organizations from the 8 coastal counties of Northwest Florida who share a common interest in creating transformational inspiration among STEM students.
Transforming Our Community’s Future by Inspiring Innovation
The HSU Educational Foundation strives to support innovative opportunities and programming that will best prepare students for careers of global demand. As our community’s greatest asset, students need a solid foundation of skills in leadership, critical thinking, and team collaboration. By combining these skills with an early exposure to the careers of tomorrow in Science, Technology, Engineering, and Math, we inspire innovation, encourage entrepreneurship, and transform our community’s future.


Our results are uploaded to our website as soon as nationally available. We currently have results from the end of our KS1 but will not have results from the end of KS2 until July 2018.
Changes to 2016 Performance Data
Key Stage 2 national curriculum test outcomes will no longer be reported using levels, and instead will be reported as scaled scores. Key Stage 2 national curriculum teacher assessments will be reported against the new interim frameworks for teacher assessment.
The headline measures will be:
- the percentage of pupils achieving the expected standard in reading, writing and mathematics
- the percentage of pupils achieving the higher standard in reading, writing and mathematics
- the school’s progress score in each of reading, writing and maths
- the pupil’s average scaled score in each of reading and mathematics
A school will be above the floor standard if:
- 65% of pupils meet the expected standard in reading, writing and mathematics (i.e. achieve that standard in all three subjects) or
- the school achieves sufficient progress scores in all of reading, writing and mathematics.


The free energy saving calculator from SEW-EURODRIVE helps you to determine the energy and CO2 saving potentials offered by energy-efficient motors.
- Quick and user-friendly
- With a few mouse clicks, you can compare the energy consumption and CO2 emission of standard motors (IE1 = Standard Efficiency) and energy-efficient motors (IE2 = High Efficiency, IE3 = Premium Efficiency) and calculate the amortization time of the required investment
- The calculation result can be downloaded in PDF
To the energy saving calculator


Applied analysis and computation are essential to research in virtually every field of science and engineering. Modern engineering is, to a large extent, computational engineering. Computational and applied mathematics (CAAM) is the fundamental discipline that underlies practice and intellectual advancement in mathematical modeling, applied analysis, the development and analysis of numerical algorithms, and the implementation and dissemination of mathematical software. CAAM provides a key enabling technology for all aspects of computational engineering and numerical simulation.
The CAAM faculty at Rice are leading researchers with international reputations. There are cutting edge research activities in inverse problems, discrete and continuous optimization, computational neuroscience, partial differential equations (PDE), PDE constrained optimization, and large scale numerical linear algebra. Our work is often highly interdisciplinary and involves interaction with all of the other departments within the School of Engineering.


I am a Professor of Mathematics and Adjunct Professor of Bioengineering at the University of Utah. I first started studying sea ice at NASA when I was in high school. Then I did research on radar propagation in sea ice, viewed as a composite material of pure ice with brine and air inclusions, at the US Army Cold Regions Research and Engineering Lab (CRREL) while I was an undergraduate at Dartmouth College. My early studies of sea ice led to a career as a mathematician specializing in theoretical models of the effective properties of composite materials. I've traveled to Antarctica five times - my first time in college, and five times to the Arctic. All my previous trips south have been on US or Australian icebreakers studying sea ice in the Southern Ocean around Antarctica. I am excited that I will finally have the chance this time to set foot on the frozen continent itself.


On Wednesday May 15th at approximately 9am, grade 7 and 8 students all across North and South America wrote the 2013 Gauss Mathematics Contest (Eastern Hemisphere wrote the contest the next day). A growing number of keen students in Grade 6 also wrote the Grade 7 contest. The Gauss Mathematics Contest is a great opportunity to inspire students who already have a strong interest in mathematics and also help to increase confidence and ability in mathematics for other students. Although the emphasis is on participation for the sake of enrichment, rather than competition, 31 students from Our Lady of Mount Carmel took on the challenge and did quite well. The University of Waterloo, Mathematics Department awarded Certificates of Recognition to all contestants, with several students being awarded Certificates of Distinction for ranking in the top 25%!!! Way to go Mathletes!!!


Date: January 4 2021
Summary: A brief overview on what arithmetic series are and some of its underlying math.
Keywords: ##zettel #arithmetic #sums #series #proof #archive
An arithmetic sequence is one which the difference between one term and the next only differs by a constant. One example is this:
where the difference between each proceeding term is the constant value, 1.
An arithmetic series is one in which values in an arithmetic sequence are summed together:
This is a visual proof of the Arithmetic Series algorithm:
A formalization of the above is:
which is equivalent to:
The latter formalization is somewhat more common and it works as gives the same values as what the size of the sequence is which is . From the visual proof, the constant comes from halving the size of each region.
(Thanks to Mark Kittisopikul, Yingbo Ma, and Benoit Pasquier for these explanations)
Zelko, Jacob. Arithmetic Sums & Series. https://jacobzelko.com/01052021044121-arithmetic-series. January 4 2021.


1. Introduction to the theory of complex optimization: computational complexity, combinatorial problems, NP-hard problems (K1-K4).
2. Evolutionary algorithms I (K1-K4).
3. Evolutionary algorithms II (K1-K4).
4. Simulated annealing (K1-K4).
5. Ant colony and bee colony algorithms (K1-K4).
6. Particle swarm optimization (K1-K4).
7. Hybrid optimization methods (K1-K4).
8. Test (1 hour).
1. Introduction to the workshops: organization, set up of work teams, assignment of topics. Introduction to available software tools to be used for solving optimization tasks (S6).
2. Presentation of literature review on selected optimization methods and their real-world applications (S9).
3. Problem definition, presentation of projects concepts proposed by every work team (S1, S2, S3, S5, S7).
4. Mastering software used for solving the selected optimization task (S6).
5. Practical realization of the projects, discussion (S2, S5, S6, S8, S9, S10).
6. Presentation of the project reports and obtained results by every work team
(S8, S9, S10).
Individual activity: 70 hours.


Tutors in Excelsior, MN
Find Private & Affordable Tutoring in the Excelsior Area!
Oberlin College - Bachelor in Arts, Biology, General , Johns Hopkins University - Master of Science, Elementary Education
Oberlin College - BA, Biology , Johns Hopkins University - MS, Elementary Education
Denison University - BA, Psychology , Ohio State University-Main Campus - PhD, Cognitive Psychology
University of Minnesota-Twin Cities - BS, Mechanical Engineering , William Mitchell College of Law - JD, Law
New York University - BA, History , Columbia University in the City of New York - MS, Journalism
University of Minnesota-Twin Cities - BS, Physics; Mathematics
University of Minnesota - BA, Political Science , University of Minnesota - M. Ed, Math Education
Bethany Lutheran College - Bachelors, Mathematics , University of Minnesota-Duluth - Masters, Applied and Computational Mathematics
Macalester College - BA, Economics
Minnesota State University-Mankato - Bachelors, Mathematics & Physics , Minnesota State University-Mankato - Masters, Mathematics &...
University of Wisconsin-Eau Claire - BA, Spanish for Business
Franklin W. Olin College of Engineering - Bachelor of Science, Engineering: Materials Science , KTH - Royal Institute of Technology -...
University of MN - BS, Physiology , Augsburg College - MS, Physician Assistant Studies
Georgetown University - BS, Foreign Service , University of Minnesota-Twin Cities - M.A., History


I have accomplished my BSc and MSc in Mathematics and Physics at the University of Warwick in the United Kingdom and I’m a researcher at the world leading company in lightfield camera technology Raytrix.
My current and future research interests include respectively analysis and modelling of point spread functions for lightfield cameras and will developing an elaborate mathematical distortion model for enhanced metric calibration.
The point spread function (PSF) of an optical system gives fundamental insight into the imaging performance of a camera. So modelling and understanding this optical property of the imaging system is significantly advantageous in a number of ways.
One major complication however is that in a lightfield camera, there is not a single PSF, but rather an array of them formed behind the microlenses imaging the object point relayed through the main lens.
Having an accurate PSF simulation model at hand allows for faster computational imaging and scene rendering for synthetic data, (as intensity images form from a convolution of the PSF with the object profile). From this, one can then test the performance of depth estimation algorithms of pixel matching and how these change with the optical setup. Thus we can ultimately optimise the system’s configuration to achieve the most accurate depth maps.
Furthermore, by knowing the depth dependent PSF, we can construct our lightfield cameras better. This is because with knowledge of the PSF, we can attain the relationship between the intensity PSF spot size at the sensor with respect to the microlens array (MLA) to sensor distance.
As lightfield camera already provides us with a depth map of our scene, then for a corresponding object depth, we can determine an initial estimate on the defocus parameter. As a result, one can then apply an iterative algorithm which alternates between estimating depth from disparity and deconvolution for image restoration of lightfield images.
As a result we obtain enhanced lateral resolution in our final microimages and therefore improved depth/3D scene reconstruction.
Mehdi Daniel Ardebili


Want a weekend stitching project? Do a little crocheting (well, okay, a lot) with a chaotic attitude, and you could even win some bubbly:
Dr Hinke Osinga and Professor Bernd Krauskopf, of Bristol University’s engineering mathematics department, used 25,511 crochet stitches to represent the Lorenz equations.
The equations describe the nature of chaotic systems – such as the weather or a turbulent river.
The academics are offering a bottle of champagne to anyone who cares to follow the pattern published in the journal Mathematics Intelligencer.
Thanks to Bob DuCharme for the pointer.


