Advertisement Hide. Convergence and approximation results for non-cooperative Bayesian games: Learning theorems. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Aliprantis, C.
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Aumann, R. Econometrica 55 , 1—18 Google Scholar. Blume, L. Debreu, G. Fifth Berkeley Symp. II , — Google Scholar. Diestel, J. Survey Providence, R. Dunford, N.
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Part I. New York: Interscience Google Scholar. El-Gamal, M. Theory 4 , — Google Scholar. Feldman, M. Himmelberg, C. Jordan, J. Kalai, E. The Richardson extrapolation can be generalized for a p-th order methods and r -value of grid ratio which does not have to be an integer as:. Thus, the above equation simplifies to:. Otherwise, it will be a third-order estimate. Richardson extrapolation can be applied for the solution at each grid point, or to solution functionals, such as pressure recovery or drag.
This assumes that the solution is globally second-order in addition to locally second-order and that the solution functionals were computed using consistent second-order methods. Other cautions with using Richardson extrapolation non-conservative, amplification of round-off error, etc For our purposes we will assume f is a solution functional i.
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This use will now be examined. We will focus on using f 1 and f 2 to obtain an error estimate. Examining the generalized Richardson extrapolation equation above, the second term on the right-hand side can be considered to be an an error estimator of f 1. The equation can be expressed as:. This quantity should not be used as an error estimator since it does not take into account r or p.
This may lead to an underestimation or overestimation of the error. One could make this quantity artificially small by simply using a grid refinement ratio r close to 1. The estimated fractional error E 1 is an ordered error estimator and a good approximation of the discretization error on the fine grid if f 1 and f 2 were obtained with good accuracy i. If such is the case, then another normalizing value should be used in place of f 1.
If a large number of CFD computations are to be performed i. We will then want to estimate the error on the coarser grid. The Richardson extrapolation can be expressed as:. Richardson extrapolation is based on a Taylor series representation as indicated in Eqn.
If there are shocks and other discontinuities present, the Richardson extrapolation is invalid in the region of the discontinuity.
Polyhedral, Tetrahedral, and Hexahedral Mesh Comparison | Symscape
It is still felt that it applies to solution functionals computed from the entire flow field. Roache suggests a grid convergence index GCI to provide a consistent manner in reporting the results of grid convergence studies and perhaps provide an error band on the grid convergence of the solution. The GCI can be computed using two levels of grid; however, three levels are recommended in order to accurately estimate the order of convergence and to check that the solutions are within the asymptotic range of convergence.
A consistent numerical analysis will provide a result which approaches the actual result as the grid resolution approaches zero. Thus, the discretized equations will approach the solution of the actual equations. One significant issue in numerical computations is what level of grid resolution is appropriate. This is a function of the flow conditions, type of analysis, geometry, and other variables. One is often left to start with a grid resolution and then conduct a series of grid refinements to assess the effect of grid resolution.
This is known as a grid refinement study. One must recognize the distinction between a numerical result which approaches an asymptotic numerical value and one which approaches the true solution. It is hoped that as the grid is refined and resolution improves that the computed solution will not change much and approach an asymptotic value i.
There still may be error between this asymptotic value and the true physical solution to the equations. Roache has provided a methodology for the uniform reporting of grid refinement studies. The GCI is based upon a grid refinement error estimator derived from the theory of generalized Richardson Extrapolation.
It is recommended for use whether or not Richardson Extrapolation is actually used to improve the accuracy, and in some cases even if the conditions for the theory do not strictly hold. The GCI is a measure of the percentage the computed value is away from the value of the asymptotic numerical value. It indicates an error band on how far the solution is from the asymptotic value. It indicates how much the solution would change with a further refinement of the grid. A small value of GCI indicates that the computation is within the asymptotic range.
The refinement may be spatial or in time.
The higher factor of safety is recommended for reporting purposes and is quite conservative of the actual errors. When a design or analysis activity will involve many CFD simulations i. DOE study , one may want to use the coarser grid h 2.
https://replongrari.gq It is then necessary to quantify the error for the coarser grid. The GCI for the coarser grid is defined as:. It is important that each grid level yield solutions that are in the asymptotic range of convergence for the computed solution. This can be checked by observing two GCI values as computed over three grids,. If a desired accuracy level is known and results from the grid resolution study are available, one can then estimate the grid resolution required to obtain the desired level of accuracy,.
The grid refinement ratio assumes that the refinement ratio r applies equally in all coordinate directions i,j,k for steady-state solutions and also time t for time-dependent solutions. If this is not the case, then the grid convergence indices can be computed for each direction independently and then added to give the overall grid convergence index,. If one generates a finer or coarser grid and is unsure of the value of grid refinement ratio to use, one can compute an effective grid refinement ratio as:. This effective grid refinement ratio can also be used for unstructured grids.
The following example demonstrates the application of the above procedures in conducting a grid convergence study. The objective of the CFD analysis was to determine the pressure recovery for an inlet. The flow field is computed on three grids, each with twice the number of grid points in the i and j coordinate directions as the previous grid.
The number of grid points in the k direction remains the same. Since the flow is axisymmetric in the k direction, we consider the finer grid to be double the next coarser grid. The table below indicates the grid information and the resulting pressure recovery computed from the solutions. Each solution was properly converged with respect to iterations. The column indicated by "spacing" is the spacing normalized by the spacing of the finest grid.
The figure below shows the plot of pressure recoveries with varying grid spacings.
Examining Spatial (Grid) Convergence
As the grid spacing reduces, the pressure recoveries approach an asymptotic zero-grid spacing value. The difference is most likely due to grid stretching, grid quality, non-linearities in the solution, presence of shocks, turbulence modeling, and perhaps other factors. We now can apply Richardson extrapolation using the two finest grids to obtain an estimate of the value of the pressure recovery at zero grid spacing,.
The grid convergence index for the fine grid solution can now be computed.


HOW TO GIVE STROKES IN MATCHPLAY
In matchplay, the difference between two players' handicaps is the number of strokes given by one player to another.
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Foursomes: each pair adds up their handicaps and takes the difference between the two totals; you then divide this difference by two to get the number of shots the higher handicap pair receives from the lower handicap pair.
Fourballs: the lowest handicap player gives strokes to the other three players based on three quarters of the difference between the lowest handicap and that of each player


A proof by Japanese mathematician Shinichi Mochizuki of an over 30-year-old problem in number theory has obtained validation, Kyoto University said Friday following controversy over his theory often labeled as too novel and complicated to understand.
The Research Institute for Mathematical Sciences of the university accepted for publication his 600-page proof of the abc conjecture, which provides immediate proofs for other theories including Fermat’s last theorem, which took almost 350 years to be demonstrated.
The abc conjecture, proposed by European mathematicians in 1985, is an equation of three integers a, b, and c composed of different prime numbers, where a + b = c, and describing the relationship between the product of the prime numbers and c.
“There are a number of new notions and it was hard to understand them,” Masaki Kashiwara, the head of the team which had examined the theory, said at a press conference.
Mochizuki, a 51-year-old professor at the university, released his study in 2012 on his website and it was run in a journal of the research institute after nearly 20 years of working it.
However, it courted controversy, with its denseness and length baffling peers who tried to confirm it. Additionally, two respected mathematicians, Peter Scholze and Jakob Stix, said in 2018 there was a flaw with Mochizuki’s proof. Stix reportedly found a “serious, unfixable gap.”
Scholze told Kyodo News after the announcement that his position has remained “unchanged” and the news “comes as a surprise” to him.
Mochizuki, who has declined requests for interviews over the years, did not appear at the press conference held by the university Friday.
Mochizuki was born in 1969 in Tokyo. He grew up in the United States where he moved when he was 5 years old.
A wunderkind, he was accepted at Princeton University in the United States at age 16 and graduated three years later.
He accepted an assistant teaching post at Kyoto University in 1992, when he was 23, and became a full professor when he was 32.
In 2005, he was one of the first recipients of the Japan Academy’s medal to honor “young” scholars aged up to 45.


