I'm a doctor of philosophy in physics who is passionate about statistical physics and its interdisciplinary applications.
I combine methods and theories from non-linear dynamics, non-equilibrium processes, and phase transitions to study various complex systems.
My work in this area includes mostly modelling and analysing socio-economic phenomena, like opinion dynamics, diffusion of innovation, or group formations.
I am also interested in energy forecasting based on time series methods and deep learning.
I hold a BSc in Theoretical Physics and a MSc in Applied Mathematics with a speciality in Mathematics for Industry and Commerce from Wrocław University of Science and Technology. I work at the same university as a Physicist. Alongside my research and teaching activities, I collaborate with several scientific journals as a reviewer. This year, I also have the pleasure to be a member of the Program Committee at the International Conference on Complex Networks and their Applications.


Every year on the 14th day of March a lot of people celebrate π day (written Pi day). Now Pi is by far one of the most interesting numbers in all of mathematics. The reason is because it belongs to a very large class of interesting numbers and it is conjectured to belong to another very interesting class. The first class is that of transcendental numbers; the second is that of normal numbers. Whatever those two words mean, I want to state now the message of this piece: trying to find meaning in the digits of Pi makes no sense.
First things first: one of the reason there’s such a craze about Pi is because of the digits after the decimal point. Let’s recall that Pi starts like this: 3.1415926536… The ellipsis serve a very specific yet very vague purpose: they indicate that the expansion continues, but it is not possible to predict what digit will appear after the last computed digit; in other words: the sequence of digits after the decimal point is “random”. This is the key word. The sort of reasoning that makes Pi interesting for a lot of people with what I call “mystical inclinations” is the fact that randomness seem to entail the possibility of finding any possible sequence of numbers in the digits of Pi; another leap of reason leads to the following “conclusion”: if we codify Joyce’s Ulysses, or Milton’s Paradise Lost, or the Book of Psalms in King James’ Bible as a string of numbers (and there are many ways of doing this) then the digits of Pi will contain them all. Yet another leap of faith leads to the “conlusion” that any possible story, work of art, biography or musical composition will be found within the digits of Pi. This is what I propose to show that is nonsense.
Now to our first digression: transendental numbers.
Different kinds of numbers
Very early in our mathematical education we get acquainted with these old chaps: 1,2,3,4,5,… (the positive integers). The ellipsis here is very precise: we know what number comes next, just add 1 to the last one appearing in the sequence. Among the many things one can do with these numbers is dividing them:
These three examples will suffice to make my point; I want to discuss what happens after the decimal point in each of these numbers. In the first one things are as simple as can be: some digits (in this case, four) and that’s it, no more. The second case is just an “infinite” string of sixes; this means we know exactly what digit to expect after the last one: the same one. The last one repeats after 6 digits, so we also know exactly what digit will appear next: we just have to count in the repeating string. By the way: this last number is one of the first of the approximations of Pi and it dates back to Archimedes in the third century before Christ.
For the three numbers above there are knowable, exact patterns to compute the decimal expansion (that is, the string of digits after the decimal point): it is either finite or it becomes repetitive after a certain number of digits (which can be VEEEERY large; try dividing 355 by 113). It so happens that this is always possible for numbers that result from the division of two integers; they are called rational numbers. These numbers, then, are the ones for which the decimal expansion is completely determined and knowable (though it may take a long time to compute it in practice). Another way of saying this is that the decimal expansion is not random.
A number that is not rational, meaning it is not the result of dividing two integers, is called irrational. Pi is irrational. So from what we said above it follows that the decimal expansion of Pi is not determined exactly, we have to settle for some approximation. Restating this: if we are given a long string of digits of Pi it is not possible to know which digit comes next just by looking at all the digits we are given, other methods are necessary (but I won’t talk about them here). A number is irrational if its decimal expansion is infinite and non-repeating.
Another irrational number is which is approximately 1.414213. The difference between this number and Pi is that can be computed as the solution to the equation (which also returns its negative); Pi on the other hand is not the solution of such an equation; of course, Pi is the solution of , but that is like saying something like “the blue sky is blue”; what I meant earlier is that Pi cannot be expressed as the solution of a polynomial equation with integer coefficients, such as
Meaning: equations in which powers of an unkown quantity appear multiplied by only integer numbers (the sort of thing one does in an algebra class). Such numbers form the first class Pi belongs to: transcendental numbers are those which are not solution to equations with integer coefficients (like the one above, or ).
The second class I referred to above, that of normal numbers, is a little more tricky, so here goes a second digression: probability.
Probability of things
If I put three yellow marbles and four blue ones on a bag, shake it and without peeking into it take a marble out, will it be yellow or blue? There is no way of knowing for certain the colour of the marble but one thing is certain: it is more “probable” that a blue one comes out, because there are more blue marbles than yellow ones. More precisely: there are seven marbles in total of which three are yellow and four are blue. In this case the probability of getting a yellow marble is
or about 42.8%; similarly the probability of extracting a blue one is or about 57.2% (I cheated here; both numbers’ decimal expansion repeat the sequence before the ellipsis; I rounded them up in percentage so they add 100%). The conclusion is that the probabilites are not equal.
If you toss a coin there are only two possible outcomes: heads or tails, each with the same probability (50-50); if you toss the coin a few times (three, five, seven) you may not notice anything particular about how many heads and how many tails appeared; but if you toss it a very large number of times (usually around twenty or thirty will suffice) the number of tails and the number of heads are more or less the same; that is, if we toss the coin seven times and register the events: heads, tails, tails, heads, heads, tails, tails, then evidently the number of times heads appears is not equal to the number of tails; however, doing it many times both events tend to even out. Another way of saying this is that we have two events with the same probability and if we repeat the experiment (tossing) long enough, the appearance of these two events will be approximately equal, that is they will appear with the same probability. This is crucial for what follows.
An irrational number is said to be normal if each digit appears with the same probability as any other digit, namely because there are ten digits: 0,1,2,3,4,5,6,7,8 and 9; I’ll explain what that means below. In the numbers above (all rational) there is no chance of normality: the digit 9 never appears in them (even in the ones that have an infinite decimal expansion); so 9 occurs with probability 0 (a mathematically pedantic way of saying it doesn’t occur… with some caveats, that is not what probability zero actually means in math). Now take the following number:
This is called Champernowne’s number and its just writing the sequence of positive whole numbers in order. Now, is the sequence of digits “random”? Not in the sense that we cannot predict what number follows, although for doing so we need to check all the digits preceding the one that we are interested in, which in practice is virtually impossible. However this number is normal, since all the digits will appear “equally often”. What this means is that if we take a very big chunk of the sequence of digits, every digit will appear with almost the same probability (recall the experiment of tossing the coins a large number of times); the bigger the chunk the more equal the probability of every digit will be. Of course the sequence doesn’t repeat itself so this number is irrational. This is a normal number.
Digressions over, back to Pi.
“O time thy pyramids”
So, “the randomness of the digits of Pi means that we can find anything in them”, right? Well, it depends what you mean by that. First of all, it is not known whether Pi is normal or not. Assuming it is normal, then it would follow that the digits of Pi are a random sequence of numbers; but remember, randomness only means in this sense that any digit will appear with the same probability as any other digit. Now, does this means it is possible to find “any given string of numbers” (and “therefore” any possible text, painting, symphony or Grateful Dead song) in the decimal expansion of Pi? Well, not really. In Champernowne’s number you will find it, but only because every single integer will appear there. Consider the number
which is not normal as we defined normality, but is normal in base 2; this means that all possible binary digits (namely: 0 and 1; the pattern is: one 1, one 0, two 1s, two 0s, three 1s, three 0s…) will appear equally often in the above number; what we actually defined above was normality in base 10. Nonetheless it is not possible to find the following string in the above number:
So normal numbers don’t necessarily enjoy this property of “having the meaning of life” within their digital expansion.
Even granting that a normal number will contain any sequence of numbers whatsoever, the question remains: how do we read those numbers? Encoding a text, a piece of music or a painting depends on the choice of code (it can be binary, hexadecimal, scrambling letters around or you can use a set of symbols of your own); so even if you can find whatever sequence you want inside the digits of Pi (as consecutive digits, of course) that leaves you with the daunting and virtually impossible task of encoding all of world literature ever written so you can “find” it in the digits of Pi (or any other normal number). The digits of Pi, by themselves, mean nothing; they’re just numbers that appear in the decimal expansion of one of the most important and useful numbers in math and science. There is no mysticism, no cosmic meaning, no interpretation of your dreams (dry or otherwise) in that string of numbers; if there is, it is you who puts it there, not Pi. I don’t believe in that sort of thing; I don’t have a problem with people that do, but I think they should not forget that those interpretations are not a property of the numbers themselves. In the film Pi the mentor of the main character (a mathematical genius obsessed with finding patterns in the digits of Pi) at one point warns him peremptorily: “once you discard scientific rigor you are no longer a mathematician, you’re a numerologist!”. Caveat credentes.
A lot of people draw the analogy of having whatever meaning encoded in Pi’s digits from Borges’ story The Library of Babel. In it, a library is described that contains any possible text; Borges mentions a volume that is “a mere labrynth of letters” but contains the sentence “O time thy pyramids” near the end; but this was Borges, he could do that sort of thing and besides, he (nor the narrator) doesn’t draw the conlusion that the library holds meaning; actually quite the contrary… but I wouldn’t deprive you of the pleasure of reading it by yourself. The idea that by randomly concatenating every individual on a set of symbols (be those decimal or binary digits, letters of the latin alphabet or the set of nonce words in a Monty Python sketch) you can obtain any possible meaning is not a new one: Jewish mysticism invented the Kabbalah (and incidentally, the theory of permutations) many centuries ago. Trying to attach meaning to things that intrinsically have none is a sort of mysticism, whether that meaning is of a mystic/religious/occult nature or not. Mysticism by itself is harmless but it breeds fanatism and reality-denial more often than not. Again, belivers beware!
Numbers are not the guardians of perennial wisdom, nor they hold the key to past, present or future; they are devices, tools we use to understand things around us. The Librarians of Babel, if there are such beings, are not numbers.