(ONE-TIME purchase, no further in-app purchases or continuing fees / tricks.)
Don’t just memorize the odds, *understand* them … This app teaches you a general method for estimating the odds for just about any Texas Hold’em scenario in your head in real time.
*** Pay once and enjoy forever ***
No in-app purchases/fees.
Continual improvement for many years.
New: A user interface upgrade for iPhone X-series models.
This is a challenging app. You need to be pretty good at doing basic math operations in your head. For example, what’s the value of:
16% * 20%
We’ve added a “Math” help button to show you how to calculate this in your head. One way is, 20% means one-fifth. One-fifth of 16% is (16% / 5), which is a little more than 3%. 3.2% to be exact. Or, convert to decimals, multiply, then convert back to %age’s.
This app uses specific hand situations, and some people don’t see the value in this because in real life play we don’t know our opponents’ cards. There’s still value in specific hand situations, because they’re the building blocks that we can use to put together and estimate the odds for more complex hand range situations. For example every good player should know the odds for specific hand situations like AJ vs. QQ pre-flop, or FlushDraw vs. TopPair on the flop. Our other apps e.g. PokerCruncher handle both specific hands and hand ranges.
“… helps players learn some of the math behind poker … teaches a general method of estimating odds for different Texas Hold’em situations.”
Many more great reviews from poker experts, pros, and coaches, and on our TwoPlusTwo forum thread.
(See our website.)
Poker trainers, odds tables, etc. are useful tools but do they help you learn why the odds are what they are or are you just memorizing specific situations? And you can’t use these tools or odds calculators in the middle of a live hand (would you really want to?; could be a “fish” tell :)). This app takes a new and different approach to poker odds – understanding instead of memorizing, so you can handle any situation that comes your way.
Our odds estimation method is a pretty simple 3-step procedure. First you count your outs and estimate your odds of improving (using for example the “Rule of 4&2”; this app shows you how). Then you consider your opponent’s counter-outs. Then you put the two together to come up with a good odds estimate.
We’ve provided about 20 common and important pre-flop and post-flop scenarios for you to practice the method on. This app will take you through the 3 steps for each practice scenario, one screen per step, and will make sure that you’re on track towards a good odds estimate each step of the way, correcting you if needed. We’ve provided info/help buttons for each sub-step.
You might be thinking, how can you cover a game as complex as Texas Hold’em in 20 scenarios? Well, we agree, you can’t! But this app is about learning a general method not many specific situations, and we picked these 20 scenarios to cover a good cross-section of common and important pre-flop and post-flop situations. If you understand how to calculate the odds for say one generic TopPair vs TopPairTopKicker scenario or one representative Pair vs LowerPair pre-flop scenario, you’ll be able to apply your knowledge to other similar situations easily.
Now, we’re not saying that you’re going to master this in 5 or 10 minutes. You’re going to have to practice the method more than a few times, think about the outs and counter-outs, and do simple math operations in your head – basic +, -, *, /; let’s not let our math teachers down :). We hope you feel that acquiring this odds skill and knowledge is more than worth the learning/practice time.
See our website for our strong free app update history over many years.
Follow us on Facebook and Twitter.
App Store reviews are greatly appreciated, thank you.
Also please check out our companion apps Hold’em Odds Quizzer and PokerCruncher.


Baccarat, is a casino game, that despite the several strands its main objective is to get a hand as close as possible to 9.
Two hands are distributed on the baccarat, for the players and for the bench. The goal is to correctly bet which hand has the highest value from 0 to 9 or tie.
Unlike many card games, the cards do not have the normal value, for example in many games the Ace is the highest card with the highest score in baccarat the Ace is worth 1, while the 10, king, jack and queen are worth 0. The rest of the cards have the value shown in their number.
How to play Baccarat
When the value is greater than 10, you will have to remove the extra digit. For example, two 8 make up the total of 16: so and according to what you need, you will take the 10 and stick with the 6.
To start the game, the player bets on both the player and the bank, a draw in the player and bank or a combination of three. After that the dealer will distribute the cards to the players and two cards to the bank. The two cards decide whether a third card is needed.
- If the player has a total of 8 or 9 on this hand, it will be given as a natural and no more cards will be dealt
- If the total of the players is 5 or less, a third card will be given
- If the player does not receive the third card and the bank has 6 or more, the bank wins
- If the player receives the third card, there are new rules for the bank.
To the bank, if the total is:
- 7 – the bank wins
- 3 and the player’s third card was not 8 …
- 4 and the player’s third card was not 0.1.8 …
- 5 and the player’s third card was 4,5,6 or 7 …
- 6 and the player’s third card was 6 or 7 … … then the bank draws a third card.
- If the total is 7 then the hand wins.
The player wins if he bets on a high hand or bets on a draw and the total is equivalent to the total of the bank. If you bet on the player or bench and the result is a draw, it is left for the next round.


How is the plan upgrade or downgrade cost calculated?
At Scalenut, we understand that your business may need changes over time, and you may wish to upgrade or downgrade your plan accordingly. To provide flexibility and ensure a fair cost calculation, we use a Pro Rata Basis approach when determining the cost of plan upgrades or downgrades. This article aims to explain how the plan upgrade or downgrade cost is calculated, so you can make informed decisions about your subscription.
Plan Upgrade or Downgrade Calculation:
When you decide to switch from one plan to another, the cost is calculated based on the Pro Rata Basis. Let's break down the calculation process using an example:
Determine the Daily Rate:
To calculate the cost, we first determine the daily rate of your current plan. For instance, if you are on the Growth plan (Monthly) priced at $79, we divide the monthly cost by the number of days in the month. Assuming there are 31 days in the month, the daily rate would be $79/31 = $2.55.
Calculate the Consumed Amount:
Next, we calculate the amount you have consumed based on the number of days you have used the current plan. Suppose you decide to upgrade to the Pro plan (Annual) after using the Growth plan for 10 days. Multiply the daily rate ($2.55) by the number of days used (10), resulting in $2.55 * 10 = $25.5. Therefore, you have consumed a total of $25.5 from the initial $79.
Adjust the Plan Cost:
To determine the final cost of the new plan, we subtract the consumed amount from the total cost of the desired plan. The Pro plan (Annual) is priced at $1072.8 with a 40% discount. Subtracting the unconsumed amount ($53.5) from the total plan cost, we get $1072.8 - $53.5 = $1019.3
Final Cost After Upgrade or Downgrade:
After 10 days of using the Growth plan, the cost of upgrading to the Pro plan (Annual) would be $1019.3. This amount reflects the remaining cost of the new plan after deducting the consumed amount from the initial price.
We aim to provide transparent and fair pricing when it comes to plan upgrades or downgrades. Our Pro Rata Basis calculation ensures that you are charged proportionately based on the number of days you have used your current plan. By understanding how the cost is calculated, you can make informed decisions about upgrading or downgrading your subscription.


For most tool post grinding operations, the material removal rate is dependent upon three factors: work diameter, depth of cut, and traverse rate. The removal is expressed in cubic in./min. and can be found using the following equation:
Removal Rate = Traverse (in./min.) x Depth (in.) x work diameter (in.) x 3.14
Example: Find the metal removal rate for an external grinding operation of 6" diameter shaft with a .001" depth of cut and a 12 in./min. traverse.
Removal Rate = 12 x .001 x 6 x 3.14 = .023 in3 / min.
Traverse feed depends upon the width of the wheel being used and the grinding operation being performed. For roughing, 3/4 the width of the wheel should be fed per work revolution. To find the traverse rates, the following formulas can be used.
- Traverse (ipr) = 0.75 x wheel width
Example: Find the traverse rate for a roughing operation using a 6 x 1/2 x 1/2 wheel
Traverse = 0.75 x ( 1/2 ) = .375 ipr
- Traverse (ft/min) = 0.75 x wheel width x SFM (work)
Example: Same as above with 50 SFM (work)
Traverse = .75 x 1/2 (50)
= 1.56 ft./min.
For finishing operations, a finer feed of 1/8 or less the width of the wheel per work revolution is used.
The depth cut used for roughing should be as much as the wheel can stand without breaking down. For finishing, the depth of cut is always light - .001" or less.


Vedic Maths is a collection of techniques/sutras to solve mathematical problem sets in a fast and easy way. These tricks introduce wonderful applications of Arithmetical computation, theory of numbers, mathematical and algebraic operations, higher-level mathematics, calculus, and coordinate geometry, etc.
Benefits of Vedic Maths:
- 1. It helps students to solve mathematical problems many times faster
- 2. It helps in making intelligent decisions to both simple and complex problems
- 3. It reduces the burden of memorizing difficult concepts
- 4. It increases the concentration of a student and his determination to learn and develop his/her skills
- 5. It helps in reducing silly mistakes which are often created by students


This book provides a comprehensive introduction to the mathematical foundations of economics, from basic set theory to fixed point theorems and constrained optimization. Rather than simply offer a collection of problem-solving techniques, the book emphasizes the unifying mathematical principles that underlie economics. Features include an extended presentation of separation theorems and their applications, an account of constraint qualification in constrained optimization, and an introduction to monotone comparative statics. These topics are developed by way of more than 800 exercises. The book is designed to be used as a graduate text, a resource for self-study, and a reference for the professional economist.
About the author
Michael Carter received his PhD in economics from Stanford University in
1980. He has taught economics and game theory in New Zealand, as well as
at various universities in Asia, Europe and North America.


RR, relative risk
The risk ratio is the probability of an outcome in one treatment comparison group divided by the probability in another.
It is commonly used as a measure of the relative effectiveness of treatments.
For example, if 20 out of 100 participants (20%) die in one treatment comparison group and 50 out of 100 (50%) die in the other group, the risk ratio is 20/50 = 0.40.
If you feel that this definition hasn't helped you to understand the term, click on our monkey to let us know.