ISMP is a triennial conference on mathematical optimization, gathering scientists from around the world to present and share their most recent advances and development in the field. It is also a unique opportunity to discuss new challenges and opportunities in practice and theory of mathematical optimization.
The 25th International Symposium on Mathematical Programming will be held in Montréal, Canada, from July 21st through July 26th, 2024.
The symposium will be hosted at Palais des congrès de Montréal159, rue Saint-Antoine Ouest
Montréal, Québec, H2Z 1H2


How to calculate a price
Each price is quite individual. To calculate approximate price for cushions,
measure cushions back length (see example on this page), then multiply it
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For your convenience we made a calculator.
Approximate price for materials and labour work
Approximate price for materials and labour work, calculated for a foot of backseat’s rear cushion:
- Fabric 50 £
- Leather + Fabric 60 £
- 100% leather 70 £
Add up cushion lengths: 3ft + 6ft = 9ft
then multiply by price per foot (45 stands for fabric finish): 9ft × 50£ = 450£
Price Calculator for cushions upholstery
Enter your sum length of cushions and select a material type:


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The purpose of this exercise is to develop an understanding of population dynamics by modeling the dynamics of a hypothetical population. After we have established basic population dynamic relationships, you will investigate these relationships in greater detail by exploring the population dynamics of a hypothetical predator - prey population of foxes and rabbits.
Fox predator catching a prey rabbit
An effective means of understanding such a system is to model it, i.e. to think about the system critically, identifying stocks, flows, converters and connectors, and create a conceptual diagram. You will be given a description of the system, and some initial parameters. You will then develop a conceptual diagram together in groups. This diagram will serve to summarize our knowledge of the system, and convert descriptive information into symbols that are easy to manipulate. From the conceptual diagram, the descriptive data, and some initial parameters, you will construct a STELLA model that describes the rabbit-fox population dynamics.
In this lab we will be considering foxes and rabbits. We need to first examine the basics of population dynamics using the rabbit population to examine how populations change over time. Start up STELLA and click on the modeling tab.
The first stock in our model is Rabbit Population.
Recall that changes in stocks are modeled with the flow tool: . Changes in population are caused by Rabbit Births, so add it to your model.
The final piece needed for a complete model is the Rabbit Birth Rate. Add this using the converter tool: . Assume that the rabbit birth rate is 4% per day or .04 and your population starts with 10 rabbits.
Use the connector tool: to complete the connections in your population model.
Define births with the following equation (this is the equation for our exponential growth model):
Rabbit Births = Rabbit Birth Rate * Rabbit Population
SAVE your model.
Specify an amount of time for the model under length of simulation:
From = 0, to = 250, DT = 1, and Unit of Time = days. Click OK.
Set up the viewing graph with the graph icon . Define your graph with the Rabbit Population stock. Run the model.
The graph that appears should look like Figure 2.
Graph of exponential population growth
The model we have created works well, but recall that all populations have limits to their growth. How can we fix the model to give a more realistic result? We can show this effect in our model by adding a carrying capacity variable and changing the equation for Rabbit Births.
Carrying capacity is the maximum size a population can grow to given the available resources.
Add a converter to the model, below and to the right of the flow icon, and name it Carrying Capacity. Connect it to Rabbit Births. What would be a good value for carrying capacity for your population?
Change your equation to include the carrying capacity component of your model (this is our equation for the logistic growth model).
Rabbit Births = Rabbit Birth Rate * Rabbit Population * (1 - (Rabbit Population / Carrying Capacity))
The model should now look like Figure 3.
Population model with carrying capacity
Run the model with the same time specs. The new graph should look like Figure 4.
Graph of population with carrying capacity
Judging by the graph, this model seems to be more in line with the way a population might increase in an environment with finite resources.
Describe two factors that would change carrying capacity in nature.
Save the first part of this model. Now we will incorporate a predator into the system. First we have to describe how the prey (rabbit) population changes and then describe how the predator (fox) population subsequently changes, since the predator is dependent on the prey species for growth and survival. Below is a written description of how this system works as well as some data on the rates of the processes. Read through this information.
Then, in a small group, draw on a piece of paper a conceptual diagram on these data.
This diagram allows you to see, using symbols, how parts of the model relate to one another. It is the intermediate step in understanding the dynamics and should serve as the template for your STELLA model.
You should have an idea of what the outcome should be. In other words, how do you think the fox population should grow in relation to the rabbit population? Sketch out a quick graph of your expectations and ask yourself "why should it look this way and how might it look different?" This should help you figure out whether or not the model is functioning well once you run it in Stella. Show your GSI your diagram and get approval before opening Stella.
Here are some tips to help exterminate the bugs in your model:
Write the prediction that you created prior to running this model about the population dynamics of foxes and rabbits. Next, describe the dynamics of both populations under the conditions outlined above and discuss why this pattern occurs and compare your results with your prediction. Submit a copy of the model, the equations and a graph of the run.
What parameter value(s) would you change to have the fox population crash (go to zero), and the rabbit population survive? Test your prediction and include your graph in your submission.
What parameter value(s) would you change to have the rabbit population crash? How would this affect the fox population? Test your prediction and include your graph in your submission.
What would you expect to happen (in terms of the dynamics of both populations) if rabbits exhibited logistic (s-shaped) growth? Test this out by modifying your model and adding this feature to the rabbit population. First, reintroduce the carrying capacity function (remember to change your rabbit birth rate equation to the logisitc growth equation). Now, set the carrying capacity to 10000. Explain the dynamics and include a graph.
Now, let's explore the impact of a long drought where the rabbit carrying capacity is much lower. Reduce the carrying capacity to 5000 from day 250 onwards. Explain the impact of a long drought on the rabbit population. Include a graph of this simulation in your submission.
For Next Week, turn in:


1st Place - Davin Sher, Immanuel College
Iconography has singlehandedly revolutionised the way we present, understand and interpret mathematics. The true role of an icon is using a symbol to represent an idea in shorthand. Utilising universal symbols to represent the same idea has connected the world in an unparalleled way. The modern world finds equality here, using mathematical notation as a sort of ‘lingua franca’ in which the native tongue of the person is nearly irrelevant to the ability to communicate, although this comparison may be misleading. Having this as one of the few true connections people from opposing circumstances can relate proves the beauty and phenomenon that is the script of maths.
When discussing a topic such as this, it is sensible to start with the icons we use to represent the concepts of each numeral. For the majority of the world, this is the Hindu-Arabic numeral system. The use of this system was one of the first instances of people representing their numerals with specific symbols that could be manipulated for different purposes.
The advantage of having these repetitive easily replicable icons for each of the base 10 integers allowed for more complicated application of the numbers, such as in algebra. When the symbol system made its way through the Roman Empire, it had to compete with the likes of the Roman numeral system. This system proved largely inefficient in comparison to the Hindu-Arabic option. Although the Romans did use different icons to represent numbers, using the same ones to represent differing base 10 numbers proved more challenging to manipulate in more complicated situations. Simple addition and subtraction was a breeze. This basic process was widely used. However, as the Western world started growing more and more advanced with its infrastructure projects, understandings of geometry and economic diversification, being able to add some numbers together wasn’t enough. Since the Hindu-Arabic number system was able to deal with these issues in a more efficient way, it grew in popularity to what it is today. The inherent utility of having these icons is inarguable. In particular, being able to conceptualise complex and mind-boggling ideas with a single stroke of a pen allows one to use the information represented by this curve in applied ways instead of using time trying to demonstrate the complexity of the phenomenon.
The popularity this system has seen has arguably made it one of the most widespread communication systems in the world. However, while it may appear as a sort of universal language, it is important to make a distinction between the idea of a language that uses iconography and a number system which uses this. A language with icons is often one that depicts sounds, words or concepts through its symbols. However, a number system exclusively acts as a more simple representation of an abstract idea. The concept of the number two is one that most are familiar with. However, being able to read a curvy line versus understanding how the number is related to the numbers on either of side of it, how it creates the concept of even numbers and the way it acts as a primary breakaway from singularity cannot be expressed through this simple symbol. So, although one may consider this system as a universal language, it is crucial to note that it isn’t entirely so.
An icon to represent a lack of existence. Something to represent nothing. The number, as we now know it, has come to change our methods of mathematics. Before the West was exposed to this numerical system, there wasn’t a commonly used method of representing this concept. Having an icon act as a symbol to display a lack of anything allows further development in various fields while also maintaining order and stability in using the Hindu-Arabic numeral system in pure mathematics. One of the ways these notions are displayed is the columnar method of separating the ones, tens, hundreds, etc. in a number. Before the proper development of this numeral system, a zero was viewed more akin to a decimal place than a number. However, once people used the oval as a placeholder between columns in order to show a lack of a ‘ten’ or a ‘one’ it grew to gain its base level understanding as a number. This shift is representative of our understanding of an icon. Since an icon is meant to be a symbol of a complex concept so it can be applicable, having an icon which has multiple different interpretations displays its versatility. It can be used in so many applications as well as be impossible to use in others since it retains this duality as a phenomenon and a value. Even though the concept of nothingness is relatively simple to comprehend, zero’s complexity can only really be simplified in its symbol. It is truly an icon that attempts to conceptualises the intangible.
The next logical extension from the way numbers are written and what we do with those numbers is algebra. Algebra is a clear proprietor of iconography here, as the whole topic is about using some symbols to represent other symbols. In this way, one could consider algebra to be an example of a ‘double icon’. Aside from algebra being an essential tool to the current understanding and teaching of mathematics today, its existence as the ideal way to solve problems makes it applicable to many different situations. Having it be so applicable may actually change the way people think. According to a 2004 study published to the Proceedings of the National Academy of Sciences of the United States of America, when children learn algebra, they are using the same regions of their brains as adults. In contrast, children who practice algebra would often gain a notable decrease in a parietal section of the brain that is storing an image of the equation. This information suggests that the brain’s response to practice of symbol manipulation can change the way the brain behaves. This diversion from storing the information as an image in one’s head to understanding the concepts shown has been made achievable through iconography in mathematics. This is important as thinking in an algebraic manner is quite essential to being able to manipulate and move around mathematical information in order to write it in the correct form or to prove a certain point.
To conclude, when one hears the word ‘icon’ they may envision a logo, a celebrity or even an application on their phones. But, when one makes the connection between the idea of an icon and mathematics, a world of deeper knowledge and insight becomes discoverable. Throughout this essay, the importance of icons within mathematics has been made apparent. However, it is also crucial to understand mathematics itself as an icon. This slightly diverges from the idea of ‘icon’ that has been present thus far. Accepting maths as an icon in and of itself allows one to fully appreciate its beauty. This, of course, is due to society’s view of maths in the modern wall. After all, in the popular culture sense of the word, an icon’s image is developed through the mainstream view of it.
Due to different factors such as complexity and the need for patience and persistence, maths is often viewed as among the more difficult subject areas one could spend their time with. In addition to this, it is one of the subjects that are hardest to conceptualisation. A process such as integration, for instance, involves the manipulation of seemingly random equations in order to return to an original all the while not presenting why the method works and what it shows us. As much as maths can try to use different tools to help us make sense of information, things like graphs and tables remain harder to process than diagrams and images. The distance from the human perception of reality from the fundamental nature of mathematics alienates many.
With this understanding of maths, many people place it in high regard. Being capable at maths is often regarded as more useful and impressive than other subject areas. It is understood as a central science. Having been crucial in the development of other areas of scientific study such as physics, chemistry and biology, maths is sometimes viewed as inescapable as well as necessary for these aforementioned difficult subjects. It is for reasons such as this and many others that maths itself has become an icon of intelligence.
As an icon of brainpower, maths has had a profound effect on the self esteem of people throughout different education systems. Those who are not naturally inclined towards maths often are left to feel stupid and less worthwhile than others. A lack of ability in this one subject can lead many highly intelligent people with lots of potential to believe they aren’t clever enough to succeed. Maths as this icon, this pinnacle of brilliance has been brought about by society, but is truly damaging to many people in this society.
 Lippman, David. “Mathematics for the Liberal Arts.” Lumen, courses.lumenlearning.com/waymakermath4libarts/chapter/the-hindu-arabic-number-system/.
 BBC “Zero” In Our Time 13 May 2004 https://www.bbc.co.uk/programmes/p004y254
 Carter, et al. “Department of Psychology, Carnegie Mellon University, Pittsburgh, PA 15213, USA. email@example.com.” Proceedings of the National Academy of Sciences of the United States of America, 2004, europepmc.org/abstract/med/15064407.