Funded PhD Studentships in the Mathematics of Information by CCIMI
The CCIMI invites applications for fully funded PhD studentships in Mathematics of Information. We held an Open Day on Thursday 15th November where prospective students came along and discovered more about our Cambridge Mathematics of Information (CMI) PhD program. Further information is available on our.
CCIMI Fully Funded PhD Studentships in the Mathematics
The advance of data science and the solution of big data questions heavily relies on fundamental mathematical techniques and in particular, their intra-disciplinary engagement.
This is at the heart of the center for the Mathematics of Information which involves mathematical expertise ranging from statistics, applied & computational analysis, to topology and discrete geometry – all with the common goal of advancing data science questions.
Alongside this, specific questions which feed into fundamental methodology development arise naturally in applications we focus on in interdisciplinary engagements.
For instance, economists and social scientists on questions about financial markets and the internet, with physicists and engineers on software and hardware development questions in the context of security, imaging, and structured data processing, as well as biomedical scientists on data science in healthcare and biology.
More info – CCIMI invitation on Funded PhD Studentships
Both this general advancement of data science, and its applications to specific questions is realized by key mathematical expertise represented in the institute including:
- inverse problems,
- convex analysis,
- sparse optimization
- stochastic analysis
- compressed sensing
- sampling theory
- approximation theory
- random matrices
- harmonic analysis
- partial differential equations
- functional analysis
- quantum computation
- discrete geometry
- graph theory.
We welcome applications for studentships relating to projects and subject areas covering all aspects of the broad field of mathematics of information. Further information on some of the projects currently being investigated by students and faculty can be found on our website, linked here.
The Cantab Capital Institute for the Mathematics of Information is hosted within the Faculty of Mathematics of the University of Cambridge and is a collaboration between DAMTP and DPMMS.
It accommodates research activity on fundamental mathematical problems and methodology for understanding, analyzing, processing and simulating data. Data science research performed in the Institute is on the highest international level, aiming to extract the relevant information from large- and high-dimensional data
Eligibility (Funded PhD Studentships)
The academic requirements for entry to this PhD are a first-class honors degree, awarded after a four-year course in mathematics or related subject, or a three-year degree with a one-year postgraduate course on advanced mathematics or related subject. For further details on how to apply for this program see the relevant entry on the website.
- The scheme is open to applicants from all countries.
Applicants must meet the following criteria:
- The academic requirements for entry to this PhD are a first-class honours degree, awarded after a four-year course in mathematics or related subject, or a three-year degree with a one-year postgraduate course on advanced mathematics or related subject.
- The usual minimum entry requirement is a first-class honours degree, awarded after a four-year course in mathematics or mathematics-related subject, or a three-year degree together with a one-year postgraduate course on advanced mathematics or a mathematics-related subject. Part III (MMath/MASt) of the Mathematical Tripos provides such a course. Note, however, that entry is competitive and a higher level of preparation may be required.
English Language Requirements:
- Applicants whose first language is not English are usually required to provide evidence of proficiency in English at the higher level required by the University.
Funding (Funded PhD Studentships)
Fully funded PhD Studentships do include University Composition Fees and maintenance for the duration of your course to match the RCUK minimum level. The scheme is open to applicants from all countries.
- Applications Deadline: Jan three, 2022
- Course Level: Studentships area unit out there to pursue Doctor of Philosophy program.
- Study Subject: Studentships area unit awarded in fields of arithmetic of data.
- Scholarship Award: office offers:
- Fully funded Doctor of Philosophy Studentships does embody University Composition Fees and maintenance for the length of your course to match the RCUK minimum level.
- The theme is hospitable candidates from all countries.
- Nationality: The theme is hospitable candidates from all countries.
- Number of Scholarships: ranges not given
- Scholarship may be taking within Britain.
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Route for application (Funded PhD Studentships)
Apply using the standard PhD application procedure via the University’s Graduate Admissions website.
It is very strongly encouraging that applications are receive(d) by 3rd January 2023.
Shortlisted candidates will be interview(ed) – the date for interviews will be communicated once shortlisting has taken place.
Please contact email@example.com in the first instance for any inquiries about the applications process, and for any inquiries regarding the PPh.D.programme.
How to Apply:
- Apply using the standard PhD application procedure via the University’s Graduate Admissions website (https://www.graduate.study.cam.ac.uk/courses/directory/mapmpdmal); and
- Send an expression of interest letter to cmi-at-maths.cam.ac.uk which explains why you are interested in this course.


Thursday, September 26 @ 15:00
Quantum mechanics is the branch of physics that studies how the universe works at the smallest scales. It has given us technologies like semiconductors, lasers, LEDs, and electron microscopes, and promises even more exciting applications like quantum computing.
The quantum world is notoriously difficult to understand, since few of our intuitions about physics apply at the subatomic scale. But not so for functional programmers! Many of the mathematical ideas that we use to understand the quantum world, like Hilbert space, operators, tensors, and group representations, can all be understood in terms of functions. In this talk, I'll use familiar concepts like composition, partial application, and currying to make the foundations of the subject clear and show how to solve some basic problems in F#. At the end, I'll show how these techniques can be used to model a simple quantum computer.
I'm the co-founder and CEO of a company called Tachyus, where we build enterprise software for the energy industry using the F# programming language. I am also passionate about scientific and mathematical programming, particularly in F# and other functional languages. I did a lot of work in my graduate degree on solving mathematical problems in quantum and particle physics using F#. I also have a forthcoming book called "Math for Programmers" teaching professional software developers calculus, linear algebra, and applications.


Math Prep Expressway
EdReady lets you test yourself in math then further provides you with a customized study plan. If you're thinking about what's next for career and college, then be smart and be prepared!
Congratulations on your choice to participate in Central Piedmont's Math Prep Expressway experience! We welcome you to CPCC and intend for your EdReady experience to:
- Prepare you for a more successful attempt on the math portions of the CPCC placement test
- Provide you with a rich learning experience that is based on independent engagement of math topics
- Assist you with gaining a realistic understanding of the amount of study time that you will need to regularly commit to spending outside of the classroom


Play today, learn for life
Your Playlearning™ adventure starts here
Modern learning for today's world
Explore learning activities like math and reading with 450+ objectives across school subjects and life skills!
Discover their superpowers
Build skills to give your child to get a head-start for school—and beyond.
What grown-ups are saying
I am so happy that I found the free trial so I could see just how many times my kids got on the app. Thank you for making a safe and educationally sound app for my kids!
This app has consistently been our favorite. The songs are engaging. Graphics are adorable. We love how much she has learned from Lingokids.
My daughter (3 years old) loves this app, she can switch between videos and games easily. The option to choose a different game or play a similar game is so good for her. She especially enjoys the songs and watching the animations.


Maths is all around us and problem solving is at the heart of the mastery approach. Look for maths problems you can solve together, making connections between what your child has been learning at school and the world around them.
Demonstrating these connections – and representing them in multiple ways – not only supports your child’s understanding and cements their knowledge; it reinforces the relevance of maths in our lives and makes it fun.
- Follow a recipe: work together to find out the quantities needed, ask your child to weigh the ingredients, discuss how you’d halve or double the recipe and discuss the ratio of ingredients.
- Talk about the weather forecast: is today’s temperature higher or lower than yesterday’s? What do the numbers mean?
- Going shopping: talk about the cost of items and how the cost changes if you buy two items instead of one. Let your child count out the coins when paying and discuss the change you get back. Use coins to explore addition, subtraction, multiplication and division.
- Planning an outing: discuss how long it takes to get to the park, and so work out what time you need to leave the house. Encourage your child to work out the best solution based on the time and distances. Discuss what shapes you see when you get there
- Telling the time, discussing the days of the week, talking about money or the coins needed to pay for items, how long things take to cook
- GROWTH MINDSET – Everyone of us can master mathematics given the opportunity.
Ideas for Counting
- Collections of objects – shells, buttons, pretty stones
- Cars on a journey, e.g. how many red cars?
- Animals in a field e.g. sheep, cows
- Stairs up to bed, steps, etc.
- Pages in a storybook
- Counting buttons, shoes, socks as a child gets dressed
- Tidy a cupboard or shelf and count the contents e.g. tins, shoes, etc.
- Counting particular vehicles on a journey e.g. Eddie Stobart lorries, motorbikes, etc.
- Keep maths practical and real life
- Money – paying for things, playing shops, purses
- Dishing up dinner – problem solving
- Games (snakes and ladders, dice)
Think about a picture and see what questions you can derive from it to ask your child. For example, using this picture, look how many questions we can ask related to maths:
Mathematics language often uses common words in a new way. For example, ‘difference’, ‘right’, ‘product’, ‘table’.
To support your child’s understanding of mathematical words, ask them to explain the words they’ve been using and what they mean. Find out what new maths vocabulary your child’s teacher is introducing so you can use it at home to complement their learning.
Always encourage your child to explain how they have gone about solving a problem, and work with them to test, prove, explain, reflect and spot patterns. Questioning and prompts can be powerful tools to boost your child’s mathematical thinking: ‘What do you think…?’ ‘Why …?’ ‘What will happen if…?’ ‘What do you notice about…?’ ‘Can you see a pattern between…?’ ‘What if we try…?’
Communicating and discussing maths problems (in a way that others can understand) demonstrates depth of understanding – another fundamental aspect of mastering mathematics.


This is interesting. Hal Higdon – uses a formula of 5 times your 10K time to predict marathon time potential. That’s a fun way of looking at our times this year after 10ks this summer as many of us prepare for fall marathons.
This turned out to be in the ballpark for me last year. I ran a 10k in 1 hr 2 min in June and did my first marathon in October in 4:57. (1:02 x 5 = 5:10 marathon prediction.) I was coming off a hip injury and 3 months of no training when I did the 10k last year, so I think this makes sense.
It’ll be fun to see how things go on Saturday.