Johannes Kepler was born on December 27, 1571, in Weil der Stadt, Wurttemberg, in what is now Germany. His father, a mercenary soldier, left the family when Kepler was five. Historians believe his father died soon afterwards. His mother was the daughter of an innkeeper and Johannes was put to work at the inn at a young age. Despite his poverty, he was able to attend Latin School at Maulbronn and at the age of twelve, enrolled in a Protestant Seminary in Adelberg. He earned a scholarship to the Lutheran University of Tübingen in 1589. By the time he received an M.A. in theology there in 1591 he had read of the Copernican model of the universe that stated the Sun, not the Earth, was the center of the Universe. Intrigued by this view, he decided to change his major studies to mathematics and astronomy. In 1594, he left the University to become a mathematics tutor in Graz, Austria where he continued his interest in astronomy. In 1596, he wrote the first influential defense of the Copernican system, the Mysterium Cosmographicum (The Sacred Mystery of the Cosmos).
In 1600, Kepler was forced out of his teaching post at Graz due to his Lutheran faith, and moved to Prague to work for the renowned Danish astronomer, Tycho Brahe. In 1601 Tycho died, and Kepler inherited his post as Imperial Mathematician to the Hapsburg Emperor. Using the precise data that Tycho had collected, Kepler discovered that the orbit of Mars was an ellipse, the first step towards his formulation of the laws of planetary motion. In 1606, he published De Stella Nova (Concerning the New Star) on a supernova (new star) that had appeared two years before. In 1609, Kepler published his book Astronomia Nova (New Astronomy) , which contained his first two laws of planetary motion. Due to his detailed calculations and data, some credit Kepler with the creation of what is now known as the scientific method.
In 1610, Kepler learned of Galileo’s use of the newly invented telescope in astronomy, which inspired him to build his own telescope. Later that year Kepler published a confirmation of Galileo’s observations of Jupiter’s moons, the Narratio de Observatis Quatuor Jovis Satellitibus (Narration about Four Satellites of Jupiter observed) , which lent further support to the Copernican model. In 1611, Kepler published Dioptrice, the first scientific discussion of the telescope.
Kepler lost his post in 1612 as Imperial Mathematician when Lutherans were expelled from Prague. He moved to Linz, Austria but had to return often to Wurttemberg where he successfully defended his mother against charges of witchcraft. In 1619, he published Harmonices Mundi (Harmony of the Worlds) , which contained his third law of planetary motion. In spite of more personal tragedies and the religious strife of the Thirty Years War, (1618-1648) Kepler continued his research, publishing the seven-volume Epitome Astronomiae Copernicanae (Epitome of Copernican Astronomy) in 1621. This important work played a major role in the eventual acceptance of Copernicus’ theories.
In 1627, Kepler completed the Rudolphine Tables, begun by Tycho Brae the previous century. These included calculations using logarithms, which Kepler developed, and provided perpetual tables for calculating planetary positions for any past or future date, forming the most concrete proof yet for the Copernican model of the Universe. Kepler also used the tables to predict a pair of transits by Mercury and Venus of the Sun, although he did not live long enough to witness the events.
Johannes Kepler died in Regensburg, Germany on November 15, 1630. His grave there was destroyed in 1632 by the Swedish army during the Thirty Years War. In poor health most of his life, and caught up in the religious turmoil of the Reformation, Kepler’s accomplishments as an astronomer, physicist, and mathematician seem even more remarkable. His greatest feat in astronomy was his explanation of planetary motion, which has earned him the title “founder of celestial mechanics” as he was the first person to identify “natural laws” in the modern sense. He was the first to prove that the ocean’s tides are due to the Moon’s gravity and pioneered the use of stellar parallax caused by the Earth’s orbit to measure the distance to the stars. Kepler was also the first to suggest that the Sun rotates about its axis, and coined the word “satellite.”
Kepler’s book Astronomia Pars Optica (the Optical Part of Astronomy) has earned him the title “founder of modern optics,” while his work Stereometria Doliorum Vianiaorum (The Stereometry of Wine Barrels) forms the basis of integral calculus. A devout Lutheran, he derived the birth year of Christ that is now universally accepted, and was the first to derive logarithms purely based on mathematics. Johannes Kepler’s most influential accomplishments in astronomy were his three Laws of Planetary Motion, which were used by Isaac Newton to develop his theory of universal gravitation:
-Kepler’s First Law: The planets move in elliptical orbits with the sun at a focus.
-Kepler’s Second Law: In their orbits around the sun, the planets sweep out equal areas in equal times.
-Kepler’s Third Law: The squares of the times to complete one orbit are proportional to the cubes of the average distances from the sun.
Kepler Crater on the Moon, Kepler Crater on Mars, asteroid 1134 Kepler, and Supernova 1604 are all named in his honor, as is the Kepler Space Observatory, NASA’s first planetary detection mission, which is due to be launched in October 2008. The Johannes Kepler University of Linz is also named for him.


More than once I’ve seen people ask questions like “In A/B Testing, how long should you wait before knowing which option is the best?”.
I’ve found that the best solution is to avoid a binary question of whether or not to use a variation. Instead, randomly select the variation to present to a user in proportion to the probability that this variation is the best based on the data (if any) you have so-far.
How? The key is the beta distribution. Let’s say you have a variation with 30 impressions, and 5 conversions. A beta distribution with (alpha=5, beta=30-5) describes the probability distribution for the actual conversion rate. It looks like this:
This shows that while the most likely conversion rate is about 1 in 6 (approx 0.16), the curve is relatively broad indicating that there is still a lot of uncertainty about what the conversion rate will be.
Let’s try the same for 50 conversions and 300 impressions (alpha=50, beta=300-50)”):
You’ll see that while the curve’s peak remains the same, the curve gets a lot narrower, meaning that we’ve got a lot more certainty about what the actual conversion rate is – as you would expect given 10X the data.
Let’s say we have 5 variations, each with various numbers of impressions and conversions. A new user arrives at our website, and we want to decide which variation we show them.
To do this we employ a random number generator, which will pick random numbers according to a beta distribution we provide to it. You can find open source implementations of such a random number generator in most programming languages, here is one for Java.
So we go through each of our variations, and pick a random number within the beta distribution we’ve calculated for that variation. Whichever variation gets the highest random number is the one we show.
The beauty of this approach is that it achieves a really nice, perhaps optimal compromise between sending traffic to new variations to test them, and sending traffic to variations that we know to be good. If a variation doesn’t perform well this algorithm will gradually give it less and less traffic, until eventually it’s getting none. Then we can remove it secure in the knowledge that we aren’t removing it prematurely, no need to set arbitrary significance thresholds.
This approach is easily extended to situations where rather than a simple impression-conversion funnel, we have funnels with multiple steps.
One question is, before you’ve collected any data about a particular variation, what should you “initialize” the beta distribution with. The default answer is (1, 1), since you can’t start with (0, 0). This effectively starts with a “prior expectation” of a 50% conversion rate, but as you collect data this will rapidly converge on reality.
Nonetheless, we can do better. Let’s say that we know that variations tend to have a 1% conversion rate, so you could start with (1,99).
If you really want to take this to an extreme (which is what we do in our software!), let’s say you have an idea of the normal distribution of the conversion rates, let’s say its 1% with a standard deviation of 0.5%.
Note that starting points of (1,99), or (2,198), or (3,297) will all give you a starting mean of 1%, but the higher the numbers, the longer they’ll take to converge away from the mean. If you plug these into Wolfram Alpha (“beta distribution (3,297)”) it will show you the standard deviation for each of them. (1,99) is 0.0099, (2,198) is 0.007, (3, 297) is 0.00574, (4, 396) is 0.005 and so on.
So, since we expect the standard deviation of the actual conversion rates to be 0.5% or 0.005, we know that starting with (4, 396) is about right.
You could find a smart way to find the starting beta parameters with the desired standard deviation, but it’s easier and effective to just do it experimentally as I did.
Note that while I discovered this technique independently, I later learned that it is known as “Thompson sampling” and was originally developed in the 1930s (although to my knowledge this is the first time it has been applied to A/B testing).