Are you ready to unlock the power of compound interest and take control of your financial future? Understanding how to calculate your investment growth can be a game-changer when it comes to maximizing your wealth. Whether you’re a seasoned investor or just starting out, this knowledge is essential to ensure your money is working as hard as possible for you. In this guide, we will break down the concept of compound interest and show you step-by-step how to calculate your investment growth. From simple interest to compound interest, we’ll explain the differences and why compound interest is the key to exponential growth. We’ll also provide you with practical examples and tools to help you visualize the potential of your investments over time. So, if you’re ready to take charge of your financial destiny and make your money work for you, let’s dive in and discover the incredible power of compound interest.
What is compound interest?
Compound interest is a powerful concept that allows your investments to grow exponentially over time. Unlike simple interest, which only calculates interest on the initial amount invested, compound interest takes into account the accumulated interest as well. This means that the interest earned in each period is added to the principal, and subsequent interest calculations are based on this new total. In other words, compound interest is interest on top of interest. This compounding effect can have a significant impact on the growth of your investments.
Understanding the concept of compound interest is crucial for anyone looking to build long-term wealth. By harnessing the power of compounding, you can potentially turn a small investment into a substantial nest egg over time. The key is to start early and let time work in your favor. Even small contributions made consistently can lead to significant wealth accumulation thanks to the magic of compounding.
The power of compound interest
The true power of compound interest lies in its ability to generate exponential growth. As your investments grow, the amount of interest earned also increases. This creates a compounding effect where your money starts working for you, instead of the other way around. Over time, the growth curve becomes steeper, and your investments can experience rapid growth.
To illustrate this, let’s consider a hypothetical scenario. Imagine you invest $10,000 at an annual interest rate of 5% with monthly compounding. After one year, you would earn $500 in interest. However, instead of withdrawing the interest, you reinvest it, bringing your new principal to $10,500. In the second year, you would earn interest on the new principal, resulting in $525. This process continues, and with each passing year, the interest earned becomes larger. By the end of 10 years, your initial investment would have grown to approximately $16,386. This is the power of compound interest in action.
Understanding the compounding period
The compounding period refers to how often interest is calculated and added to the principal. It can have a significant impact on the growth of your investments. The more frequent the compounding, the more interest you will earn. For example, monthly compounding will generate more interest compared to annual compounding.
To understand the impact of compounding periods, let’s consider two scenarios. In the first scenario, you invest $10,000 at an annual interest rate of 5% with monthly compounding. In the second scenario, you invest the same amount at the same interest rate, but with annual compounding. After 10 years, the investment with monthly compounding would grow to approximately $16,386, while the investment with annual compounding would only reach around $16,289. This shows that even a small difference in compounding frequency can lead to a significant disparity in investment growth.
How to calculate compound interest
Calculating compound interest may seem daunting at first, but it’s actually quite simple once you understand the formula. The formula to calculate compound interest is:
A = P(1 + r/n)^(nt)
– A is the future value of the investment
– P is the principal (initial investment amount)
– r is the interest rate (expressed as a decimal)
– n is the number of compounding periods per year
– t is the number of years the money is invested
Let’s break down the formula using an example. Suppose you invest $5,000 at an annual interest rate of 6% with quarterly compounding. You want to calculate the value of your investment after 5 years. Plugging the values into the formula, we get:
A = 5000(1 + 0.06/4)^(4*5)
Simplifying the equation, we have:
A = 5000(1.015)^20
Using a calculator, we find that the future value of your investment after 5 years would be approximately $6,416.40.
Examples of compound interest calculations
To further illustrate the power of compound interest, let’s consider a few examples.
You invest $1,000 at an annual interest rate of 8% with monthly compounding. After 20 years, your investment would grow to approximately $4,661.39.
You invest $10,000 at an annual interest rate of 3% with quarterly compounding. After 10 years, your investment would grow to approximately $13,439.85.
You invest $500 at an annual interest rate of 6% with annual compounding. After 15 years, your investment would grow to approximately $1,197.96.
These examples demonstrate how compound interest can significantly boost the growth of your investments over time. The longer you stay invested and the higher the interest rate, the greater the impact of compound interest.
The impact of different interest rates and compounding periods
The interest rate and compounding period have a direct impact on the growth of your investments. A higher interest rate or more frequent compounding can accelerate the growth of your money.
To understand this concept, let’s compare two scenarios. In the first scenario, you invest $10,000 at an annual interest rate of 4% with monthly compounding. In the second scenario, you invest the same amount at the same interest rate, but with quarterly compounding. After 20 years, the investment with monthly compounding would grow to approximately $24,117.14, while the investment with quarterly compounding would only reach around $23,847.23. This shows that a higher compounding frequency can result in slightly higher investment growth.
Similarly, if we compare two scenarios with the same compounding frequency but different interest rates, we can observe a significant difference in investment growth. For instance, if you invest $10,000 at an annual interest rate of 3% with monthly compounding, your investment would grow to approximately $18,061.41 after 20 years. However, if you increase the interest rate to 5%, the investment would grow to approximately $26,533.88. This demonstrates the exponential growth potential of compound interest when paired with higher interest rates.
The Rule of 72: A quick way to estimate investment doubling time
The Rule of 72 is a handy rule of thumb that allows you to estimate the time it takes for an investment to double based on the compound interest rate. The formula is simple:
Years to Double = 72 / Interest Rate
For example, if you have an investment with an annual interest rate of 6%, it would take approximately 12 years for your investment to double (72 / 6 = 12). This rule provides a quick and easy way to gauge the potential growth of your investments over time.
Strategies to maximize compound interest growth
Now that you understand the power of compound interest, you may be wondering how to maximize its growth potential. Here are a few strategies to consider:
1. Start early: The earlier you start investing, the longer your money has to compound and grow. Time is one of the most critical factors in generating substantial wealth through compound interest.
2. Increase your contributions: By consistently contributing more money to your investments, you can accelerate the growth of your portfolio. Even small increases in contributions can have a significant impact over time.
3. Take advantage of tax-advantaged accounts: Utilize tax-advantaged accounts such as 401(k)s or IRAs to maximize the growth of your investments. These accounts offer tax benefits that can help your money grow faster.
4. Diversify your investments: Spread your investments across different asset classes to minimize risk and maximize returns. Diversification can help you weather market fluctuations and increase the overall growth potential of your portfolio.
5. Reinvest your earnings: Instead of withdrawing your earnings, reinvest them to take advantage of the compounding effect. By reinvesting your dividends or interest payments, you can accelerate the growth of your investments.
6. Regularly review and adjust your portfolio: Keep an eye on your investments and make necessary adjustments to ensure they align with your financial goals. Regular reviews can help you identify under-performing investments and make informed decisions to optimize your portfolio’s growth.
By implementing these strategies, you can harness the full potential of compound interest and maximize the growth of your investments.
Compound interest vs simple interest: Key differences
While compound interest is a powerful wealth-building tool, it’s essential to understand the key differences between compound interest and simple interest.
Simple interest is calculated based on the principal amount and the interest rate, without taking into account any accumulated interest. Unlike compound interest, simple interest does not have a compounding effect. This means that the interest earned remains constant throughout the investment period.
On the other hand, compound interest considers the accumulated interest and calculates future interest based on the new principal amount. This compounding effect can lead to exponential growth, making compound interest a more favorable option for long-term wealth accumulation.
In summary, compound interest allows your investments to grow exponentially over time, while simple interest only calculates interest based on the initial principal. Understanding these differences is crucial when making investment decisions and maximizing your wealth-building potential.
Understanding how to calculate your investment growth is essential for anyone looking to take control of their financial future. Compound interest is a powerful tool that can turn a small investment into significant wealth over time. By harnessing the compounding effect, your money can work for you, generating exponential growth and securing your financial well-being.
In this guide, we’ve explored the concept of compound interest, its calculation formula, and practical examples. We’ve also discussed the impact of different interest rates and compounding periods, as well as strategies to maximize compound interest growth. By applying this knowledge and taking advantage of the power of compound interest, you can unlock the full potential of your investments and achieve your financial goals.
So, are you ready to take charge of your financial destiny? Start harnessing the incredible power of compound interest today and watch your investments grow beyond your expectations. Remember, time is on your side, and with the right knowledge and strategies, you can unlock the doors to financial freedom.