Kearns Mathematics/Statistics Scholarship
1. Applicant must be a Mathematics Major. Special consideration will be given to applicants planning on meeting the Applied Mathematics and Statistics Area Requirements.
2. Applicant must be a full-time undergraduate student with Junior or Senior class level standing who will qualify for graduation within the next two semesters.
3. Applicant must have a minimum 3.0 cumulative GPA.
4. Applicant must be a U.S. citizen.
5. Applicant must be an active participant in community and/or campus student activities.
6. Applicant must require financial assistance in order to continue studies at CSUS.
No discrimination shall be made on the basis of sex or ethnicity.


Crash Casino Game Algorithm Demystified: How It Works
Crash Casino game, also known as Crash, is a popular online gambling game that has gained immense popularity in recent years. The game is a thrilling combination of luck and strategy, where players place bets and try to predict when the game will crash. But have you ever wondered how the algorithm of the Crash Casino game actually works? In this article, we will demystify the algorithm behind this exciting game.
At its core, the Crash Casino game algorithm is based on a random number generator (RNG). This RNG ensures that every game outcome is completely random and unbiased, providing a fair playing field for all participants. The algorithm uses complex mathematical formulas to generate these random numbers, ensuring that they cannot be predicted or manipulated.
When the game begins, players place their bets and wait for the multiplier to increase. The multiplier represents the potential payout for each player. As the multiplier increases, so does the risk of the game crashing. Players must decide when to cash out and collect their winnings before the game crashes. The challenge lies in predicting the optimal moment to cash out, as the game can crash at any time.
The algorithm behind the Crash Casino game calculates the multiplier based on a variety of factors. These factors include the number of bets placed, the total amount of money wagered, and the rate at which players are cashing out. The algorithm constantly adjusts the multiplier to ensure that the game remains balanced and fair.
One important aspect of the algorithm is the concept of "bust probability." This represents the likelihood of the game crashing at any given moment. The bust probability is influenced by the multiplier and the amount of money wagered. As the multiplier increases, the bust probability also rises, making it riskier for players to continue betting.
Another factor that affects the algorithm is the house edge. The house edge is the statistical advantage that the casino has over the players. In the Crash Casino game, the house edge is typically around 1%, meaning that the casino is expected to make a profit of 1% of the total bets placed over time. The algorithm ensures that this house edge is maintained, allowing the casino to generate revenue while still providing a fair and enjoyable gaming experience.
It is important to note that while the algorithm of the Crash Casino game is designed to be fair and random, there is still an inherent risk involved in playing. The outcome of each game is unpredictable, and players should always approach gambling responsibly and set limits on their bets.
The algorithm behind the Crash Casino game is based on a random number generator and utilizes mathematical formulas to determine the multiplier and bust probability. It ensures that each game outcome is fair and unbiased. Understanding the algorithm can help players make informed decisions, but it is important to remember that gambling always carries a risk. Play responsibly and enjoy the excitement of the Crash Casino game!


TouchDRO Arbitrary Hole Circle Function
The Arbitrary Hole Circle function is very similar to the Bolt Hole Circle function but instead of creating holes that are spaced equally, it can be used to create a set of holes with arbitrary spacing.
The function has two modes - Basic and Advanced:
- Basic mode can be accessed by pressing the "Arbitrary Hole Circle Function" button in the function strip or "Arbitrary Hole Circle" on the "Add Sub-Datum(s)" menu
- Advanced mode can be accessed by long-pressing the "Arbitrary Hole Circle Function" button in the function strip or "Arbitrary Hole Circle" on the "Add Sub-Datum(s)" menu.
In basic mode, the function will create a circle in the machine's default projection plane with the center located at the current spindle/cutter position (current absolute position for the relevant axes).
Default Projection Planes
The project plane is set as follows:
- Milling machine - XY, 0.0 degrees parallel to positive X direction
- Lathe - XZ, 0.0 degrees parallel to positive Z direction
Please note: in basic mode, the third axis dimension is omitted. For example, milling machine sub-datums will not have Z set to any value.
The function accepts the following parameters:
Circle radius can be any positive decimal number or an arithmetic expression. This value is required.
Circle radius dimension can be set in inches or millimeters regardless of the currently selected axis units. By default, units will be set based on the TouchDRO default system that is set in the application preferences.
Hole Angle (α)
Hole angle field, in conjunction with the "+" button is used to add between 2 and 100 holes at arbitrary angles. The field can accept decimal values between -360.0 and 360.0, as well as basic arithmetic equations.
In advanced mode, the Arbitrary Hole Circle function offers more control over the projection plane of the Bolt Hole Circle, position of the center, and the value of the third axis.
In advanced mode, the function accepts the following additional parameters:
Machine plane on which the circle will be created. Default values for the plane are XY for a milling machine and XZ for a lathe. Starting angles are set as follows:
- XY - 0.0 is parallel to the positive Y direction
- XZ - 0.0 is parallel to the positive Z direction
- YZ - 0.0 is parallel to the positive Z direction
X, Y, and Z Position fields define the location of the Bolt Hole Circle in two or three dimensions. Depending on the selected projection plane, two of the three input fields will be enabled and are required. The third field is optional and can be enabled if desired.
By default, enabled fields are pre-filled with the current readings for the relevant axes. If a value is changed, it can be restored by pressing the "synchronize" button to the right of the field.
Units for each field are set to the current units for each individual axis.
How To Use
Drill Three Pairs of Holes
Goal: drill three pairs of holes, as shown in the sketch below.
- Bring up the "Arbitrary Hole Circle" function
- Enter 25/2 into the Circle Radius field
- Click on "mm" radio button to switch to millimeters (if needed)
- Enter the following values into the "Angle α" field; press the "+" button after each entry:
- Press "Create Circle"
The result will look similar to the screen fragment below:


Each year, the Knowles Science Teaching Foundation (KSTF) awards teaching fellowships to exceptional young people committed to teaching physical sciences, biological sciences, or mathematics in U.S. high schools. These fellowships are designed to meet teachers’ needs from the time they begin working on their teaching credentials through the early years of their careers.
KSTF Teaching Fellowships combine extensive financial and professional support. The total award for each fellow is valued at nearly $150,000 over the course of the five-year fellowship. Fellows receive tuition assistance while participating in a teacher credentialing program, monthly stipends, and grants for professional development and teaching materials.
Applicants should have received their most recent content (i.e., science, mathematics, or engineering) degree within five years of the start of the fellowship. Individuals in the final year of an undergraduate or master’s degree are also eligible. Applicants must be enrolled or plan to enroll in a recognized teacher education program that leads to a secondary science or mathematics teaching license. Applicants need not be U.S. citizens but must be able to study and work in the U.S. for the term of the fellowship.
KSTF Teaching Fellowships are awarded based on four selection criteria: science or mathematics content knowledge, commitment to teaching, professional ability, and leadership.


What FitAll Does
FitAll is a general-purpose, nonlinear regression analysis (curve fitting) program that can be used to fit a set of experimental data to any of the defined functions (models).
FitAll fits data to continuous, single-valued functions with the general form:
Yi = f (Xij, Kj, Pj)
and implicit functions with the general form:
Yi = f (Yi, Xij, Kj, Pj),
Yi is the dependent variable (measured value of the function for the ith data point).
Xij is the set of j independent variables for the ith data point.
Kk is a set of k constants that can be modified at runtime.
Pp is the set of p parameters that are evaluated.
Linear Least Squares (lls): minimizes the sum of the squares of the deviations between the observed and calculated values of a function that is linear in its parameters.
Nonlinear Least Squares (nls): minimizes the sum of the squares of the deviations between the observed and calculated values.
Nonlinear Least Absolute Deviations (nlad): minimizes the absolute deviations between the observed and calculated values.
Three different regression analysis methods can be used to obtain the best fit:
The regression analysis can be weighted using four different weighting schemes.
The data can be smoothed using the Boxcar and Fast Fourier Transform methods.
FitAll reports the standard deviation of the overall fit as well as that of each of the resolved parameters.
All results and reports can be printed, saved to a file, copied to the clipboard or transferred to a MS Word or LibreOffice Writer.
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One questions many “for sale by owner” sellers ask is “how can I determine if a potential buyer can afford to buy my house?” In the real estate industry this is referred to as “pre-qualifying” a buyer. You might think this is a complex process but in reality it is actually quite simple and only involves a little math. Before we get to the math there are a few terms you should understand. The first is PITI which is nothing more than an abbreviation for “principal, interest, taxes and insurance. This figure represents the MONTHLY cost of the mortgage payment of principal and interest plus the monthly cost of property taxes and homeowners insurance. The second term is “RATIO”. The ratio is a number that most banks use as an indicator of how much of a buyers monthly GROSS income they could afford to spend on PITI. Still with me? Most banks use a ratio of 28% without considering any other debts (credit cards, car payments etc.). This ratio is sometimes referred to as the “front end ratio”. When you take into consideration other monthly debt, a ratio of 36-40% is considered acceptable. This is referred to as the “back end ratio”.
Now for the formulas:
The front-end ratio is calculated simply by dividing PITI by the gross monthly income. Back end ratio is calculated by dividing PITI+DEBT by the gross monthly income.
Let see the formula in action:
Fred wants to buy your house. Fred earns $50,000.00 per year. We need to know Fred’s gross MONTHLY income so we divide $50,000.00 by 12 and we get $4,166.66. If we know that Fred can safely afford 28% of this figure we multiply $4,166.66 X .28 to get $1,166.66. That’s it! Now we know how much Fred can afford to pay per month for PITI.
At this point we have half of the information we need to determine whether or not Fred can buy our house. Next we need to know just how much the PITI payment is going to be for our house.
We need four pieces of information to determine PITI:
1) Sales Price (Our example is 100,000.00)
From the sales price we subtract the down payment to determine how much Fred needs to borrow. This result brings us to another term you might run across. Loan to Value Ratio or LTV. Eg: Sale price $100,000 and down payment of 5% = LTV ration of 95%. Said another way, the loan is 95% of the value of the property.
2) Mortgage amount (principal + interest).
The mortgage amount is generally the sales price less the down payment. There are three factors in determining how much the PI& interest) portion of the payment will be. You need to know 1) loan amount; 2) interest rate; 3) Term of the loan in years. With these three figures you can find a mortgage payment calculator just about anywhere on the internet to calculate the mortgage payment, but remember you still need to add in the monthly portion of annual property taxes and the monthly portion of hazard insurance (property insurance). For our example, with 5% down Fred would need to borrow $95,000.00. We will use an interest rate of 6% and a term of 30 years.
3) Annual taxes (Our example is $2,400.00)/12=$200.00 per month
Divide the annual taxes by 12 to come up with the monthly portion of the property taxes.
4) Annual hazard insurance (Our example is $600.00)/12=$50.00 per month
Divide the annual hazard insurance by 12 to come up with the monthly portion of the property insurance.
Now, let’s put it all together. A mortgage of $95,000 at 6% for 30 years would produce a monthly PI
Putting it all together
From our calculations above we know that our buyer Fred can afford PITI up to $1,166.66 per month. We know that the PITI needed to purchase our house is $819.57. With this information we now know that Fred DOES qualify to purchase our house!
Of course, there are other requirements to qualify for a loan including a good credit rating and a job with at least two years consecutive employment. More about that is our next issue.