How well do time series (intrinsic) and cross-sectional (relative) momentum work for different types of currency exchange rates? In their April 2017 paper entitled “Momentum in Traditional and Cryptocurrencies Made Simple”, Janick Rohrbach, Silvan Suremann and Joerg Osterrieder compare the effectiveness of time series and cross-sectional momentum as applied to three groups of currency exchange rates: G10 currencies; non-G10 conventional currencies; and, cryptocurrencies.
To measure momentum they employ three pairs (one fast and one slow) of exponential moving averages (EMA) spanning short, intermediate and long horizons.
AHL Explains - Momentum
When the fast EMA of a pair is above (below) the slow EMA, the trend is positive (negative). They extract a momentum signal for each exchange rate from these three EMA pairs by:
- For each EMA pair, taking the difference between the fast and slow EMA.
- For each EMA pair, dividing the output of step 1 by the standard deviation of the exchange rate over the last three months to scale currency fluctuations to the same magnitude.
- For each EMA pair, dividing the output of step 2 by its own standard deviation over the last year to suppress series volatility.
- For each EMA pair, mapping all outputs of step 3 to signals between -1 and 1.
- Averaging the signals across the three EMA pairs to produce an overall momentum signal.
The time series portfolio holds all currencies weighted each day according to their respective prior-day overall momentum signals.
The cross-sectional portfolio is each day long (short) the three currencies with the highest (lowest) overall momentum signals.
Key performance metrics are annualized average gross return, annualized standard deviation of returns, annualized gross Sharpe ratio (assuming risk-free rate 0%) and maximum drawdown. Using daily foreign currency exchange rates for 23 conventional currencies and seven cryptocurrencies versus the U.S.
dollar as available through late March 2017, they find that:
- For G10 currencies (data start July 1974):
- The time series (cross-sectional) portfolio generates annualized average gross return 22% (7%), with annualized standard deviation 41% (24%) and annualized gross Sharpe ratio 0.53 (0.27).
- Maximum drawdown of the time series (cross-sectional) portfolio is about -17% (-35%).
This drawdown for the cross-sectional portfolio commences in 2001 and has not yet recovered its high water mark.
- For 13 non-G10 conventional currencies (data start January 1995):
- The time series (cross-sectional) portfolio generates annualized average gross return 31% (21%), with annualized standard deviation 38% (28%) and annualized gross Sharpe ratio 0.82 (0.78).
- Maximum drawdown of the time series (cross-sectional) portfolio is about -25% (-35%).
- For seven cryptocurrencies (data start March 2015):
- The time series (cross-sectional) portfolio generates annualized average gross return 299% (359%), with annualized standard deviation 185% (242%) and annualized gross Sharpe ratio 1.62 (1.48).
- Maximum drawdown of the time series (cross-sectional) portfolio is about -40% (-70%).
- Across the samples:
- Time series portfolios offer higher risk-adjusted performances returns than corresponding cross-sectional portfolios.
- Strategies work well during calm markets but suffer crashes during unusual events.
In summary, evidence indicates that time series momentum generally works better than cross-sectional momentum across different groups of currency exchange rates on a gross risk-adjusted basis.
Cautions regarding findings include:
- As noted in the paper, return calculations are gross, not net.
Accounting for trading frictions (likely substantial due to daily portfolio reformation) would reduce all returns. Since turnovers may vary by currency and by strategy (time series or cross-sectional), net findings may differ from gross findings.
- There may be an issue of performing all required calculations and executing portfolio reformation in a timely manner on a daily basis.
- Assuming a risk-free rate of 0% in calculating Sharpe ratio seems unreasonable, especially for the sample of G10 currencies that commences in the mid-1970s.
- Signal generation rules are elaborate, suggesting potential for snooping bias in rule construction and/or parameter settings and therefore overstatement of expected returns.
- As noted in the paper, the sample period for cryptocurrencies is extremely short in terms of variety of currency market and economic conditions.
See also “When Carry, Momentum and Value Work”.


The first time you insert an equation, select Use MathType, or turn it on in Preferences:. To make equation authoring easier, the equation editor is in math mode by default, so it isn't necessary to add math mode commands to your equations. If an equation is by itself on a line of text, the equation centers based on the equals sign.
This section will cover how to typeset mathematics. It will also cover how to handle complicated equations and multiple equation environments. For many people the most useful part of LaTeX is the ability to typeset complex mathematical formulas.
This article will detail how to work with math mode in LaTeX and how to display equations, formulas, and mathematical expressions in general. LaTeX uses a special math mode to display mathematics. Save the document press Ctrl-S or click File, then Save as 'mymath' don't include the quote marks in the name in a folder of your choice.
One of the greatest motivating forces for Donald Knuth when he began developing the original TeX system was to create something that allowed simple construction of mathematical formulae, while looking professional when printed. The fact that he succeeded was most probably why TeX and later on, LaTeX became so popular within the scientific community. Typesetting mathematics is one of LaTeX's greatest strengths. It is also a large topic due to the existence of so much mathematical notation.
Mathematician Al Maneki and Alysha Jeans, an electrical engineer working in Virginia, draw upon their experiences as blind professionals as they describe an option that has exciting possibilities. Blind and visually impaired students in the fields of mathematics, science, and engineering often encounter difficulties when they need to present mathematical material to sighted instructors and classmates. Fortunately, advances in digital technology offer interesting possibilities.
There are two major modes of typesetting math in LaTeX one is embedding the math directly into your text by encapsulating your formula in dollar signs and the other is using a predefined math environment. You can follow along and try the code in the sandbox below. I also prepared a quick reference of math symbols. To make use of the inline math feature, simply write your text and if you need to typeset a single math symbol or formula, surround it with dollar signs:.
Go to home page. Let's examine the contents of a simple LaTeX file which has been used as a first example in this tutorial. First we must take a quick look at LaTeX syntax.
TeX is an advanced typesetting system which was majorly developed by Donald Knuth. TeX is mainly popular because of its ability to handle complex technical text and in displaying mathematical formulae. So, what is LaTeX? In short, it is a document preparation system which is predominantly used for technical or scientific writing and publishing.


Author: Kevin Mekulu
- Monte Carlo simulation is one of the most important algorithms in quantitative finance
- Monte Carlo simulation can be utilized as an alternative tool to price options ( the most popular option pricing model is based on the Black-Scholes-Merton formula)
How Does Monte Carlo Simulation Work?
Before demonstrating the implementation of the Monte Carlo algorithm, it’s important to fully comprehend the science behind it. Simply put, Monte Carlo simulation generates a series of random variables that have similar properties to the risk factors which the simulation is trying to simulate.
The simulation produces a large number of possible outcomes along with their probabilities. In summary, it’s used to simulate realistic scenarios (stock prices, option prices, probabilities…).
Note: Monte Carlo simulations can get computationally expensive and slow depending on the number of generated scenarios.
Next, I will demonstrate how we can leverage Monte Carlo simulation to price a European call option and implement its algorithm in Python.
Pricing a European Call Option Using Monte Carlo Simulation
Let’s start by looking at the famous Black-Scholes-Merton formula (1973):
Equation 3–1: Black-Scholes-Merton Stochastic Differential Equation (SDE)
S(t) = Stock price at time t
r = Risk free rate
σ = Volatility
Z(t) = Brownian motion
Our goal is to solve the equation above to obtain an explicit formula for S(t).
We utilized Euler Discretization Scheme to solve the stochastic equation above. The solution is given by the expression:
Equation 3–2: Euler Discretization of SDE
Let’s apply the logarithm function to equation 3–2 above which will allow a faster implementation in Python (the vectorization process using the numpy package in Python would easily ingest the log version of the solution above).
Equation 3–3: Euler Discretization of SDE (log version)
Monte Carlo Implementation in Python
We will utilize the numpy package and its vectorization properties to make the program more compact, easier to read, maintain and faster to execute. The source code below is available here.
# Monte Carlo valuation of European call options with NumPy (log version)
from numpy import *
from time import time
# star import for shorter code
t0 = time()
S0 = 100.; K = 105.; T = 1.0; r = 0.05; sigma = 0.2
M = 50; dt = T / M; I = 250000
# Simulating I paths with M time steps
S = S0 * exp(cumsum((r – 0.5 * sigma ** 2) * dt
+ sigma * math.sqrt(dt)
* random.standard_normal((M + 1, I)), axis=0))
# sum instead of cumsum would also do
# if only the final values are of interest
S = S0
# Calculating the Monte Carlo estimator
C0 = math.exp(-r * T) * sum(maximum(S[-1] – K, 0)) / I
# Results output
tnp2 = time() – t0
print(‘The European Option Value is: ‘, C0) # The European Option Value is: 8.165807966259603
print(‘The Execution Time is: ‘,tnp2) # The Execution Time is: 0.9024488925933838
The graph below displays a plot of the first 10 simulated paths. Those simulated paths represent different outcomes for the price of the underlying asset (index level).
import matplotlib.pyplot as plt
Figure 3–1: Simulated index levels by time steps
Next, let’s investigate the frequency of the simulated index levels at the end of the simulation period.
plt.rcParams["figure.figsize"] = (15,8)
Figure 3–2: Histogram of all simulated end-of-period index values
Next, let’s look at the histogram of all simulated end-of-period option values.
import numpy as np
plt.rcParams["figure.figsize"] = (15,8)
plt.hist(np.maximum(S[-1] - K, 0), bins=50)
plt.xlabel('option inner value')
Figure 3–3: Histogram of all simulated option values
We notice something very interesting in the latter plot. The European call option expires worthless the majority of the time. This might be useful information for an astute options trader! But that’s a discussion for another day.
Hilpisch, Y. (2015). Python for Finance: Analyze Big Financial Data
Originally posted here


|Our group brings tools from nonlinear dynamics, statistics, and decision-sciences to study problems in monsoon dynamics, climate dynamics, and climate change policy.
- Nonlinear climate dynamics:   An important research theme in our group is understanding monsoons from a dynamical systems perspective, using techniques in nonlinear dynamics, model-reduction, and time-series analysis.
- Statistical methods in climate:  We work on development and application of statistical methods, with ongoing work in spatial statistics, renewable energy integration, and learning directed graphs from data to understand climate teleconnections.
- Climate change economics and policy:  Related to global warming, we have studied origins of path independence between cumulative CO2 emissions and global warming and comparing mitigation of different climate forcers having different atmospheric lifetimes. We have integrated simple climate modeling and economics to study mitigation, and are collaborating on economics of climate change.