Math grade= 10% Classwork/Journal Entries, 30% Quizzes, 60% Test
1. Spell it
2. Draw it
3. Expanded form
4. Add 10
5. Subtract 10
6. Add 100
7. Subtract 100
8. Next 3 even numbers
9. Next 3 odd numbers
Math games to practice math facts:
Use a deck of playing cards (without face cards and jokers) and split in half between two players. The player does not look at their stack cards. Both players lay down one card at the same time face up. The first player to multiply the two numbers and calls out the answer, wins that pair of cards. The pair of cards go in a separate stack until all cards have been used. The player with the most cards at the end, wins the game.
You can also add or subtract the two numbers that are laid down.
Be sure to practice current and previous math skills using our GoMath! link that you will find under Student Links on our school website.
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Free Year-4 worksheet printables. Educational resources developed by teachers for parents and teachers. Covering 8 subjects and dozens of topics.
Primary Resources - free worksheets, lesson plans and teaching ideas for primary and elementary teachers.The process of data handling is when you gather, record and present information and data in a way that is more helpful for others. This can be in graphs, charts or tables. It is also sometimes known as statistics. In the UK, children start to learn about data handling in KS2.Year 4 Maths Worksheets (age 8-9) By the end of Year 4 children will be using standard written methods of addition and subtraction, and will be developing their written methods of multiplication and division. They will have a sound knowledge of all times tables up to 10x10 and use this to work out corresponding division facts. All this will be based on sound mental methods of calculating.
Year 4 Mental Arithmetic. Statistics (Handling Data) for Year 5 (age 9-10) We have a great selection of worksheets for Data Handling, or Statistics as it will be known from September in the new Programme of Study, including plenty of line graphs. The new statutory requirements state that pupils should be taught to: solve comparison, sum and difference problems using information presented in.
Year 4 Maths. Showing top 8 worksheets in the category - Year 4 Maths. Some of the worksheets displayed are Year 4 maths number place and value workbook, Year 4 entry into year 5 25 hour revision booklet english, Year 4 practice sats mathematics pack, Y4 addition and subtraction, Downsend school year 5 easter revision booklet, Name teacher numeracy year 7 8, Fun math game s, Big maths beat that.
Help your child prepare for the Year 4 times tables test with this practice multiplication tables check. Login or Register to add to your saved resources Subscribe now now to instantly download this content, plus gain access to 1000s of worksheets, learning packs and activities exclusively available to members.
Explore the Statistics and Data Analysis Worksheets in Detail. Tally Mark Worksheets. Let's go back in history and learn a fun way to count with this batch of Tally worksheets, featuring colorful and engaging activities to count and read tally marks, spinner board activities, classifying and counting tally marks, word problems and a lot more. Pictograph Worksheets. The assemblage here provides.
Welcome to IXL's year 4 maths page. Practise maths online with unlimited questions in more than 200 year 4 maths skills.
These year 4 maths worksheets are colourful and child friendly and they aim to be easily accessible to children so that children can work independently on them. They are free to download and print off without registration or paying a monthly fee. We are constantly adding new resources to our website so why not have a look around and see what else you like.
Year 4: Statistics Pupils should understand and use a greater range of scales in their representations. Pupils should begin to relate the graphical representation of data to recording change over time. New Maths Curriculum (2014): Year 4 objectives.
This teaching resource could be used in a variety of ways when teaching data representation and interpretation.Some suggestions include: pre- and post-testing; independent classwork; revision; homework. This teaching resource pack includes worksheets addressing the following concepts.
Free worksheets, online interactive activities and other resources to support children learning about handling data.
They are asked to use the data in the table to answer some questions. Database exercise 3. A paper based exercise which could be given to students to check their understanding of databases or as a piece of homework. Students are presented with a small table of data where the data has been coded. Students are asked to consider the usefulness of.
Year 4 Flexible Maths Blocks Select Units for your Blocks Our Learn at Home pack for Year 4 Maths is now available. Explore all our Learn at Home packs. Teach selected units from the blocks below in an order that corresponds with your maths scheme. Alternatively, follow our recommended pathway. Learn at Home Packs Coverage tracker LKS2 Maths in Topics Autumn Spring Summer. Autumn Blocks Place.
Probability Worksheets. These printable math worksheets will help students learn about probability of random events. Probability Spinners (Basic) FREE. Use the pictures of the spinners to determine the probability of outcomes for events. 4th through 7th Grades. View PDF. Filing Cabinet. Logged in members can use the Super Teacher Worksheets filing cabinet to save their favorite worksheets.
A fantastic PDF Homework Booklet for Year 4 Mathematics. Click Here to download a SAMPLE PDF showing some pages from this product This practical learning tool includes 40 single-sided A4 homework sheets, covering topics on the Year 4 National Curriculum. Delivered to you electronically. We recommend one homework sheet to be set each week, with.
Welcome to the statistics and probability page at Math-Drills.com where there is a 100% chance of learning something! This page includes Statistics worksheets including collecting and organizing data, measures of central tendency (mean, median, mode and range) and probability. Students spend their lives collecting, organizing, and analyzing data, so why not teach them a few skills to help.


Many college students who take calculus fail to earn a C or better. Could ‘active learning’ help turn things around? Math departments fail too many calculus students and do not adequately prepare those they pass.
We hear it all the time. And we’ve had enough. Because we believe that the idea of “math people” is the most self-destructive idea in America today. The truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career. Worse, you may be helping to perpetuate a pernicious myth that is harming underprivileged children—the myth of inborn genetic math ability.
Math is a prerequisite for most STEM fields, and the reason many students abandon STEM careers. In higher levels of mathematics, gender imbalances persist: In 2015, about 76% of math doctorates were awarded to men. This figure should prompt alarm in mathematics departments across the country — and encourage focus on an area that is shockingly neglected in discussions of equity: teaching methods in classrooms.
Most women working in math today are used to being the “only” one. The only girl who ignored the teacher who said boys are better at math. The only girl on the high school math competition team. The only female student in advanced college math courses. The only woman in her graduate school cohort, or […]


Did you ever feel like rolling something nicely on a level surface but you are bored of using spheres and cylinders? Well then, we got the perfect answer for you, solids of constant width. These are solids which when rolled over a level surface have their geometric center at a constant height in all situations.
Oloids happen to be a quite famous and rather artsy example as of such a solid. An Oloid is a three-dimensional curved geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the edge of the other circle. The distance between the circle centers equals the radius of the circles. While it may look complex it is actually pretty easy to model. Another interesting family of solids is the steinmetz solids which is a solid obtained from the intersection of two or three cylinders of equal radius at right angles to each other and one can also see that these roll too.
Read more here on Wikipedia.
A really nice animation of the oloid on YouTube.
Oloids and more.
Steinmetz solid in action.
About that Steinmetz solid.


I’ve written a few small projects that I’ve released and have semi-maintained. Some of these are from my PhD, while others are work related. All are currently hosted on GitHub.
- brandseyer2 is an R library that wraps the DataEQ JSON API, providing a convenient interface for analysing and visualising a client’s account data in R. It’s licensed under the MIT license.
- CL-HEAP provides various implementations of heap data structures (a binary heap and a Fibonacci heap) as well as an efficient priority queue. The Fibonacci heap has interesting run time constraints, with many operations occurring in constant or amortised constant time, making it ideal for use in implementing other algorithms, such as Dijkstra’s shortest path and Prim’s minimum spanning tree algorithms. The project is licensed under the GPLv3, and written in Common Lisp.
- L-MATH is a Common Lisp library for simple linear algebra in geometric applications. Vector and matrix classes are available, as are linear interpolation functions and various operations related to creating rotation matrices. It also has some spline support (Hermite and Bézier curves, B-Splines and Catmull-Rom splines)! The code is licensed under the GPLv3, with the Classpath linking exception.