The NEET 2022 results will shortly be made public by the National Testing Agency (NTA). To learn how to calculate NEET percentile, continue reading.
Soon, the National Eligibility comp Entrance Test results will be released. The results will be uploaded either on nta.ac.in or ntaneet.nic.in. The precise result timings have not yet been disclosed by NTA, the organisation in charge of organising NEET 2022. Before the publication of the NEET results in 2022, candidates can learn how to compute their percentile, nevertheless.
A candidate must first be aware of his NEET scores and NEET rank in order to compute his NEET percentile. Let’s now look at how to calculate these two components of the NEET 2022 outcome. Before looking at the NEET results, candidates should be informed that, according to the NTA, general category students must receive at least 50 percentile marks, whilst students in reserved categories must receive at least 40 percentile in order to be eligible for NEET counselling.
How to calculate NEET rank?
The NTA determines a candidate’s NEET rank and publishes it in the rank list. However, candidates can compare their NEET score from the prior year to their NEET rank to determine their NEET 2022 rank before the results are made public. It is also useful to be aware that NTA has said on its official website that it ranks candidates using a variety of tie-breaking procedures after taking into consideration their raw NEET scores.
How to calculate NEET 2022 score
Candidates can use the NTA-provided answer key to compute their NEET score. To determine which questions you successfully answered, students can compare their responses to those listed in the NEET answer key. Each correct response earns 4 points, while each incorrect response loses 1 point. Questions with several answers marked but no response will not receive any points.
- Total NEET Marks = 180 MCQs X 4 = 720 Marks
- Your NEET score = (Total number of correct answers x 4) (Total number of incorrect answers x 1)
How to calculate NEET percentile?
Marks based NEET percentile
A student can easily calculate their NEET percentage with his NEET score and NEET topper’s score in that particular year.
NEET percentile = your NEET score * 100 / topper’s NEET score
Rank based NEET percentile
- Students can also calculate their NEET percentile along with their NEET rank and the number of candidates who appeared for NEET in that particular year.
- NEET percentile = [(total number of candidates appeared in NEET rank) / total no. of appearing candidates] x 100
- This method is quite difficult compared to the previous one.


Prof. Ken Hayami (National Institute of Informatics)
Dr. Yasunori Aoki (Uppsala University)
Cluster Newton Method Source Code:
COPYRIGHT © 2014-2022 Yasunori Aoki, Ken Hayami and Akihiko Konagaya
Released under the MIT license:
Here is the procedure to make the MATLAB code work with version R2015b. To make the MATLAB code work for version R2015b, please change the following lines: In main.m, change line 3 from matlabpool open to poolobj = parpool; and remove line 17 matlabpool close In PBPK_model.m, change line 46 from function [ans]=RHS(t,u) to function [answer]=RHS(t,u) and change line 49 from ans=h(x,u,t)'; to answer=h(x,u,t)'; Then running main.m should perform as usual.
Please provide feedback if you have any questions or suggestions.
info-elsi [at] cbiri.cbi-society.info
The following MATLAB codes of Aoki’s original Cluster Newton Method and Arikuma Irinotecan PBPK model are released under the MIT license:
In zipfile: inverse_prob_PBPK_model_start_here
Yasunori Aoki, Ken Hayami, Hans De Sterck, Akihiko Konagaya: Cluster Newton Method for Sampling Multiple Solutions of Underdetermined Inverse Problems:Application to a Parameter Identification Problem in Pharmacokinetics, SIAM J. Scientific Computing, 36 (1), B14-B44 (2014); available at 10.1137/120885462 and NII-2011-002E
 Arikuma T, Yoshikawa S, Azuma R, Watanabe K, Muramatsu K, Konagaya A.: Drug interaction prediction using ontology-driven hypothetical assertion framework for pathway generation followed by numerical simulation, BMC Bioinformatics, vol. 9 (Suppl 6), S11 (2008); available at doi:10.1186/1471-2105-9-S6-S11
* Arikuma irinotecan PBPK model was originally written by Takeshi Arikuma in GNU Octave in 2008, and then rewritten and improved by Yasunori Aoki in Matlab in 2011.
The following MATLAB codes of Yoshida’s Cluster Newton Method, Gaudreau’s Cluster Newton Method are published as some derivatives of the original cluster newton method under the Academic Free License, version 3.0 specified below. The following references should be cited when publishing related topics with these codes, respectively.
Yoshida K, Maeda K, Kusuhara H, Konagaya A: Estimation of feasible solution space using Cluster Newton Method: application to pharmacokinetic analysis of irinotecan with physiologically-based pharmacokinetic models, BMC Systems Biology 2013, 7 (Suppl 3): S3 (2013); available at doi:10.1186/1752-0509-7-S3-S3
Philippe Gaudreau, Ken Hayami, Y. Aoki, Hassan Safouhi, Akihiko Konagaya:
Improvements to the cluster Newton method for underdetermined inverse problems.J. Computational Applied Mathematics283:122-141(2015); available at doi:10.1016/j.cam.2015.01.014 and NII-2013-002E
In zipfile: invprob_140127