Conceptualizing the two crystal ball problem
You have two identical crystal balls and have to determine the most efficient way to figure out the lowest floor in a given building where the crystal balls will break when dropped.
Setting up the problem
If all you had was one crystal ball, the only way to guarantee success in finding the answer would be with a linear search, or simply dropping the ball from each story, starting at the first, until you found the solution. Because we have 2 crystal balls though, we have more freedom to jump around. Suppose that the building in the problem has 120 floors. With a linear search, your worst case scenario would be 120 drops. If you moved in intervals of 2 floors though, your worst case scenario would be 61 steps. 60 steps forward, 1 step backward to see if the previous floor was the breaking point or if the floor at step 60 was the breaking point. If you moved in intervals of 3, your worst case scenario would be 42 steps. 40 steps forward, 1 step backward to the floor above the previous interval and 1 more step forward to complete scanning the entire interval.
To write this out mathematically, you could use use the following variables:
n = total number of floors
m = step interval
And the equation would look like this:
n / m + (m - 1) = total number of steps in the worst case
For the example using intervals of 3, the equation would be applied as:
120 / 3 + (3 - 1) => 40 + 2 => 42
So in order to figure out the optimal, or most efficient, interval size, we want to minimize the total number of drops and we can do this by finding the derivative of the function of n / m + (m - 1) which is the square root of n. Another way to find this solution without the use of calculus is to approximate m - 1 as m and set n / m = m. Solve for m, which is also the square root of n.
Given this minimization, the optimal way to find the lowest floor at which one of the crystal balls would break is by traversing the building in intervals of the square root of n.


The standard fees on all donations processed through WeDidIt are:
Processing (WeDidIt) fee: 1-4% for Subscription clients
Transaction (WePay) fee: 2.2% + 30 cents per transaction (3.5% for Amex)
Example: Donor gives $100 to an organization with a 1% processing fee
Processing fee: $1
Transaction fee: $2.2 + $0.30 = $2.5
Net amount received: $100 - $1 - $2.5 = $96.5
Donors have the option to cover the transaction (WePay) fees for their donations. Clicking the 'Cover transaction fees' button will automatically calculate and tell the donor how much their payment will increase in order to cover the fee.
When donors opt to cover the transaction fee, the fee is applied to the total of their payment, which is their donation + the standard transaction fee.
So for a $100 donation, the standard transaction fee is $2.5. This means that the total amount that the fee is applied to is $102.5, not $100 - The donor pays the fee on the fee as well. This is to get the organization as much of the donation as possible.
2.5% of $102.5 = $2.56.
So the total the donor will pay is $100 + $2.56 + $0.30 = $102.86.
The WeDidIt fee is also applied to the $102.86 total, so 1% = $1.03.
Net amount received: $103.18 - $3.18 - $1.03 = $99.27.


We built a simple tool recently to review so-called value spreads over time. (1)
This tool maps out the median valuations for the top decile and bottom decile “cheap stock” portfolios (e.g. EBIT/TEV or sales/price).
Why might this be useful?
This tool allows one to identify the “valuation” spread between the cheapest stocks and the most expensive stocks in the universe. Some research suggests this can be a useful prediction device.
The analysis to build the data works as follows:
- Identify Universe: Top 1500 stocks by market cap (US and EAFE)
- Identify Valuation: Calculate a valuation metric (e.g., EBIT/TEV) for all firms in the universe
- Decile Splits: Sort the 1500 stocks into 10 buckets, 150 stocks each, equal-weight, rebalance monthly
- Calculate Median Valuation: For each decile, calculate the median valuation metric (e.g., EBIT/TEV)
- Plot the data
We create the following time series (assuming EBIT/TEV is the value metric):
- US Value = Top Decile EBIT/TEV (1)
- US Glamour = Bottom Decile EBIT/TEV (10)
- US Spread = US Value – US Glamour
- EAFE Value = Top Decile EBIT/TEV (1)
- EAFE Glamour = Bottom Decile EBIT/TEV (10)
- EAFE Spread = US Value – US Glamour
Here is a chart of the spreads since 1992 for the US and EAFE. You can dig in to the tool for the raw data and additional breakouts on the data.
Spreads aren’t crazy and fairly in line with historical norms. Surprising.


Expressways To Math®
Level 1 and A-Vip
Expressways To Math® makes use of the same A-VIP techniques used in Expressways To Reading®. A-VIP is used to embed into permanent memory the fundamental facts of calculation. Expressways To Math® then exercises and integrates the accumulating memory base so that the learner can apply it to developing math skills through computer games, recitation to music and rhythms, and contests.
Level I simultaneously integrates number theory, counting, addition, subtraction, multiplication and division as applied to practical situations. The goal is to commit to memory the 400 math facts, to understand counting and number theory, and to be able to recall the facts for quick mental applications. When the math facts are thoroughly mastered, students often generate computational speeds that rival electronic calculators.
Level 2 and Level 3
Expressways To Math® Level II involves telling time, handling money, inequalities, types of fractions and their processes, decimal numbers, ruler fractions, percentages, measurements in the English and metric systems, basic plane geometric forms, averages, estimates, ratio, probability, exponents, circles, areas of circles, volumes of solids, negative numbers, temperature and some advanced lessons for logical thinkers. Each lesson has an interactive instructional introduction before a student is directed to perform the exercises.
Level III consists of the section of word problems from QuikComp®.


Welcome to Target Defence Academy here we have detailed syllabus for NDA. NDA written examination is divided into two papers namely Mathematics and General Ability Test (GAT). The boys and girls who are seeking to apply for NDA must begin their preparation for the written exam by following the detailed NDA Syllabus which has been discussed in this article.
Important Information about NDA Exam:
Name of the Exam
National Defence Academy (NDA)
Union Public Service Commission, UPSC.
Frequency Of Exam
Twice in a Year (April & September)
Written Test consisting of Objective Type QuestionsIntelligence & Personality Test (SSB)
Mode of exam
Papers in NDA Exam
MathematicsGeneral Ability Test
Mathematics: 300 MarksGAT: 600 Marks
Total No. of Questions
Mathematics: 120GAT: 150
Mathematics: -0.83GAT: -1.33 marks
2.5 Hours For Each Paper (5 hours total)
Language of Question Paper
English &Hindi English Language paper has to be attempted in English.
After getting to know the NDA Exam Pattern it is important for the candidates to know the Syllabus of NDA Exam. The NDA Syllabus is discussed below.
NDA Maths Syllabus:
The chapters covered for NDA Maths Syllabus are given in the table below.
Sets, Venn diagrams, De Morgan laws, Cartesian product, relation, equivalence relation. Real numbers, Complex numbers, Modulus, Cube roots, Conversion of a number in Binary system to Decimals, and vice-versa. Arithmetic, Geometric and Harmonic progressions. Quadratic equations, Linear inequations, Permutation and Combination, Binomial theorem, and Logarithms.
Concept of a real-valued function, domain, range, and graph of a function. Composite functions, one-to-one, onto, and inverse functions. The notion of limit, Standard limits, Continuity of functions, algebraic operations on continuous functions. Derivative of function at a point, geometrical and physical interpretation of a derivative-application. Derivatives of sum, product, and quotient of functions, a derivative of a function concerning another function, the derivative of a composite function. Second-order derivatives. Increasing and decreasing functions. Application of derivatives in problems of maxima and minima
Matrices and Determinants
Types of matrices, operations on matrices. Determinant of a matrix, basic properties of determinants. Adjoint and inverse of a square matrix, Applications-Solution of a system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method.
Integral Calculus and Differential Equations
Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric, exponential, and hyperbolic functions. Evaluation of definite integrals—determination of areas of plane regions bounded by curves—applications.Definition of order and degree of a differential equation, formation of a differential equation by examples. General and particular solution of differential equations, solution of the first order, and first-degree differential equations of various types—examples. Application in problems of growth and decay.
Angles and their measures in degrees and radians. Trigonometric ratios. Trigonometric identities Sum and difference formulae. Multiple and Sub-multiple angles. Inverse trigonometric functions. Applications-Height and distance, properties of triangles.
Vectors in two and three dimensions, magnitude, and direction of a vector. Unit and null vectors, the addition of vectors, scalar multiplication of a vector, scalar product, or dot product of two vectors. Vector product or cross product of two vectors. Applications—work done by a force and moment of a force and in geometrical problems.
Analytical Geometry Of Two and Three Dimension
Rectangular Cartesian Coordinate system. Distance formula. Equation of a line in various forms. The angle between two lines. Distance of a point from a line. Equation of a circle in standard and a general form. Standard forms of parabola, ellipse, and hyperbola. Eccentricity and axis of a conic. Point in a three-dimensional space, the distance between two points. Direction Cosines and direction ratios. Equation two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. The angle between two lines and the angle between two planes. Equation of a sphere.
Statistics and Probability
Probability: Random experiment, outcomes, and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events. Union and Intersection of events. Complementary, elementary, and composite events. Definition of probability—classical and statistical—examples. Elementary theorems on probability—simple problems. Conditional probability, Bayes’ theorem—simple problems. Random variable as function on a sample space. Binomial distribution, examples of random experiments giving rise to Binomial distribution.
The topic-wise question distribution in Maths
Geometry, Matrices & Determinants
Statistics & Probability
NDA General Ability Test Syllabus:
The General ability test comprises two papers English and General Knowledge. The English paper consists of 200 Marks while the General Knowledge is of 400 Marks.
NDA Syllabus- English
The question paper in English will be designed to test the candidate’s understanding of English. The syllabus covers various aspects like as given below:
No. of Questions
No. of Marks
Ordering of Words in a Sentence
NDA Syllabus- General Knowledge
Check the topic-wise number of questions and marks distribution in the table below.
No. of Questions
No. of Marks
History & Freedom Movement
NDA Syllabus- Physics
NDA Syllabus For Physics
Physical Properties and States of MatterModes of transference of HeatMass, Weight, Volume, Sound waves and their propertiesSimple musical instrumentsRectilinear propagation of LightDensity and Specific GravityReflection and refractionPrinciple of ArchimedesSpherical mirrors and LensesPressure BarometerHuman EyeMotion of objectsNatural and Artificial MagnetsVelocity and AccelerationProperties of a MagnetNewton’s Laws of MotionEarth as a MagnetForce and MomentumStatic and Current ElectricityParallelogram of ForcesConductors and Non-conductorsStability and Equilibrium of bodiesOhm’s LawGravitationSimple Electrical CircuitsElementary ideas of workHeating, Lighting, and Magnetic effects of CurrentPower and EnergyMeasurement of Electrical PowerEffects of HeatPrimary and Secondary CellsMeasurement of Temperature and HeatUse of X-RaysGeneral Principles in the working of Simple Pendulum, Simple Pulleys, Siphon, Levers, Balloon, Pumps, Hydrometer, Pressure Cooker, Thermos Flask, Gramophone, Telegraphs, Telephone, Periscope, Telescope, Microscope, Mariner’s Compass; Lightning Conductors, Safety Fuses.
NDA Syllabus- Chemistry
NDA Syllabus For Chemistry
Preparation and Properties of Hydrogen, Oxygen, Nitrogen and Carbon Dioxide, Oxidation and Reduction.Acids, bases and salts.Carbon— different formsPhysical and Chemical ChangesFertilizers—Natural and ArtificialElementsMaterial used in the preparation of substances like Soap, Glass, Ink, Paper, Cement, Paints, Safety Matches, and GunpowderMixtures and CompoundsElementary ideas about the structure of AtomSymbols, Formulae, and simple Chemical EquationAtomic Equivalent and Molecular WeightsLaw of Chemical Combination (excluding problems)ValencyProperties of Air and Water
NDA Syllabus- General Science
NDA Syllabus For General Science
Common Epidemics, their causes, and preventionDifference between the living and non-livingFood—Source of Energy for manBasis of Life—Cells, Protoplasms, and TissuesConstituents of foodGrowth and Reproduction in Plants and AnimalsBalanced DietElementary knowledge of the Human Body and its important organsThe Solar System—Meteors and Comets, Eclipses. Achievements of Eminent Scientists
NDA Syllabus- History, Freedom Movement
NDA Syllabus For History
Forces shaping the modern world;RenaissanceExploration and Discovery;A broad survey of Indian History, with emphasis on Culture and CivilisationFreedom Movement in IndiaFrench Revolution, Industrial Revolution, and Russian RevolutionWar of American Independence,Impact of Science and Technology on SocietyElementary study of Indian Constitution and AdministrationConcept of one WorldElementary knowledge of Five Year Plans of IndiaUnited Nations,Panchsheel,Panchayati Raj, Democracy, Socialism and CommunismRole of India in the present worldCo-operatives and Community DevelopmentBhoodan, Sarvodaya,National Integration and Welfare StateBasic Teachings of Mahatma Gandhi
NDA Syllabus- Geography
NDA Syllabus For Geography
The Earth, its shape and sizeOcean Currents and Tides Atmosphere and its compositionLatitudes and LongitudesTemperature and Atmospheric Pressure, Planetary Winds, Cyclones, and Anticyclones; Humidity; Condensation and PrecipitationConcept of timeTypes of ClimateInternational Date LineMajor Natural Regions of the WorldMovements of Earth and their effectsRegional Geography of IndiaClimate, Natural vegetation. Mineral and Power resources;Location and distribution of agricultural and Industrial activitiesOrigin of Earth. Rocks and their classificationImportant Sea ports and main sea, land, and air routes of IndiaWeathering—Mechanical and Chemical, Earthquakes and VolcanoesMain items of Imports and Exports of India
NDA Syllabus For Current Events
Knowledge of Important events that have happened in India in recent yearsProminent personalities of both Indian and International level, Cultural activities and sports activities, Current important world events.
Intelligence and Personality Test
The SSB procedure consists of two stages. Only those candidates who clear written exam are permitted to appear for SSB round. The details o SSB exam are given below:
NDA SSB Personality Test
Officer Intelligence Rating (OIR)Picture Perception Description Test (PP&DT).