Commands are entered in prefix notation, meaning the operators come first. For example, (+ 1 2) evaluates to 1 + 2 = 3.
Besides basic arithmetic, other operators include:
define - This is how you declare variables. Example: (define a 1). Now if you call a, the value is 1. The return value of a define expression itself is 0.
set! - This is how you update variables. Example: (set! a 2). If a were previously defined as something else, now if you call a, the value is 2. The return value of a set! expression itself is 0.
lambda - This is how you declare a function. Example: (lambda (x y) (+ x y)). This creates a function which takes parameters x and y and returns x+y. If you call a define on it with a name, you can then call the defined name to use the function. Otherwise you can just pass the parameters after, as such: ((lambda x (+ x 1)) 2). This would return 3.
"="" - The equality operator. Example: (= 1 2) tests if 1=2. If it does, it'll return #t, meaning true. If it's false (which it is), it'll return #f, meaning false. There's also <, >, <=, >= for testing other conditions.
if - The if operator for doing conditional statements. Example: (if (= 1 1) 1 2). Tests if 1=1, if it's true (which it is), it'll return 1, if it's false, it'll return 2.
quote - This can be shorthanded by a ' in front of an expression. It returns the expression as a literal. Example: (quote (1 2 3)) or '(1 2 3) return [1, 2, 3]. Use this when you want to use expressions in operations, like (set! a '(dog)) sets the variable a to "dog"
cons - For concatenation. Example: (cons 1 '(2 3)) concatenates 1 to the front of [2, 3], resulting in [1, 2, 3].
car - Returns the first value in a list. Example: (car '(1 2 3)) returns 1.
cdr - The opposite of car, returns everything but the first value in a list. Example: (cdr '(1 2 3)) returns [2, 3].
begin - If you want to get fancier, use this to chain expressions. Example: (begin (set! a 1) (set! b (+ a 2)) (* b a)) will perform a = 1; b = a + 2; b * a = 3


Crackers And Goo
Crackers and Goo uses patterns and mental math strategies to teach children critical thinking skills. Crackers and Goo shows children multiple math strategies for solving problems, teaching them to look at any problem from multiple perspectives and come up with creative solutions.
Children will find Crackers and Goo a fun and challenging application because it uses pattern puzzles to highlight mental math strategies. Parents and teachers alike will appreciate the math hints and equations that are built into the game to provide children with the learning support they need to improve their number sense. Crackers and Goo was specifically designed for the Common Core Standards of Mathematics.
Crackers and Goo has over 6,000 games in 16 different levels of math called skill steps, with many games of increasing complexity at each step. This provides children with enough variety and repetition for deep understanding and long-term retention of the math skills they learn.
Children from ages 4 and will learn and practice the following skills with Crackers and Goo:
- Counting and skip-counting by 1’s, 2’s, 5’s, 10’s, and 100’s
- Pattern recognition and sorting
- Addition within 10, 20, 100, and 1,000
- Mental arithmetic strategies, including grouping
- Repeat-addition, leading to multiplication
- Rounding and multiplying
- Expanding and multiplying
- Algebraic concepts
Features of Crackers and Goo include:
- Over 6,000 different games and 100 hours of play
- Lots of action: players form patterns with flying crackers while dodging goo splats! A math problem must be solved to win the game once the correct patterns are completed.
- 16 different steps of play, each with over 5 hours of play and focusing on increasingly complex math skills
- Demo mode showing how to play the game
- Multiple players with separate tracking of scores and stats: each player has a customized name, sneaker avatar, and game play progress.
- Hints for completing patterns in every game
- Mental math strategy hints to help with more complex games
See more at www.crackersandgoo.com


Posted on May 15, 2013
Math games often make up part of our summer curriculum. We normally do a scaled-back school schedule through the summer. I like to find different ways for the children to learn math facts than the usual drill sheets and flashcards. I was looking around recently for some fun math games and came across a few that I am planning to use and thought I would share them with you.
1. Dr. Mikes Math Games for Kids – a great resource with printable games
2. Math.com – a list of math games that can be done with things you already have around the
3. Kids Activities Blog – 13 fun math games
4. Hip Hip Hooray Magazine – a list of math games
5. Confessions of a Homeschooler – math game printables
I am excited to do some of these activities with our kids this summer. Do you have other fun math games you like to do with your children?


The Powerball is now just two days away.
But it’s still not over yet.
If you want to know when the jackpots are going to go up and down, you have to wait until the jackpottoes are drawn.
That’s because the lottery office is counting the number of tickets available for each jackpot.
As of Tuesday, the lottery said it has 3,976,865 tickets available.
The top-ranked Powerball jackpot, at $2.5 billion, is worth $7.7 million.
But the jack pot for the lowest-ranked jackpot is $400 million.
Those are the numbers the lottery is trying to keep up with.
So when the lottery draws a jackpot it says, we’ll get that number of times as fast as possible.
The number of numbers on the board is divided by 2.5 to determine how many times it will be drawn.
If there are more than two numbers on that line, the jack will be flipped.
That means that for the next three days, the odds of winning a Powerball are always going to be lower than the odds for winning a $300 bill.
That includes the fact that, for the $400-million jackpot in 2022, only one ticket is available.
If it’s a $1,000 Powerball, you only have to win a $10 bill for the entire three-day period.
If, on the other hand, the $600-million Powerball has no numbers on it, you still have to be lucky for the rest of the three days.
But for the top-ranking jackpot and the $1 million jackpot the odds are still going to come down, and then down a lot.
So it’s going to take a lot of luck for the Powerball to get the jack, and it will take a little bit of luck to get a $200 bill, too.
That is what the lottery calls the odds at which the jack is being drawn, or the number that is the smallest number that can get it right.
That number is called the chance of winning.
But there are a lot more numbers on a PowerBall jackpot than there are numbers on $300 bills.
The lottery says that in 2019, the probability of winning is 1 in 30,000.
For 2022, that number is 1.26, or 0.00001 in 1,000,000 trillion.
That was the same probability as winning the lottery’s biggest jackpot ever, the one for $1.4 billion in 2012.
But this year, the winning number has dropped by $200 million to 0.0005, or 1 in 100.
This year, it’s dropping by another $200 to 0,0008, or 3 in a trillion.
But what about the bigger jackpot for 2022?
That one is even smaller: $2 billion.
The biggest jackpots ever have been won with a $500 bill.
In 2022, the largest jackpot was for a $50 bill.
So the lottery says it will keep drawing jackpots for $2,000 bills until 2020, and will have no luck for $10 bills until 2022.
But if it draws a $600 bill, you can always hope.
And if you’re lucky enough to win $600, you’ll get to keep it.
The Powerballs are the biggest prize in sports.
They are the only prize you get when you buy tickets to see your favorite team play a game.
There’s an average of 1,500 Powerball tickets sold every night at the Palace of Auburn Hills, and there are roughly 4,000 in circulation at the MGM Grand in Las Vegas.
The first Powerball ticket was sold on Aug. 15, 2006, at 9:19 p.m. at the Omni in downtown Las Vegas, and tickets have been sold every day since then.
The next Powerball will be sold Aug. 19, 2019.
The jackpot of $2 million is now $2 trillion.
The other three jackpots have been split evenly between three states: Arizona, California and Nevada.
The winner of each jackup has to pick up the remaining $1 billion in prizes.