Sports betting is an exciting mix of sport and the potential to win big! Figuring out your potential payouts is a vital part of this thrilling hobby. Whether you’re a pro or starting out, understanding how to calculate your winnings can enhance your betting experience.
To work out potential payouts, consider several factors. These include the bookmaker’s odds (decimals, fractions, or moneyline), the amount wagered, and the type of bet placed. The odds show the chance of a result occurring.
For decimal odds, multiply your wager by the decimal odds. For example, £100 on a team with odds of 2.5 = £250 (£100 x 2.5). Fractional odds = multiply your bet by the fraction + 1. So, £50 on a team with fractional odds of 5/1 = £300 (£50 x (5/1 + 1)).
For moneyline odds, positive and negative numbers are used. To calculate potential winnings for positive odds, divide your bet by 100 and multiply by the odds. For instance, £200 on an underdog with moneyline odds of +250 = £700 (£200 / 100 x 250 + 200).
Tip: Don’t forget to read the terms and conditions of each bet to ensure accurate calculations and no surprises when it’s time to collect your winnings.
Remember, calculating potential payouts is key to sports betting. By getting a grip on the odds and using these calculations, you can judge the profitability of a bet and make the right decision. Good luck!
Understanding Odds in Sports Betting
Understanding Odds in Sports Betting
Odds are an integral part of sports betting, providing crucial information about the potential payouts. To make informed betting decisions, it is essential to understand how odds work and what they represent.
To simplify a complex topic, let’s use a table that demonstrates the different types of odds commonly used in sports betting:
|Type of Odds||Definition||Example|
|Fractional||Shown as a fraction such as 5/1, indicating the potential profit relative to the stake.||Betting £1 at 5/1 odds would yield a £5 profit if the bet wins, plus the initial stake.|
|Decimal||Shown as a decimal such as 6.00, representing the total payout including the initial stake.||Betting £1 at 6.00 odds would lead to a total payout of £7 if the bet wins (including the initial stake).|
|Moneyline||Commonly used in American sports, it shows the amount that would be won on a $100 bet for positive odds or the amount that needs to be bet to win $100 for negative odds.||Odds of +150 would mean a $150 profit on a $100 bet, while odds of -200 would require a $200 bet to win $100.|
Understanding these different odds formats enables bettors to compare them effectively, be it at a bookmaker’s shop or online. Whether fractional, decimal, or moneyline odds, they all serve the same basic purpose of indicating potential payouts.
Now, here’s a pro tip: It’s crucial to shop for the best odds among different bookmakers. By doing so, you can maximize your potential payout and increase your chances of long-term profitability.
Understanding different types of odds: it’s like decoding a secret language, except instead of finding treasure, you’re just trying to figure out if your team will win or if it’s time to drown your sorrows in ice cream.
Explaining the Different Types of Odds (Decimal, Fractional, American)
Exploring the world of sports betting needs an understanding of the different types of odds. These are decimal, fractional and American. Each has its own intricacies and nuances.
Decimal odds are popular in Europe and Australia. This shows how much you can win per unit you bet. For example, betting £10 on a match with decimal odds of 2.50 means a potential win of £25.
Fractional odds are popular in the UK. They show the potential winnings as a fraction of the stake. So, betting £10 on a horse race with 5/1 odds would give you £50.
American odds are used in America. They come in two forms – positive and negative. Positive shows how much you’d win from a £100 bet, while negative shows how much you need to bet to win £100. For instance, American odds of +250 means a £100 bet could win you £250.
Bookmakers sometimes offer unique variations or combinations tailored to specific events or markets. Oddsmakers consider various factors such as team form, player injuries, weather conditions and historical data when setting betting lines. They help in determining the accurate probabilities for each outcome.
Calculating payouts with decimal odds can be like doing math in a foreign language – though not understood, you still hope for a jackpot.
Calculating Potential Payouts Using Decimal Odds
When it comes to calculating potential payouts in sports betting using decimal odds, there are a few key steps to keep in mind. By understanding how decimal odds work and employing a simple formula, you can quickly determine the potential payout for any given bet.
To demonstrate this process, let’s create a table that outlines the necessary calculations and showcases the results. This will help you visualize the steps involved and make the process more accessible.
Bet Amount (£)Decimal OddsPotential Payout (£)102.5025.00201.7535.00503.00150.00
In this table, the “Bet Amount (£)” column represents the amount of money you are placing on a bet. “Decimal Odds” indicates the odds given for the specific event or match, presented in decimal format. Finally, the “Potential Payout (£)” column provides the amount you could potentially win if your bet is successful.
To calculate the potential payout, you simply multiply the “Bet Amount (£)” by the “Decimal Odds.” For example, if you bet £10 with decimal odds of 2.50, your potential payout would be £25. Similarly, a £20 bet with odds of 1.75 would result in a potential payout of £35. This formula allows you to determine your potential winnings with ease.
It’s important to note that decimal odds incorporate the initial stake into the potential payout. Unlike fractional odds, where the potential profit is separate from the stake, decimal odds provide a clearer picture of the total amount you could receive if your bet is successful.
In the world of sports betting, understanding how to calculate potential payouts is crucial for making informed decisions. Being knowledgeable about decimal odds and their calculations can give you an edge when it comes to evaluating potential betting scenarios. Knowing how to bet on the World Cup can help ensure you make the most of every bet placed.
So, the next time you place a bet, take a moment to consider the potential payout using decimal odds. Armed with this information, you can make more informed choices and potentially increase your winnings.
Remember, gambling should be done responsibly, and it’s always prudent to consult reliable sources for accurate odds and information.
Decimals may look intimidating, but calculating potential payouts with them is easier than finding a decent pizza topping for pineapple lovers.
Step-by-Step Guide on Calculating Potential Payouts with Decimal Odds
Calculating potential payouts with decimal odds is key for sports bettors. It helps you figure out the profitability of your bets and make smart decisions.
Follow these 6 steps to work out the payouts:
|1||Find the decimal odds. For example, if the odds are 2.5, you’ll get £2.50 for each £1 you bet.|
|2||Decide how much you want to wager.|
|3||Multiply your stake by the decimal odds to get your total potential payout.|
|4||Add back your original stake to the total payout.|
|5||Weigh the risk-reward ratio: Higher odds mean higher risk and reward.|
|6||Make an informed call. Use the analysis to decide whether to place a bet or not.|
It’s important to remember that sports betting outcomes can’t always be predicted. Do your research before betting.
John was a betting enthusiast who had a knack for accurately predicting football matches. He saw a game with decimal odds of 4.0 and his favorite team was playing. He put down £50 and his team won.
This meant a total potential payout of £200 (4 x £50), including a profit of £50 from his original stake. This teaches us the importance of understanding decimal odds and using them to evaluate payouts.
Calculating Potential Payouts Using Fractional Odds
Calculating Potential Payouts Using Fractional Odds
Fractional odds are commonly used in sports betting to calculate potential payouts. These odds represent the ratio of the potential profit to the amount staked. To determine the potential payout using fractional odds, you need to understand the format and make some simple calculations.
To illustrate this, let’s consider a hypothetical football match between Team A and Team B. The bookmaker offers the following fractional odds for each team: Team A at 2/1 and Team B at 3/2.
Using these odds, we can create a table to calculate the potential payouts:
|Team||Fractional Odds||Staked Amount||Potential Profit||Potential Payout|
In this table, we assume a £10 stake for each team. To calculate the potential profit, we multiply the stake by the numerator of the fractional odds and divide it by the denominator. For Team A, the potential profit is (£10 * 2)/1 = £20. Similarly, for Team B, the potential profit is (£10 * 3)/2 = £15.
To determine the potential payout, we add the potential profit to the stake. For Team A, the potential payout is £10 (stake) + £20 (profit) = £30. For Team B, the potential payout is £10 (stake) + £15 (profit) = £25.
It’s important to note that the potential payout represents the total amount you would receive if your bet is successful, including the return of your initial stake.
Understanding how to calculate potential payouts using fractional odds is essential for making informed betting decisions. By knowing the potential payout, you can assess the potential risk and reward of a particular bet.
In the world of sports betting, calculating potential payouts using fractional odds has been a tried and tested method for decades. Bookmakers have used this system to provide transparent information to bettors, enabling them to make knowledgeable choices when placing their bets. So, the next time you consider placing a bet, make sure you understand how to calculate the potential payouts using fractional odds.
Fractional odds may seem confusing at first, but once you wrap your head around them, you’ll be calculating potential payouts like a mathematically-inclined sports fanatic with a dark sense of humor.
Step-by-Step Guide on Calculating Potential Payouts with Fractional Odds
Figuring out payouts with fractional odds requires careful steps. Here’s a guide:
|1. Find Fractional Odds:|
|– Look for the fractional odds given by the bookie, like 2/1 or 5/2.|
|– The first number is potential profit & the second number is the stake.|
|2. Calculate Potential Profit:|
|– Divide the first number by the second and add 1. E.g. 2/1 = 2/1 + 1 = 3.|
|– Multiply this by your stake to get potential profit.|
|3. Work Out Total Payout:|
|– Put your original stake and potential profit together for total payout.|
|– E.g. bet £10 on 2/1 odds = £30 profit, so £10 + £30 = £40 total payout.|
|4. Learn Different Odds Formats:|
|– Get used to formats such as decimal and moneyline.|
|– Converting between formats helps compare different markets.|
|5. Think About Margin & Value:|
|– Bookies add margin to guarantee profit no matter the result.|
|– Find bets with value, i.e. odds higher than expected.|
It’s not just calculatin’ payouts with fractional odds, other elements are needed for success. Strategy & research are key, plus practice makes perfect when it comes to identifying value bets. Calculating payouts with American odds is simpler than understanding NFL draft picks!
Calculating Potential Payouts Using American Odds
Calculating Potential Payouts Using American Odds can be a complex task, but fear not, we’ve got you covered. To help you navigate through the intricacies of sports betting, let’s explore how to calculate your potential payouts using American odds.
Let’s begin with a comprehensive table that will assist us in understanding the calculations involved. Take a look at the table below:
In the table above, we have different types of bets along with their corresponding odds, stake, profit, and potential payout. The calculations follow a specific formula that takes into account the odds and the amount wagered.
Now, let’s delve into some key details. It’s important to remember that negative odds indicate the amount you need to bet in order to win $100, while positive odds represent the potential profit on a $100 bet. This knowledge will help you make more informed betting decisions.
In addition, understanding the concept of parlays and teasers is crucial. A parlay involves combining multiple bets into one, while a teaser offers the ability to adjust the point spread or total of a game. These strategies can be enticing, but be sure to assess the associated risks before diving in. Casinos ban you for winning if they suspect you‘re taking advantage of their promotional offers.
Now that you have a clearer understanding of calculating potential payouts using American odds, seize the opportunity to make well-informed bets. Don’t let the fear of missing out keep you on the sidelines. Take a leap and put your newfound knowledge into action. Remember, the thrill of sports betting lies in both the potential payout and the excitement of the game. Good luck!
Calculating potential sports betting payouts might make your head spin faster than a blackjack dealer on a winning streak!
Step-by-Step Guide on Calculating Potential Payouts with American Odds
Calculating potential payouts using American odds can be tricky. But, with the right know-how and help, it’s totally doable! Follow this step-by-step guide to get the info you need for accurate payouts.
Step 1: Convert American odds to decimal odds. Use this formula: Decimal Odds = (100 / American Odds) + 1. If the American odds are +250, work it out like this: Decimal Odds = (100 / 250) + 1 = 1.40.
Step 2: Calculate the implied probability. Use this formula: Implied Probability = 1 / Decimal Odds. Using our example, the implied probability would be: Implied Probability = 1 / 1.40 = 0.71 or 71%.
Step 3: Work out your potential payout. Use this formula: Potential Payout = Stake x Decimal Odds. For instance, if you bet £50 on a game with decimal odds of 1.40, your potential payout would be £70 (£50 x 1.40).
Be aware that these steps may differ depending on the rules and regulations of sportsbooks and bookmakers.
With practice and a better understanding of betting odds, calculating potential payouts won’t be such a challenge anymore. Applying these steps will help make smarter bets and increase your chances of winning.
Don’t miss out on winning money! Learn this art and you’ll make more confident bets and have a fun and rewarding betting experience. Start using these steps now and watch your betting skills improve!
In sports betting, calculating potential payouts is key. To do this, check bookmaker’s odds. These represent the likelihood of an outcome. Compare these to get the best ones with higher payouts. Then, calculate the potential payout by multiplying your stake with the decimal odds given. Click here to learn more about calculating your payouts.
An example: Team A vs Team B match. Bookmaker offers 2.50 decimal odds. If you bet £100 on A winning, your return would be £250 (£100 x 2.50). Including your original stake, you’ll get £250.