Resource page for the book: Applied Adaptive Statistical Methods
By T. W. O'Gorman
Published by SIAM in 2004.The book contains a detailed description of an adaptive testing method, which is based on permutations of independent variables. It also includes adaptive confidence intervals, and adaptive estimation procedures. To obtain SAS macros or an R function for computing adaptive tests of significance for this book click here.
To obtain SAS macros for computing adaptive confidence intervals, using the approach described in the book, click here.
To obtain SAS macros for computing adaptive estimates click here.
To obtain an executable file for performing variable selection with case-control data click here.
If you are interested in adaptive testing please do not hesitate to contact me at my e-mail address (firstname.lastname@example.org) .


The post is not in this menu.
When math makes sense, kids leap way ahead. Mathnasium’s specially trained math instructors will teach your child how to understand math in an individual setting. Our unique approach enables us to effectively explain math concepts well and lend a helping hand to every student.
Every day, students around the world attend Mathnasium learning centers to boost their math skills. We are highly specialized; we teach only math. Members usually attend two or three times a week for about an hour. Our goal is to significantly increase your child’s math skills, understanding of math concepts, and overall school performance, while building confidence and forging a positive attitude toward the subject.
Our approach is to use sophisticated techniques to determine, with great accuracy, what a student knows and does not know. Next, we tailor-make a personalized and prescriptive learning program. Each student follows the program with the help of specially trained Mathnasium math tutors who provide instruction—and lots of warm encouragement. For proof of progress, we rely on the student’s report card, independent tests, and parent testimony to measure the speed and magnitude of improvement in math skills, numerical thinking, and attitude. Multiple independent studies have found Mathnasium to be effective 100 percent of the time, increasing student performance on standards-based tests in 20 sessions or fewer. Student skills jumped at least a grade level and in most cases, multiple grade levels.
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She sang for her pi — and threw it, too.
A student whose jingle helped her memorize more than 125 digits of the infinitely long number that begins with “3.14” was allowed to smear a real pie in the face of her math teacher during a competition that was part of the March 14 Pi Day festival at Bedford-Stuyvesant Collegiate school on Gates Avenue. Even she was surprised by the feat.
“It is exciting to know that I was able to memorize a larger amount of pi,” said 14-year-old Yolanda Dunbar. “I am impressed with myself.”
As a reward, Dunbar got to throw a pie — the edible kind — in the face of her math teacher. The second- through fourth-place winners were awarded the opportunity to challenge the teacher of their choice to a pie-eating contest.
Dunbar said she learned half the digits of pi by listening to a song she found on Youtube and half by rote memorization.
This is the second year Dunbar won. Back in the sixth grade, she beat out her classmates and creamed then-math-teacher Caitlin Webster, who says the annual party is an excuse to inject some fun into the sometimes-dry subject.
“We celebrate Pi Day to create some joy and excitement around math,” said Webster. “It is something that is fun and not strictly challenging.”
The school hosts a handful of other wacky math events. For example, they celebrate a Fun Facts Friday, where teachers dressed as ninjas randomly storm classrooms and students have to correctly multiply to help free their space-traveling tiger mascot, Cosmo, whom the ninjas have supposedly kidnapped.


Happy New Year to all the friends and families of Class R!
It is safe to say that this is not how we expected the Spring term to start but we at St. Mary's hope you all remain safe and well throughout this time.
Each day there will be learning activities uploaded to the website for your child to complete. I ask that four observations are uploaded to tapestry each week consisting of one phonics, one literacy, one mathematics and one piece of work that the child is proud of. I understand the strain that home learning places on many families so please do not feel that each observation needs a length description - just the photo and a caption is sufficient.


KAMI GRAPH SHEET A4 60G 20CM X 24CM - 30S SKOT6030
Product Code : SKOT6030
Product Name : KAMI Graph Sheet A4 60g 20cm x 24cm - 30S
Brand : KAMI
- Kami brand graph sheet
- Size: 60gsm 200mm x 240mm.
- Specifications: printed with fine lines making up a regular grid.
- The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves.
- Measured graph/grid paper for use in a variety of math learning situations.