Statistics in Trade: Part 1
Different Type of Datasets:
In the economy, we mainly deal with Four main categories of Data:
1. Time Series Data: X(t)
The data which change by time. like the stock price which changes frequently, every minute, hour, day or…
2. Cross-Sectional Data: X(i)
These data sets are taken at a specific period of time. like the Unemployment rate in the year2021. The data taken are obtained based on random sampling.
3. Pooled Cross-sectional:
the data has both cross-sectional and time-series features.
4. Panel or longitude Data:
This category of data is taken, the same cross-section at different times.
Sample and Population:
It measures the strength of the linear relationship between two numerical variables (x,y), only concerned with the strength of the relationship and no causal effect.
Cov (X,Y) >0 — -> same direction of the two variable movements
Cov (X,Y) =0 — -> independent
Cov (X,Y) <0 — -> One moves in the opposite of the other one.
but we can not find anything related to the causation in this case.
Coefficient of correlation:
To find a correlation between the stock price and trade we can use the coefficient of correlation. The presence of correlation shows a casual relationship between the two data series.
how closely the two variables are changing together?
Correlation includes causations, but causation is one of the influences of that.
Econometrics investigates by how much y changes if the X changes. in econometrics, we are dealing with finding the reasons.
On the website tylervigen.com you can find different samples of the case.
Norm.INV(probability, mean, max)
This the function on excel that used for making normally distributed data.
Note: The correlation coefficient may be misleading with a small number of data. to check if the number of data selected was trusted or not, we should consider the significance of the data. in case we use the Random Probability and make two different sets, those two sets have a correlation of 0.
A Statistical distribution describes the number of times each possible outcome occurs in a sample.
a normal distribution is a continuous distribution in which there is most likely to be at the mean, and the data are very less likely to be away from the mean.
The data in the normal distribution are distant from the mean value for 99% of data is less than 3 times standard deviation. 66% of the data are distributed is around the mean value.
In order to make the process of using the data more handy, we can make the normalization on the data sets with the normal distribution. By normalizing, in fact, we move the data distribution curve in a direction that the mean of the new curve becomes 0 and squeeze or push the data in a way, that the Standard deviation becomes 1. In that way, the values instead of x in horizontal access are called to be z-scores. The interesting thing about the Z-score is that it shows the difference that the data has from the mean value as a multiplier of the Standard Deviation.
In normalization, we make the mean equal to zero and the SD=1.
we also change the X-axis to a new variable we can have Z axis.
In probability and statistics, Student’s t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally-distributed population in situations where the sample size is small and the population’s standard deviation is unknown.
The t-distribution plays a role in a number of widely used statistical analyses, including Student’s t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student’s t-distribution also arises in the Bayesian analysis of data from a normal family.
If we take a sample of observations from a normal distribution, then the t-distribution with (n-1) degrees of freedom can be defined as the distribution of the location of the sample mean relative to the true mean, divided by the sample standard deviation, after multiplying by the standardizing term Sqrt(n). In this way, the t-distribution can be used to construct a confidence interval for the true mean.
The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student’s t-distribution is a special case of the generalized hyperbolic distribution.
in any research, we have a main Hypothesis, while for some statistical tests, we have to use Null and Alternative Hypotheses. The Null Hypothesis is the main hypothesis that we have in our test. It is the possible answer that we believe is going to be the correct answer and we plan for making the tests in a way to find if it is true or not.
The alternative Hypothesis, on the other side, is the hypothesis that we are going to reject in our analysis.
Example: Education Affects Wages:
Null Hypothesis (H0): Education Affects the Wage Positively.
Alternative Hypothesis (Ha): Education has no effect or negative effects on the wage.
Alpha is the possibility of rejecting the null hypothesis. i.e. we approved that the H0, ( the positive effect of Education on wage is true), but there is 5% of data that are not covering this hypothesis, this is called to be the significance level of our experiment.
Power of Test:
If the test Null hypothesis is rejected, then there is a possibility that we have some data that are not approving the rejection. This is called Beta. the value 1-Beta is called the power of the test.
In statistics, we deal with a lot of different tests to check the validity of the Hypothesis. In most of the Hypothesis testing, we are looking for confidence of 95% on our results, so the standard level of Alpha (significance) that should be used is always around 5%.
Pearson Correlation Test:
The test is designed to know that how correlated are two different variables. In fact, It is a measure of how linear a set of data is. The values range from -1 to 1. If the best fit line is positive, then the coefficient is positive; if negative, then the coefficient is negative. If the slope of the best fit line is 0 (horizontal) or infinity (vertical), then the coefficient is 0.
Considerations of Application:
Under heavy noise conditions, extracting the correlation coefficient between two sets of stochastic variables is nontrivial, in particular where Canonical Correlation Analysis reports degraded correlation values due to the heavy noise contributions. A generalization of the approach is given elsewhere.
Pierson tests in simpler situations when we are dealing with Normal Distribution (Either Bivariate or not) & With no correlation between the datasets can be modified to the Students t-distribution test.
How we convert the data in the T-Test:
r is the critical value that we use in the calculations. consider the case, that we are dealing with 2 dataset each with 1000 data, then the value of r will be smaller till the time that it reaches a constant value.
One tailed and two tailed considerations on the Student t- test:
The main keywords that we have to know in implementing this test are the concepts of one-tailed or two-tailed data.
In the case of our Hypotheses based on the validity of Correlation is made, i.e.
In this case, we have to check the r for one-tailed data.
In case that the correlation coefficient is not equal to zero, then we use the to two tail data.
Table of critical values:
Here also there is a tool that calculates the values for differnt tests.
Critical Values Calculator
This simple calculator allows you to calculate critical values for the z, t, chi-square, f and r distributions…
Application of the Normal Distribution in Trade:
when dealing with finding the Stop Loss or Take profit, I can base my calculations on the 15 minutes Trade Volume data.


Robert Q. Berry III, Ph.D. will join the University of Arizona on July 1 as the Dean and Paul L. Lindsey and Kathy J. Alexander Chair of the College of Education. In this role, he will continue to build upon the College of Education's mission to promote learning, research, and transformative action to create a more inclusive and equitable community and world.
Dr. Berry will join the College of Education from the University of Virginia, where he is currently the inaugural Associate Dean of Diversity, Equity, and Inclusion in the School of Education and Human Development and the Samuel Braley Gray Professor of Mathematics Education. Additionally, Dr. Berry is the immediate Past-President of the National Council of Teachers of Mathematics (NCTM), and he was recently elected to the National Academy of Education. His full CV can be viewed here.
Equity issues in mathematics education are central to Berry’s research efforts with four related areas: a) Understanding Black children’s mathematics experiences; b) Measuring standards-based mathematics teaching practices; c) Unpacking equitable mathematics teaching and learning with issues of social justice, and d) Exploring interactions between technology and mathematics education. Berry is the lead developer of a mathematics classroom observation instrument, Mathematics Scan, which measures standards-based mathematics teaching practices.
In addition, he co-edited the 2020 book High School Mathematics Lessons to Explore, Understand, and Respond to Social Injustice. His articles have appeared in the Journal for Research in Mathematics Education, Journal of Teacher Education, Educational Studies in Mathematics, and the American Educational Research Journal. Berry has authored more than 100 publications and has lectured and made major presentations worldwide. Berry is a two-time recipient of NCTM’s Linking Research and Practice Publication Award and received the University of Virginia’s All-University Teaching Award in 2011.
Berry is a first-generation college graduate who received his Bachelor of Science degree from Old Dominion University, his master’s degree from Christopher Newport University, and a Ph.D. from the University of North Carolina at Chapel Hill.
He will succeed Dr. Bruce Johnson, who returns to research and teaching in the College of Education following a successful five-year appointment as Dean.