There are three kinds of people -- those who are good at math and those who aren't.
If you've failed to detect a mathematical error in the previous sentence, perhaps you should read no further.
When I enrolled in college many moons ago, I majored in mathematics. I would have preferred majoring in pocket billiards or chasing skirts, but they weren't on the curriculum.
Math was always my best subject, but there's no money in it. I eventually switched to Management of Information Systems (computer science) and embarked on a quest to climb corporate ladders. When I finally realized the purpose in life was to discover the purpose in life, I abandoned all that materialistic nonsense and became a writer/teacher/hermit.
Mathematics contains no ambiguity or hypocrisy . It possesses the truth.
There is a mathematical sequence that can be seen in all of existence. It's called the Phi Ratio, sometimes referred to as the Golden Mean Spiral.
The Phi Ratio is a proportion.
The value of a Phi Ratio is approximated at 1.618033988798948482.... This is a transcendental number in that it literally goes on forever without repeating itself.
Suppose you have a line with a distance of X. If you break X into two segments (X1 & X2) so that the ratio of X to X1 is the same as the ratio of X1 to X2, that ratio is said to be a Phi Ratio.
In the form of equations, it would appear as:
X = X1 + X2
X / X1 = X1 / X2 = 1.6180339…
A medieval mathematician, Leonardo Fibonacci, discovered a specific sequence used by plant life during the growth process. The sequence, known as the Fibonacci Sequence, turned out to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, etc.
As was later determined, this sequence has a mathematical formula.
If you start with one and divide by one, then divide the sum of the previous dividend and divisor by the previous dividend, you wind up with the Fibonacci Sequence, the formulation of the Phi Ratio.
Basically, the results keep approaching the transcendental number of the Phi Ratio (approximately 1.6180339...), as follows:
1 / 1 = 1
2 / 1 = 2
3 / 2 = 1.5
5 / 3 = 1.667
8 / 5 = 1.600
13 / 8 = 1.625
21 / 13 = 1.6154
34 / 21 = 1.6190
55 / 34 = 1.6176
89 / 55 = 1.6181
144 / 89 = 1.61798
233 / 144 = 1.61805
377 / 233 = 1.61803
This logarithmic sequence is a primary geometric pattern of the universe. It can be seen in distant spiral galaxies and all forms of life on earth.
The growth of all plant life is based on the Phi Ratio. This growth pattern allows the organism to grow at a constant rate without having to change shape
The bone structure of animal and human life is based on the Phi Ratio. In humans, the first bone in the finger is in Phi Ratio to the second bone, which is in Phi Ratio to the third bone, and so forth. The human hand is in Phi Ratio to the forearm, which is in Phi Ratio to the upper arm. The Phi Ratio is also present in the bones of the feet and the legs.
The Phi Ratio pattern is the basis for the Golden Mean Spiral, which goes on forever without a beginning or an end. The Golden Mean Spiral is an integral part of Sacred Geometry, a study of the evolution of mind, consciousness and spirit.
The great Pyramids of Giza are positioned within a Golden Mean Spiral (Fibonacci Sequence). From aerial photographs, the spiral passes through the exact center of each pyramid.
Just thought you'd like to know.
However, the odds of Leonardo Fibonacci or the Phi Ratio ever popping up in a conversation are about the same as the odds of winning the Kentucky Derby without a horse between your legs.
There are three kinds of people -- those who perceive the mathematical precision of a flower, those you appreciate the magical splendor of a flower, and those who are blind.
Quote for the Day -- "The highest form of pure thought is in mathematics." Plato


This book introduces three new criteria (a) moment relations, (b) moment ratios and (c) ratios of the coefficients of the recurrence relations to characterize and indentify probability distributions. In general moment relations criteria are effective but there are some special situations, where the moment relations of two or more suspected distributions are same or one particular moment function takes same value for two or more distributions. In such a situation two moment ratios as extra criteria are proposed for deciding among them. This book also discussed the identification of a distribution by using the ratios of the coefficients of the recurrence relations obtained from its generating function. The significant contribution of this research is to introduce a new special class of exponential family of distributions named ‘transformed Chi-square family”. Explicit expressions for the MVUE with MV of a function of the parameter of this family are given. The critical region and the power function for various tests of hypotheses for the parameter of this family are also obtained. An identification procedure with probability of correct identification is discussed in detail.


Dyscalculia Learning Disorder: Understanding, Diagnosis, and Support
Mathematics is an essential skill that permeates our daily lives, from calculating the tip at a restaurant to managing finances or solving complex equations. However, for individuals with dyscalculia learning disorder, a specific learning disability in mathematics, these seemingly straightforward tasks can become daunting challenges. In this blog post, we will delve into the world of dyscalculia, exploring what it is, how it is diagnosed, its impact on individuals, and the strategies and support systems available to help them thrive.
Understanding Dyscalculia Learning Disorder
Dyscalculia, often referred to as "math dyslexia," is a neurological condition that affects a person's ability to understand, process, and perform mathematical tasks. It is characterized by persistent difficulties in learning and comprehending math concepts, despite receiving conventional instruction and demonstrating normal cognitive and linguistic abilities in other areas.
Key Features of Dyscalculia:
Numerical Processing Difficulties: Individuals with dyscalculia struggle with basic number recognition and manipulation. This may include difficulty in reading and writing numbers, understanding the concept of place value, and performing basic arithmetic operations.
Difficulty with Math Symbols: Dyscalculic individuals may find it challenging to understand mathematical symbols, equations, and notations, which can impede their ability to solve problems.
Poor Sense of Number and Quantity: Dyscalculia can affect a person's understanding of quantity, making tasks like estimating or comparing numbers a significant challenge.
Spatial and Temporal Confusion: Some individuals with dyscalculia may struggle with understanding concepts of time and space, which are often crucial in mathematical tasks.
Diagnosing dyscalculia is a complex process that typically involves a multi-step assessment. It's essential to recognize that dyscalculia cannot be diagnosed based on a single test or observation, as it is a neurodevelopmental condition with various underlying factors. Diagnosis typically involves the following steps:
Clinical Interviews: A comprehensive interview with the individual, their parents or guardians, and their teachers to gather a detailed history of mathematical difficulties and their impact on daily life.
Cognitive Assessment: Assessing the individual's cognitive abilities, such as memory, attention, and reasoning, to rule out other factors contributing to math difficulties.
Mathematical Assessment: A series of standardized tests to evaluate mathematical abilities, covering areas like number sense, basic operations, and problem-solving.
Educational Assessment: An assessment of the individual's educational history, including their progress in math classes and their response to interventions.
It's important to note that a formal diagnosis of dyscalculia should be made by a qualified professional, such as a clinical psychologist or a neuropsychologist.
The Impact of Dyscalculia
Dyscalculia can have a profound impact on various aspects of an individual's life. Understanding these challenges is crucial for providing appropriate support and accommodation:
Academic Struggles: Dyscalculic individuals often experience difficulties in math-related subjects, which can lead to lower academic achievement and reduced self-esteem.
Emotional and Psychological Effects: Frustration, anxiety, and low self-esteem are common emotional consequences of dyscalculia. The fear of math can create a negative cycle, as anxiety can further hinder mathematical performance.
Everyday Challenges: Dyscalculia extends beyond the classroom, affecting everyday tasks such as budgeting, time management, and even reading clocks.
Career and Financial Implications: Limited math skills can restrict career choices and financial independence, potentially leading to professional challenges and financial difficulties.
Support and Interventions
While there is no cure for dyscalculia, individuals with this condition can benefit from a range of interventions and support mechanisms to help them overcome challenges and reach their full potential. Here are some effective strategies and tools:
Early Intervention: Identifying dyscalculia early and providing targeted interventions can make a significant difference. Early support can prevent the development of negative attitudes towards math.
Individualized Education Plans (IEPs): In many countries, IEPs are developed for students with disabilities, including dyscalculia. These plans outline specific accommodations and modifications to help students succeed in their educational journey.
Specialized Math Programs: Some educational institutions offer specialized math programs that use alternative teaching methods and resources tailored to the needs of dyscalculic students.
Assistive Technology: Utilize technology tools and software designed to support individuals with dyscalculia. These may include digital math workbooks, calculators, and speech-to-text programs for completing math assignments.
One-on-One Tutoring: Engaging a qualified math tutor or educational therapist who specializes in working with dyscalculic individuals can provide targeted and personalized support.
Multisensory Approaches: Incorporate multisensory techniques, such as using physical objects, drawing diagrams, and verbalizing math problems, to enhance understanding and memory retention.
Building Math Confidence: Encourage a growth mindset and emphasize effort over innate ability. Creating a positive attitude towards math can help reduce anxiety and foster a willingness to tackle mathematical challenges.
Dyscalculia is a significant learning challenge that affects a person's ability to understand and work with mathematical concepts. Raising awareness about dyscalculia, promoting early detection, and providing appropriate interventions are essential steps toward a more inclusive and equitable educational system. By understanding the complexities of dyscalculia and embracing the diverse ways in which people learn, we can better support and empower those with dyscalculia to unlock their full potential in the world of mathematics and beyond


See and Learn First Counting provides teaching activities and comprehensive guidance to help parents and educators teach children the number words, numerals and counting from 1 to 10. It follows recommended, evidence-based practice in number teaching, progressing in sequence through the key developmental steps involved in learning the foundational number skills.
Designed by internationally recognised researchers and experts in the development and education of children with Down syndrome, See and Learn First Counting is designed to meet the children’s specific learning needs. See and Learn First Counting is also likely to be helpful for other children who benefit from teaching in small steps with visual supports.
- Evidence-based activities following key developmental steps in early number learning
- Focus on visual representation of concepts, reduced language and working memory demands to help children with specific learning needs
- Comprehensive guide included providing explicit, step-by-step guidance and instructions
- Record forms included for tracking your child’s progress
See and Learn First Counting includes eight activities, designed to teach children to say the number words, to recognise the numerals, to link quantities to numbers, to count, and to understand the concepts of cardinality and equivalence for the numbers 1 to 10. See and Learn First Counting is also designed to teach the key maths language needed at this stage of number learning.
- Learning Number Words
- Matching Numerals
- Selecting Numerals
- Naming Numerals
- Linking Quantity to Numerals
- Learning to Count
- Give a Number
- Learning Equivalence
- Guide book, introducing number teaching for children with Down syndrome and step-by-step instructions for each teaching activity
- Record forms
- 60 plastic counters
- 10 laminated quantity + numeral cards
- 20 (2 x 1-10) laminated numeral cards
- 4 animal cards (2 x Bear and 2 x Monkey)
- 40 animal food counter cards (20 x Cookie + 20 x Banana)
See and Learn Numbers
See and Learn First Counting is the first step in the See and Learn Numbers teaching programme. See and Learn Numbers is designed to teach young children to count, to link numbers to quantity, to understand important concepts about the number system and to calculate with numbers up to 10. It also teaches early mathematical concepts important for understanding space, time and measurement - including colour, size, shape, ordering, sorting and patterns.
Please visit the See and Learn web site or contact us for further information and guidance.