Mathematical methods in economics
- 283 Pages
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New York University Press, Croom Helm , New York, N.Y, London
|LC Classifications||HB135 .S36 1984|
|The Physical Object|
|Pagination||vii, 283 p. :|
|LC Control Number||84011502|
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The author has provided solutions to selected problems so that the book will function as an effective teaching tool on introductory courses in mathematics for economics, quantitative methods and for mathematicians taking a first course in economics.
Mathematics in Economics has been developed from a course taught jointly by Ken Binmore Cited by: 4. Mathematical Methods in Economics book.
Mathematical Methods in Economics. DOI link for Mathematical Methods in Economics. Mathematical Methods in Economics book. By Norman Schofield.
Edition 1st Edition. First Published eBook Published 5 March Pub. location London. Imprint by: User Review - Flag as inappropriate This book provides mostly definitions and virtually NO examples on how to execute economic and mathematical problems.
Furthermore it assumes you remember everything from past courses in trigonometry, precalculus, and calculus; offering no review chapter at the begging of the touches mathematical methods for economics on a very superficial level.1/5(1). This book is a good and fairly clear guide to mathematical applications in Economics and Finance.
Plenty of explanations and worked examples in order to help the student solve problems. However, this is not a text for people who are completely unfamiliar with algebra, trigonometry Mathematical methods in economics book precalculus/5(33).
Alpha C. Chiang, Kevin Wainwright-Fundamental Methods of Mathematical Economics, 4th Edition-McGraw-Hill () pd 03 April () Post a Review. di erential equations and numerical methods. The eld of mathematical nance is only 50 years old, uses leading-edge mathematical and economic ideas, and has some controversial foundational hypotheses.
Mathematical nance is also data-rich and even advanced results are testable in the market. Using ideas illustrated daily in nancial news, the book. Inﬁnitesimal Methods in Mathematical Economics Robert M. Anderson1 Department of Economics and Department of Mathematics University of California at Berkeley Berkeley, CAU.S.A.
and Department of Economics Johns Hopkins University Baltimore, MDU.S.A.
Description Mathematical methods in economics FB2
Janu 1The author is grateful to Marc Bettz¨uge, Don Brown, Hung. matics, statistics, and mathematical economics. With this volume we hope to present a formulary tailored to the needs of students and working professionals in economics.
In addition to a selection of mathematical and statistical formulas often used by economists, this volume contains many purely economic results and theorems. Mathematical Economics Material You Must Know as a Bare Minimum. You'll certainly want to read a good undergraduate "Mathematics for Economists" type book.
The best one that I've seen happens to be called Mathematics for Economists written by Carl P. Simon and Lawrence Blume.
It has a quite diverse set of topics, all of which are useful tools. Mathematical Methods of Economics Joel Franklin California Institute of Technology, Pasadena, California WThe American Mathematical Monthly,AprilVol Number 4, pp.
– hen Dr. Golomb and Dr. Bergquist asked me to give a talk on economics,my. Mathematical methods in economics by Norman Schofield,Taylor & Francis Group edition, in EnglishCited by: Mathematics has become indispensable in the modelling of economics, finance, business and management.
Without expecting any particular background of the reader, this book covers the following mathematical topics, with frequent reference to applications in economics and finance: functions, graphs and equations, recurrences (difference equations), differentiation, exponentials and logarithms. Mathematical Methods for Economics (2nd Edition) 2nd Edition by Michael Klein (Author) out of 5 stars 20 ratings.
ISBN ISBN Why is ISBN important. ISBN. This bar-code number lets you verify that you're getting exactly the right version or edition of a book. The digit and digit formats both by: Fundamental Methods of Mathematical Economics, 4th Edition | Alpha C.
Chiang, Kevin Wainwright | download | B–OK. Download books for free. Find books. If you are a beginner then read: (1). Mathematics for economists by Taro Yamane (2). Mathematics for Economists by C.P.
Simon (3). Fundamental Methods of Mathematical Economics by A.C. Chiang and K. Wainwright B. If you want to look into mathem. Mathematical Methods in Economics Hardcover – November 1, by Norman Schofield (Author) › Visit Amazon's Norman Schofield Page.
Find all the books, read about the author, and more. See search results for this author. Are you an author. Learn about Author Central Author: Norman Schofield. Mathematical Methods for Economic Analysis∗ Paul Schweinzer School of Economics, Statistics and Mathematics Birkbeck College, University of London Gresse Street, London W1T 1LL, UK Email: [email protected] Tel:Fax: Chiang's Fundamental Methods of Mathematical Economics is an introduction to the mathematics of economics.
It starts with a review of algebra and set theory then goes on through calculus, differential equations, matrix algebra, integration. It serves well as a transition from very basic economics up to graduate level by: This book offers an introductory text on mathematical methods for graduate students of economics and finance–and leading to the more advanced subject of quantum mathematics.
The content is divided into five major sections: mathematical methods are covered in the first four sections, and can be taught in one : Springer. Facts is your complete guide to Fundamental Methods of Mathematical Economics.
In this book, you will learn topics such as Linear Models and Matrix Algebra, Linear Models and Matrix Algebra, Comparative Statics and the Concept of Derivative, and Rules of Differentiation and Their Use in Comparative Statics plus much : CTI Reviews.
student of economics must possess a good proficiency in the fundamental methods of mathematical economics. One of the significant developments in Economics is the increased application of quantitative methods and econometrics. A reasonable understanding of econometric principles is indispensable for further studies in economics.
Mathematical economics and game theory approached with the fundamental mathematical toolbox of nonlinear functional analysis are the central themes of this text. Both optimization and equilibrium theories are covered in full detail.
The book's central application is the fundamental economic. Introduction. In this book we use the language of mathematics to describe situations which occur in economics. The motivation for doing this is that mathematical arguments are logical and exact, and they enable us to work out in precise detail the consequences of economic by: Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics.
By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods.
An excellent book which should find wide use. — Mathematics this classic volume, a noted economist and teacher has combined a modern text for graduate courses in mathematical economics with a valuable reference book of analytical economics for professional.
ECONOMICS DEPARTMENT. ECON MATHEMATICAL METHODS IN ECONOMICS Thayer Watkins. Texts: Michael W. Klein Mathematical Methods for Economics. Course Syllabus. Topics: By the method of construction b cannot fit into any place in the sequence because if b=x J we would have the contradiction that its J-th digit was not equal to its J-th digit.
Fundamental Methods of Mathematical Economics book. Read 32 reviews from the world's largest community for readers.
As in the previous edition, the purpo 4/5. Mathematical Methods For Economics book.
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Read reviews from world’s largest community for readers. Mathematical Methods for Economics uses an applications /5. This formalism can often deter graduate students. The progression of ideas presented in this book will familiarize the student with the geometric concepts underlying these topological methods, and, as a result, make mathematical economics, general equilibrium.
Originally published inMathematical Techniques in Finance has become a standard textbook for master’s-level finance courses containing a significant quantitative element while also being suitable for finance PhD students. This fully revised second edition continues to offer a carefully crafted blend of numerical applications and theoretical grounding in economics, finance, and.
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This book describes a system of mathematical models and methods that can be used to analyze real economic and managerial Download the eBook Mathematical Methods and Models in Economic Planning, Management and Budgeting - Galimkair Mutanov in PDF or EPUB format and read it directly on your mobile phone, computer or any device.The book also explores the analogs of the duality theorem, the equivalence of game problems to linear programming problems, and also the inter-industry nonlinear activity analysis model requiring special mathematical methods.
The text will prove helpful for students in advanced mathematics and calculus.Originally published in Since the logic underlying economic theory can only be grasped fully by a thorough understanding of the mathematics, this book will be invaluable to economists wishing to understand vast areas of important research.
It provides a basic introduction to the fundamental mathematical ideas of topology and calculus, and uses these to present modern singularity theory.
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Amanda J. Bernard, CPA, CFE, CMA
Principal, Maillie LLP
Maillie LLP utilizes IDEA™ data analysis software, a powerful data mining tool used to perform data analysis quickly, by importing, analyzing and reviewing a variety of data. If you would like us to perform any of these data analysis procedures or would like more information on how you can use data analytics to improve your organization, please contact us at firstname.lastname@example.org.
Benford’s Law is a mathematical theory of leading digits, meant specifically for real-life numerical data sets distributed in a non-uniform way. The theory is based on probability of occurrences. A logical assumption seems to be that each number, 1-9, would appear as the leading digit of a number consistently at 11.1% (or 1/9) of the time. However, Benford’s Law actually states the number 1 will appear as the first digit of a number about 30% of the time, while the larger digits occur in the first position less frequently, with 9 appearing as the first digit less than 5% of the time.
Why does it work? Believe it or not, there is a certain logic behind Benford’s Law. A number that begins with 1 needs to increase by 100% to become a 2, while a number that begins with 5 needs to increase by only 20% to become a 6, and a number that begins with 8 needs to increase by only 12.5% to become a 9, and so on.
The history of Benford’s Law dates back as far as 1881; the phenomenon was then made popular in 1938 by the physicist Frank Benford after he tested it on data sets from 20 different domains, including surface areas of rivers, population sizes, molecular weights, numbers contained in Reader’s Digest, street addresses, and death rates. Today, Benford’s Law is used to detect possible red flags of fraud in financial and other data based on the assumption that people who make up numbers tend to distribute their digits fairly uniformly. The made-up figures simply do not follow the expected Benford’s distribution.
An example of fraud that can be easily detected using Benford’s Law is avoidance of purchase approval policies. One method used to avoid obtaining approval on a purchase transaction is to make sure the cost does not exceed the established threshold requiring approval by splitting larger transactions into multiple smaller ones. For example, if the threshold for approval is $5,000 and the expense data shows a spike in transactions beginning with the number 4, it could indicate someone is avoiding the organization’s purchase approval procedures.
Other data sets where Benford’s Law is especially useful include credit card transactions, customer balances and refunds, vendor disbursements, purchase orders, travel and entertainment expenses, and many more.
Maillie LLP can help you use data analytics, including these Benford’s Law examples, to analyze your financial data for potential red flag indicators of fraud. Contact us today for more information.