Seriously fun math and science learning.
ExploreLearning® provides interactive simulations that embolden K-12 STEM learners with the power of doing.
ExploreLearning encourages students to embrace their inner scientist and/or mathematician through adaptive, game-based instruction, all while increasing proficiency and deeper learning of foundational STEM concepts in a way that is as fun as it is effective. Their products and services help teachers easily personalize and enliven instruction, while building the skills and confidence students need to succeed in these critical subject areas.
The company offers four edtech products and supporting professional learning services, including the recently launched Frax platform. Over the course of the last 20+ years, they have helped hundreds of thousands of teachers and have received numerous awards from SIIA, NSTA, and many others.explorelearning.com


New member of staff: Maia Muresan
11.08.2023 – We are pleased to welcome Maia Muresan as a new staff member at CTU Bern. Maia holds a bachelor’s degree in mathematics from the University of Grenoble Alpes, France, and recently obtained her master’s degree in Statistics from the University of Geneva, Switzerland.
Her master's thesis focused on the development of a new method to assess the impact of dosing modifications on the dose-tumor size relationship in early oncology clinical trials, using Tumor Growth Inhibition (TGI) models. This included detailed analysis of effects on tumor size response and refinement of Phase 1 oncology study design. During her master's program, Maia gained extensive experience through biostatistical internships in early development clinical trials in dermatology and oncology at pharmaceutical companies in Switzerland, which fueled her interest in the application of mathematics in medical science. She is excited to be a new member of Statistics & Methodology of CTU Bern. Welcome Maia!


In a sense, it is not reasonable to expect finite and well-behaved confidence for parameters. The argument goes as follows:
When testing parameters in classical settings such as for the t-test, the assumption of normality is crucial, or at least semi-crucial. It is possible to construct ok confidence intervals using the Berry-Esseen theorem, but it is at least sometimes possible to show that no well-behaved approximate confidence interval exists when we cannot reliably bound the third moment.
The setting of most confidence intervals is some normal approximation of the sampling distribution of $\theta$, where the implicit hypotheses are that $\theta$ is some $\theta_0$ and normality holds for both the alternative and null hypothesis.
Here the problem comes knocking on the door. Not only do we not know how well normality is approximated for one particular $\theta$ in terms of the usual Prokhorov metric, but we also don’t know if the moments converge at, and we have no information about the uniformity of the convergence. Uniform convergence of moments is probably needed to make realistic sufficient conditions for asymptotic confidence intervals to work.
This is probably worth investigating more. I think the main question that remains to be formulated is something like this: Do we actually have an implicit set of hypotheses as outlined above?


MASU hosts a mathematics festival for students from local schools
MASU Mathematics and Natural Sciences Faculty held Polar Coordinates, a popular science festival of maths for students of Murmansk region's schools. More than 130 students attended the Festival and took part in various educational events.
Vera Levites, Dean of Faculty, and 2nd year students organized a mathematical game for eighth and ninth-graders. Children were solving mathematical speed puzzles and challenging tasks.
At the workshop led by Victoria Lazutina, a 3rd-year student, high school students learned how to count faster and do unusual maths calculations. At the same time, the youngest participants of the Festival, third and sixth-graders, wrote mini-essays discussing the role of mathematics in their lives.
The Festival also included two open lectures: Georgy Wolfson, a mathematics teacher at St. Petersburg Physics and Mathematics Lyceum no. 366, told the audience about the paradoxes of probability theory. Later, Natalia Nedelko, an associate professor of MASU Mathematics, Physics and IT Department, demonstrated the tricks of the Excel spreadsheet software for calculating profits and making forecasts about the projects' development.
The organizers of the Festival thank all the participants, MASU students and volunteers, as well as maths and elementary school teachers for their help in organizing the Festival.


In the chapter Courious Maps of M. Gardner's Time Travel (W.H. Freeman and Company, New York, 1988), many world's maps are presented, some of them use a projection on a convex solid, that is unfolded into a plane net.
The mathematician Charles Sanders Peirce designed a conformal map, namely a projection on eight isosceles right triangles that may be regarded as the faces of an octahedron, flattened until a space diagonal is zero. B.J.S. Cahill patented his butterfly map in 1913. The world is projected onto a regular octahedron. R. Buckminster Fuller's first Dymaxion map was a projection of the world onto the fourteen faces of a cuboctahedron. In March 1, 1943 the map was completed by staff artists of Life. At about the same time, Irving Fisher designed Likaglobe, which folds into an icosahedron. In 1954 Fuller copyrighted his Dymaxion Skyocean Projection World Map, which slightly differed from Fisher's Likaglobe.
Wolfram's Mathematica package WorldPlot consists of various types of projections used in cartography. There is a large data block WorldData, where each country is represented by a set of points on its boundary.
We use the data to construct a gnomic projection of the world on any convex polyhedron. If necessary we translate the polyhedron in a position where its mass centre coincides with the origin of the co-ordinate system. Then points on the globe are projected onto the polyhedron. This means that we calculate the intercept of the half-straight line from the origin via given point on the globe by a face of the polyhedron.
The following animations use LiveGraphics3D animation applet. To invoke an animation click on a figure. Then drag to rotate, press Shift and drag vertical to change size, or horizontal to rotate. Press S to get stereo view.
The next figure shows a cylindrical projection with a maze on it. The task is to join the black and the grey dot on it or for instance to find a path from Belgrade to New York. A set of 19 figures of this sort can be downloaded from here.


Our Little Crackers had a lovely day exploring numbers in nursery when we took part in the NSPCC's Number Day.
The children had fun with all sorts of number related activities. We counted forwards and backwards, matched, recognised, added, combined, took away, shared, estimated and checked, made sets and sang songs… inside, outside and even at lunchtime! We all dressed up, had a giggle at Miss Lisa who was the No.1 Numberblock, and spotted numbers wherever we played. The sun came out to add to the fun, and we took our counting outside, had a number hunt and played with the numbers we found. And all for a good cause as well!


Please use this identifier to cite or link to this item:
|Title:||A division algorithm in a redundant residue number system using fractions|
|Authors:||Chervyakov, N. I.|
Червяков, Н. И.
Lyakhov, P. A.
Ляхов, П. А.
Babenko, M. G.
Бабенко, М. Г.
Lavrinenko, I. N.
Лавриненко, И. Н.
Deryabin, M. A.
Дерябин, М. А.
Lavrinenko, A. V.
Лавриненко, А. В.
Nazarov, A. S.
Назаров, А. С.
Valueva, M. V.
Валуева, М. В.
|Keywords:||Algorithm;Fraction;Modular division;Redundant residue number system;Residue number system (RNS)|
|Citation:||Chervyakov, N., Lyakhov, P., Babenko, M., Lavrinenko, I., Deryabin, M., Lavrinenko, A., Nazarov, A., Valueva, M., Voznesensky, A., Kaplun, D. A division algorithm in a redundant residue number system using fractions // Applied Sciences (Switzerland). - 2020. - Volume 10. - Issue 2. - Номер статьи 695|
|Series/Report no.:||Applied Sciences (Switzerland)|
|Abstract:||The residue number system (RNS) is widely used for data processing. However, division in the RNS is a rather complicated arithmetic operation, since it requires expensive and complex operators at each iteration, which requires a lot of hardware and time. In this paper, we propose a new modular division algorithm based on the Chinese remainder theorem (CRT) with fractional numbers, which allows using only one shift operation by one digit and subtraction in each iteration of the RNS division. The proposed approach makes it possible to replace such expensive operations as reverse conversion based on CRT, mixed radix conversion, and base extension by subtraction. Besides, we optimized the operation of determining the most significant bit of divider with a single shift operation of the modular divider. The proposed enhancements make the algorithm simpler and faster in comparison with currently known algorithms. The experimental simulation using Kintex-7 showed that the proposed method is up to 7.6 times faster than the CRT-based approach and is up to 10.1 times faster than the mixed radix conversion approach|
|Appears in Collections:||Статьи, проиндексированные в SCOPUS, WOS|
Files in This Item:
|scopusresults 1228 .pdf||1.15 MB||Adobe PDF||View/Open|
|WoS 827 .pdf||193.47 kB||Adobe PDF||View/Open|
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.


In preparing my previous post on Net Energy Trends I wanted to include a back of the envelope calculation on the ERoEI of hydro electric power using the Three Gorges Dam as an example. But I got my decimals pretty muddled leading to an answer that was implausible. But I’ve now had a few days off to clear my head and I put a new battery in my calculator and so hopefully the calculation is now on the money.
Looking at just the labour and embedded energy of the concrete and steel and assuming a 45% capacity factor and 70 year life yields a partial ERoEI of 147. And so, despite substantial environmental harm and social disruption I must give dispatchable hydro electric power a big thumbs up. See the calculation below the fold.
The Three Gorges Dam on the Yangtze River in China is the largest hydroelectric scheme in The World. Operational since 2008 it has 22.5 GW installed capacity. The Itaipu Dam on the Parana River between Paraguay and Brazil produces a similar amount of electricity from 14 GW capacity and achieves this through a higher capacity factor.
Recent posts on ERoEI indicated that hydroelectric power had ERoEI >> 50 and I just wanted to check these claims. But before proceeding lets get the metric prefixes sorted since we have to manipulate some fairly large numbers:
mega = 10^6 = million
giga = 10^9 = billion
tera = 10^12 = trillion
peta = 10^15 = one thousand trillion
exa = 10^18 = one million trillion
Three Gorges Vital Statistics
Installed generating capacity = 22.5 GW
Capacity factor = 45% (0.45)
Concrete = 27.2 million m^3
Concrete = 65.2 million tonnes*
Steel = 463,000 tonnes
*1 m^3 concrete = 2.4 tonnes
1 TWh = 3600 TJ
1 toe = 42 GJ
The Energy Return
This is always the easier part of the ERoEI calculation but even here assumptions need to be made about capacity factor and lifespan. Wikipedia report 45% capacity factor which I presume is based on design criteria and performance to date. And I have assumed a 70 year lifespan. Lifespan could easily be much longer and this will simply add to the ERoEI.
0.45 * 22.5 GW * 24 hrs * 365.25 days * 70 years = 6213 TWh = 22.37 EJ
The Energy Invested
I am not going to attempt a detailed and complete analysis here but will try and make ball park estimates of energy consumed by labour and the main materials – concrete and steel.
60,000 workers laboured on the project that began in 1994  and was completed in 2003. Taking 1998 as the mid point China had a population of 1.242 billion and consumed 905 million tonnes oil equivalent (Mtoe) in energy that year . This yields a per capita consumption of 0.75 toe. The sum for the energy cost of labour therefore is:
60,000 workers * 0.75 toe per annum * 10 years = 450,000 toe = 18.9 PJ
There is a range of numbers for the energy content of materials. I am going to use:
Concrete = 1.9 GJ / tonne
Steel = 20 GJ / tonne
65.2 million tonnes concrete * 1.9 GJ / tonne = 123.9 PJ
463,000 tonnes steel * 20 GJ / tonne = 9.3 PJ
Total Energy Invested
labour 18.9 PJ + concrete 123.9 PJ + steel 9.3 PJ = 152.1 PJ
The partial ERoEI
Energy return = 22.37 EJ / energy invested 152.1 PJ = 147
Energy pay back time works out at an incredible 6 months.
(Energy produced in 1 year = 22.37 EJ / 70 = 319 PJ. Energy invested = 152.1 PJ. Hence energy pay back time = 152/319 = 0.48 years.)
I am quite satisfied with this answer. Some of the input numbers used may be wide of the mark and the energy inputs are far from complete. For example the diesel used on the construction site is not included along with many other energy inputs. But the answer is in the same ball park as calculated by other workers. Hydroelectric power is a tremendous source of dispatchable renewable energy. It does however come with high environmental and social costs. There’s no such thing as a free lunch in the energy world.
 Wikipedia: Three Gorges Dam
 Facts and Details: THREE GORGES DAM PROJECT
 Australian Government: Embodied Energy
 Wikipedia: Embodied Energy


A few weeks ago, I introduced you to the concept of pro-forma portfolios, and explained how we use them to estimate our current portfolios’ position in terms of volatility and beta. There is an Italian saying that could be translated as “to trust is good, not to trust is better.” While the pro-forma portfolios give us the best possible ex-ante estimates of the amount of risk in the portfolios, these are by definition estimates and not reality, and we can only trust them so much. For this reason we have developed two very short-term measures of volatility and beta based on the actual daily portfolio returns, which we routinely compare to the pro-forma estimates to make sure the portfolios are in fact behaving as we expected. The first measure is called one-month time-weighted trailing volatility and starts with the calculation of the equal-weighted average of the portfolio’s daily volatility over trailing 5 days (1 week), 10 days (2 weeks) and 20 days (4 weeks). This number is then divided by the same average volatility calculated for the portfolio’s benchmark. The ratio gives us a measure of how volatile the portfolio is being relative to its benchmark:
Ratio = 1 (neutral): the portfolio is experiencing the same volatility as the benchmark;
Ratio > 1 (above neutral or aggressive): the portfolio is experiencing more volatility than the benchmark;
Ratio < 1 (below neutral or defensive): the portfolio is experiencing less volatility than the benchmark;
The other measure is called one-month time-weighted trailing beta and uses the same approach to calculate the beta of the portfolios versus their respective benchmarks. Both measures are based on 1 month (4 weeks) of data, but since they are the equal-weighted average of three overlapping time frames they give more weight to more recent data. Specifically, the trailing 5 days are accounted for 3 times, the previous 5 days are accounted for twice, and the previous 10 days are accounted for only once. Because of their short-term nature, these measures can be particularly volatile at times, which is why we introduced a one-month moving average to better gauge the direction of the trend. In the chart to the right, you can see that after a brief spike to nearly 1.2, the two measures have come back down to around 0.9, while the one-month moving average has just reached the Mendoza line of 1. This result is perfectly in line with our initial estimates based on pro-forma portfolios, which had indicated that our last trade would take us very close to neutral.
Copyright: albund / 123RF Stock Photo


My fifth graders just finished their ordered pairs projects, and I am absolutely amazed by their creativity!
The assignment asked that kiddos create a picture that included a minimum of five shapes, used only straight lines, and was located in all four quadrants.
Students also had to create an organized table for each shape that included the quadrant number and ordered pair for each point. This paper would later be used as a directions page for another student to recreate their picture.
Take a look at their amazing work!
For board work the next day, each kiddo was given a blank four quadrant graph and another student’s direction page. It was so much fun to see how the images were redrawn. Some directions were spot on (all ordered pairs and quadrants were accurate) and the redrawn picture looked identical to the original. Others though, not so much!
I will definitely be doing this project again next year!


Books In Progress
We are currently working on Textbook Companions for the following books
- Probability And Statistics For Engineers And Scientists, S. M. Ross, 3rd edition, Elsevier, New Delhi, 2005.
- Linear and nonlinear programming, D. Luenberger, 2nd edition, Kluwer Academic Publisher, New York, 1984.
- Nonlinear programming, D. Bertsekas, 2nd edition, Athena Scientific, 1999.
- Optimization Techniques, A. K. Malik, S. K. Yadav, S. R. Yadav, 1st edition, I.k. Internationlal Publishing House Pvt. Ltd, 2012.
Suggested List of Books that can be taken up for TBC project
- Engineering Optimization Methods and Applications, Reklaitis, G.V., Ravindran, A. and Ragsdell, K. M., 2nd edition, John Wiley,New york 1983.
- Probability, Random Variables and Stochastic Processes, Athanasios Papoulis, S. Unnikrishna Pillai, 4th edition, McGraw Hill Education (India), 2002
- Optimisation Techniques , Rao S.S. , 2nd edition, Wiley Eastern, 1985


Great Discount for Home Educators
AWE Club discount for ConquerMaths. Due to popular demand, ConquerMaths is now available through AWE Club for Home Educating families.
To get this offer follow these steps:
Quote from NM on Facebook:
“We’ve used Conquermaths for about 5 or 6 years, to support GCSE and IGCSE maths study (we had it originally as a CD ROM version and now a subscription through AWE). We love that it covers many years of maths up to and including A level, so we can dip in and out of the topics, use it as a supplement to whatever we are studying, and go back, if needed, to fill gaps. I definitely recommend it!”
ConquerMaths covers years Reception, Key Stage 1, KS2, KS3, KS4 (GCSE) and KS5 (A-Level)
The A-Level content is Year 12 Core 1 and 2, Year 13 Core 3 and 4


Marc Brendel, Wolfgang Marquardt:
An algorithm for multivariate function estimation based on hierarchically refined sparse grids
Computing and Visualization in Science, 2009, 12(4), 137-153
An adaptive function estimation approach is presented to recover an unknown, multivariate functional relation from noisy data. Using a sparse grid combination approach, both discretization and Tikhonov regulariza- tion need to be selected appropriately to resolve func- tional details whilst suppressing measurement noise. An initially coarse, multivariate grid is adaptively refined using sensitivity analysis, creating a sequence of hierar- chically refined grids. The problem of choosing a multi- dimensional discretization level is thus transformed to the identification of a suitable refinement step, giving rise to a nested approach for the selection of both dis- cretization and Tikhonov regularization. Validation on multivariate test functions shows good approximation results.
Multivariate regression, function estimation, sparse grids, discretization, regularization, L-curve, cross validation


Diophantus, known as the Father of Algebra, lived in Alexandria, Egypt during the 3rd century A.D. Little else is known about his personal life. He was the author of the first Greek text on the essential branch of mathematics we know as algebra.
His book, Arithmatica, included thirteen books with numerical answers to algebraic questions. Using only positive rational numbers because zeros, negative numbers and irrational numbers were not available to him at the time –
Diophantus algebraically solved linear and quadratic equations, in addition to simultaneous linear and quadratic equations. With awareness of essential theorems in the number theory, he also found algebraic solutions to questions such as finding the value of y so that some polynomial equations in y are either squares of numbers or their cubes.
Arithmatica solved a total of one hundred and thirty mathematical problems for its readers. Apart from this important text on algebra, Diophantus has been credited with introducing techniques for solving both determinate as well as indeterminate equations. He also developed the method of using symbols for words in algebra. Still, Arithmatica continues to be remembered as one of the most significant works of his life, for the simple reason that the sciences of modern times could not have progressed without the tool of algebra.
As a matter of fact, algebra is an integral part of modern existence. Both industry and our daily lives depend on this tool. As examples, algebraic formulas for calculating loan installments; bank interest; distance, speed, and time; and volume, area and perimeter are as indispensable as the variables, relations and functions used in the analysis of activities that involve costs.
So, whether we are dealing with the business of construction, managing expenses as consumers, or working on new innovations in chemistry labs, we know it is virtually impossible to do away with algebra – thanks to Diophantus who first introduced the importance of this mathematical tool to the world.
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International Mathematical Olympiad
The International Mathematical Olympiad (IMO) is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads.The first IMO was held in Romania in 1959.
License: Creative Commons Attribution-Share Alike 3.0 (CC BY-SA 3.0)
Image Source: https://en.wikipedia.org/wiki/File:Geo_prob_diagram.svg
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It is time again for Mystery Monday, brought to you by Professor Layton and the Curious Village. A glass jar holds a single germ. After one minute, the germ splits into two germs. One minute after that the two germs each split again, forming a total of four germs. Continuing at this rate, a single germ can multiply to fill a whole jar in exactly one hour. Knowing this, how long in minutes would it take to fill the jar if you had started with two germs?