Since I write for both adults and children, I often visit classrooms. And when I do, I usually bring early 19th century school books with me, to show children what text books were like two hundred years ago, when students had to supply their own books, and often books were shared. (I have one Webster’s speller that was once re-covered in wood – which is now cracked and split – to protect it.)
Most of the books I have are concerned with reading, writing, and speaking. So a couple of weeks ago when I was at an antique show I was happy to pick up a copy of an arithmetic book. It was a little later than my early books — it was published in 1863 — but I love it. The title, Arithmetical Problems or Questions in Arithmetic, for the Use of Advanced Classes in Schools, says it all. It was written by W.H. Farrar, the principal of the Woonsocket (Rhode Island) High School. And it’s a collection of one thousand – yes — one thousand – “word problems. — all of which are actually practical applications of arithmetic.
At some time a student — perhaps the Charles T. Haynes who signed it in 1871 — started the book, and penciled in the answers to the problems. But he gave up at #99.
Since there’s a lot of talk today about our schools, and how our students are falling behind those in other countries, I thought today I’d share (without solutions) ten of the “arithmetical problems” in this little book. You might even want to share them with a young friend or relative.
Here then, for your edification and amusement ….
1. What must I pay per hogshead for molasses, that I may keep it 8 months, when money is worth 6 percent, and then sell it at $42 per hogshead, and gain 12 per cent?
2. The foot of a ladder, 60 feet long, remaining in the same place, the top will just reach a window 40 feet high on one side of the street and another 30 feet high on the other side. How wide is the street?
3. If a certain number be diminished by 7 and the remainder be divided by 8, and the quotient be multiplied by 5, that product increased by 4, the square root of the sum extracted, and 3/4 of that root be cubed, and that cube be divided by 9, the last quotient will be 24. What is that number?
4. Bought 200 yards of cambric for 90 pounds, but, it being damaged, I am willing to lose 7 pounds 10 shillings on it. What must I demand per ell English?
5. How many square yards of paper in a book of 672 pages, the leaves being 6 1/2 by 10 1/4 inches?
6. A house is 36 feet long and 28 feet wide, and the ridge is 12 feet above the beam. The roof projects one foot over the ends and eaves in all directions. How many shingles will be required to cover the roof, 6 shingles being allowed to the foot, and 10 per cent for waste?
7. A field is 72 chains long and 131 rods 10 feet wide. How many acres does it contain?
8.A stone weighs 120 pounds in the air and 100 pounds in water. What is the specific gravity of the stone, water being 1000?
9. If 63 men can build a wall, 45 1/3 feet long, 6 7/12 feet high and 3 1/8 feet thick, in 34 days of 11 1/3 hours each, in how many days of 8 1/2 hours each will 21 men build a wall 98 3/4 yards long, 2 1/2 yards high, and 1 1/4 yards thick?
10. A man lost in a speculation 1/4 of his money. He then gained a sum equal to 1/3 of what he then had. Afterwards he lost 1/5 of what he then had, and then gained a sum equal to 1/4 on what he had left, when he found he had $1200. How much had he at first?
And they didn’t even have calculators ……
Okay, we’re talking cruel and unusual punishment here! I hated word problems in math. One of the hardest things for me to do as a principal was tutor some 4th graders in math problems. Oh, my goodness. I confess to not even reading all of these. They mess with my brain so bad. Of course some of that is the vocabulary, but even if I knew what some of those terms were, I’d be out of luck. LOL
As someone who taught math for thirty years, I have to say that although these problems could not be solved by the average college student today, some of them are pedagogically unsound. Problem 2, for example, is an ingenious use of the Pythagorean theorem. The ladder forms two right triangles when it is leaned against the windows on each side of the street with the length of the ladder being the hypotenuse and the height of the windows being one other leg. Solve for the missing led in each triangle, add them together, and you get the width of the street. If a student can see that is the solution, she has mastered mathematical reasoning. But she may not get the answer because the actual calculation requires finding square roots, both of which are irrational numbers. The problem would be be so elegant if different lengths had been chosen. After all, the width of any actual street is always a rational number. One wonders whether the intent was to instruct or torment! Thanks for this post. It was fun.
Do we get the answers too?
These are problems I would have to pass on to my math-whiz children. What a fun post, Lea.
So, anyone know what is meant by “when money is worth 6 percent”? A hogshead is a barrel or cask of liquid (about 63 gallons or 300L). An ell is 45 inches. A schilling is 0.05 pounds and there are 12 pence per schilling. A chain is 66 feet and there are 4 rods in a chain. A mile is 80 chains. A furlong is 10 chains. An acre is 1 chain by 10 chains or 4840 square yards or 43560 square feet. With that information these are all easily solvable without the use of a calculator. The square roots needed to solve problem two can be estimated knowing that 40 squared is 1600, 50 squared is 2500, and 60 squared is 3600. Hope that helps…I have answers to the first 5 and only used a calculator to check my work at the end. As an interesting side note, calculus had already been around for over 100 years at the date of publication of this book…so there was much more advanced math being done without the aid of calculators or computers at that time. I will work answers to the others when I have time in the next day or two.


Recently nearly exact expressions for the distortion in a commonly used family of Pulse Width Modulators (PWMs) known as Asynchronous Sigma Delta Modulators (ASDMs) were presented. Such an ASDM consists of a feedback loop with a schmitt-trigger (or a comparator), and a continuous time loop filter. However these previous results are not yet practically applicable because the effect of unavoidable loop delay (e.g. in the schmitt trigger) was not taken into account. Therefore we now present a more general theory that is also valid when there is a nonzero loop delay. A comparison of the resulting equations with computer simulations demonstrated a very good matching, confirming the validness of the theory. This way, a designer can now easily understand the relationship between the loop filter dynamics and the linearity of an ASDM.


I found it remarkably hard to figure out how to monitor convergence for the expectation maximization (EM) estimtation algorithm. Elementary textbook presentations often just say “until convergence”, which left me scratching my head. More advanced presentations often leave you in a sea of generalized maximization routines and abstract functionals.
Typically, EM is phrased for maximum likelihood estimation (MLE) problems where there are no priors. Given data and parameters , the goal is to find the parameters that maximize the likelihood function .
Likelihood and Missing Data
Usually EM is used for latent parameter problems, where there are latent variables which are treated like missing data, so that the full likelihood function is actually . For instance, might be mixture component indicators, as in soft (EM) clustering. Typically the full likelihood is factored as .
Even though the expectation (E) step of EM computes “expectations” for given current estimates of and the data , these “expectations” aren’t used in the likelihood calculation for convergence. Instead, the form of likelihood we care about for convergence marginalizes away. Specifically, the maximum likelihood estimate is the one that maximizes the likelihood with marginalized out,
Monitoring Likelihood or Parameters
There’s more than one way to monitor convergence. You can monitor either the differences in log likelihoods (after marginalizing out the latent data) or the differences in parameters (e.g. by Euclidean distance, though you might want to rescale). Log likelihood is more task-oriented, and thus more common in the machine learning world. But if you care about your parameters, you may want to measure them for convergence, because …
Linearly Separable Data for Logistic Regression
In data that’s linearly separable on a single predictor, the maximum likelihood coefficient for that predictor is infinite. Thus the parameters will never converge. But as the parameter approaches infinity, the difference its (absolute) growth makes to log likelihood diminishes (we’re way out on the extremes of the logistic sigmoid at this point, where the slope’s nearly 0).
Convergence with MAP?
Textbooks often don’t mention, either for philosophical or pedagogical reasons, that it’s possible to use EM for general maximum a posterior (MAP) estimation when there are priors. Pure non-Bayesians talk about “regularization” or “shrinkage” (specifically the ridge or lasso for regression problems) rather than priors and MAP estimates, but the resulting estimate’s the same either way.
Adding priors for the coefficients, even relatively weak ones, can prevent estimates from diverging, even in the case of separable data. In practice, maximum a posteriori (MAP) estimates will balance the prior and the likelihood. Thus it is almost always a good idea to add priors (or “regularize” if that goes down better philosophically), if nothing else to add stability to the estimates in cases of separability.
Maximization Step with Priors
In EM with priors, the maximization step needs to set , the parameter estimate in the -th epoch, to the value that maximizes the total probability, , given the current “expectation” for the latent parameters based on the the data and previous epoch’s estimate of . That is, you can’t just set to maximize the likelihood, . There are analytic solutions for the maximizer in many conjugate settings like Dirichlet-Multinomial or Normal-Normal, so this isn’t as hard as it may sound. And often you can get away with increasing it rather than maximizing it (leading to the so-called generalized EM algorithm, GEM).
Convergence with Priors
Well, you could just monitor the parameters. But if you want to monitor the equivalent of likelihood, you need to monitor the log likelihood plus prior, , not just the log likelihood . What EM guarantees is that every iteration increases this sum. If you just monitor the likelihood term , you’ll see it bouncing around rather than monotonically increasing. That’s because the prior’s having its effect, and you need to take that into account.


Everybody loves palindromes, right? I do, at least. That’s why I was excited
when I learned that every number can be a palindrome when it’s written in the
appropriate base. There is a trivial proof for this property: any number N > 3
is a palindrome in base N-1 because it may be written “11”. So here is my
solution, in Haskell, to this CodeChef
problem, that finds the smallest base
that makes any given number a palindrome.
This is certainly not the fastest solution possible. Indeed it is downright
naive. But hopefully the logic is clear.


Multi-Sensory Math Manipulatives
Thank you to the Goddard Education Foundation and generous donors. Innovative Teacher Grants are engaging Goddard Public Schools Students in new and exciting ways each day. In Kristina Scott’s Elementary Math classes, her Innovative Teacher Grant is putting math in the palm of her student’s hands (literally!).
Mrs. Scott’s Innovative Teacher Grant provided TouchMath into her small group instruction classes. TouchMath is a research-based method of instruction with the ability to bridge the gap between concrete and representational mathematics. These manipulatives provide students with another unique learning experience that helps them achieve learning goals. “This is that One More Thing as a teacher I needed to help some of my students ‘get it’, says Mrs. Scott.” She goes on to say, “It helps bridge the gap visually using these manipulatives and adds to a variety of different tools we have to offer concepts based on the needs of each student.”
In the short time utilizing these new manipulatives, Mrs. Scott has seen a positive impact in students, not only by improving their math skills but also by increasing their self-esteem and confidence that comes with understanding math concepts. First Grade student, Vada Moore said “Making the numbers felt good. It made me feel like I was doing it on my own. I know how to draw these numbers now.” The excitement of Mrs. Scott's rising mathematicians was evident in her math class as her students’ smiles lit up the classroom, smiles that captured the moment they “got it”.
Without our generous donors, resources like TouchMath would not be possible for our Goddard USD Students. The Goddard Education Foundation is dedicated to creating opportunities and expanding possibilities for Goddard students to be successful learners, like Vada Moore a rising first-grade Mathmetician.