PySDDR combines the interpretability of a statistical model with the power of deep neural networks in an easy-to-use python package.
Typically, statistical models and machine learning models are on opposite ends of the data analysis spectrum:
Statistical models, at one end of the spectrum, are readily interpretable, but have the disadvantage of lacking the expressive power to deal with complex data such as images. Deep learning techniques, at the other end of the spectrum, allow modeling of almost any complex relationships, but the resulting models are hard-to-interpret black boxes. A new technique that combines the strength of both worlds is semi structured deep distributional regression (SDDR).
SDDR enhances the expressive power of standard statistical modelling with deep learning while maintaining interpretability. It is based on generalized additive models (GAMs), which are a widely used statistical tool to model how an observed variable depends on a set of features. ‘SDDR extends GAMs with the modeling power of deep neural networks’, says Lisa Barros de Andrade e Sousa, Helmholtz AI consultant at Helmholtz Munich.
In their effort to democratize AI, the Helmholtz AI consultant team at Helmholtz Munich has joined efforts with Dr. David Rügamer (LMU), the researcher that invented the SDDR technique, and developed the python package PySDDR to make this innovative new tool available to the Python community.
A good example of a PySDDR application would be predicting the price of a vacation rental apartment’, says Christina Bukas, also a member of the Helmholtz AI consultant team. ‘With PySDDR, the probability distribution of the price of the apartment can be modeled not only based on tabular features like the size of the apartment, its location or the quality category of the bathroom, but is also estimated from photos of the apartment.’
PySDDR, along with the R implementation of SDDR developed at LMU, allows any scientist, statistician or data scientist to use this exciting new technique and tap into the power of the union from statistics and machine learning.
- GitHub - https://github.com/HelmholtzAI-Consultants-Munich/PySDDR
- Paper preprint: https://arxiv.org/abs/2104.02705


The infinity of numbers
mixed technique, steel, digital motion, mirror steel, numerals, 50 × 50 × 25 cm
purchased 2020 with the support of the Czech Ministry
Two kinetic objects made of polished steel. The surface of one of the objects displays prime numbers from 2 to 97 plus the number 1, arranged in a meander-shaped structure. On the second object, the remaining numbers of the series are arranged in a similar configuration. These meandering sections move up and down in a regular rhythm, causing the numerals to emerge and then recede again. The numbers and their rhythmic up-and-down motion present us with interpretative questions. Prime numbers are unique in that they can only be divided by 1 or by themselves. Their indivisibility by other numbers may evoke sensations of exceptionality and originality. Prime numbers can thus be perceived as exceptional phenomena, or personalities which emerge periodically during the course of human history only to recede again, having left an important imprint on which society’s values are built. By using numbers, the artist can communicate on a generally comprehensible, semantically neutral level, unencumbered by more specific visual signs. Viewers can bring their own levels of interpretation to the work, decoding the processes of the numbers and their interrelations.


One of the most famous anecdotes from Antiquity deals with the philosopher-mathematician Pythagoras (c.570-c.495), who discovered the theorem that is named after him, and sacrificed an ox – or even one hundred oxen – to celebrate this. The joke that ever since the oxen are afraid of scientific progress has been used a bit too often by scientists dismissing critical reviews.
For several reasons, this anecdote is problematic. In the first place, because it is probably one of those unhistorical tales attributed to Pythagoras. Another example is his legendary visit to the ancient Near East, which is referred to for the first time in the second century CE, when Apuleius says that the Samian sage was “believed by some to have been a pupil of Zoroaster” (Apology, 31). In his Refutation of All Heresies (1.2.12), Hippolytus of Rome (early third century CE) implies that he had read this story in a book by Aristoxenus of Tarentum, a contemporary of Alexander the Great. Yet, even if Hippolytus’ is right (which is doubtful), this means that Pythagoras’ eastern trip is unmentioned by earlier authors describing Pythagoras’ life and opinions, even though Herodotus, Plato, and Aristotle had many opportunities to discuss it. The story is almost certainly invented, just like Pythagoras’ visit to India.
The same applies to the theorem that in right-angled triangles the square on the hypothenuse is equal to the sum of the squares on the sides containing the right angle. Pythagoras and his pupils were interested in mathematical proof, certainly, but the first to attribute the theorem to the Samian sage is Proclus (412-485), who lived almost one thousand years after Pythagoras (On Euclid I, 426.6-14 [Friedlein]).
A second problem is that the principle was already well-known prior to Pythagoras. Several cuneiform texts from the twenty-first and twentieth century BCE prove the that the ancient Babylonians not only knew that a²+b²=c², but also knew that this principle was generally applicable. There is a difference in the way Babylonians and Greeks proved this rule, but it is possible to overstate Pythagoras’ importance.
J. Høyrup, ‘The Pythagorean “Rule” and “Theorem” – Mirror of the Relation between Babylonian and Greek Mathematics’ in: J. Renger (red.): Babylon. Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne (1999).


The Mathematics of Doing, Understanding, Learning and Educating for Secondary Schools (MODULE(S2)) project creates and supports the use of undergraduate mathematics course materials that provide opportunities for users to develop deep mathematical knowledge of geometry, statistics, algebra, and modeling, specifically as it relates to doing the work of teaching.
MODULE(S2) materials have been used by 60+ faculty at 50+ universities in the US and Canada who have taught 1000+ prospective teachers and university students in university mathematics courses.
The project promotes effective instruction through asynchronous professional development activities on a faculty community Canvas site.
MODULE(S2) investigates the impact of instruction with the materials on prospective teachers’ knowledge–thereby informing research efforts in teacher education.
The MODULE(S2) project began through the work of the Mathematics Teacher Education Partnership (MTE-P), which seeks to create a gold standard for the preparation of secondary mathematics teachers across its over 90 member universities.
Rationale and Research: Evidence has mounted in recent decades that to implement sound mathematics and statistics instruction, teachers must possess understandings of content that are specifically applied in the work of teaching (mathematical knowledge for teaching – MKT*). To develop MKT, prospective teachers need opportunities to use mathematical knowledge to recognize, respond to, and understand mathematical issues that arise in the context of teaching. Findings from plan-do-study-act cycles in the project indicate an increase in prospective secondary teachers’ MKT when they learn with MODULE(S2) materials. MODULE(S2) is now expanding research efforts to examine the conditions of instruction that impact prospective teachers’ MKT, development of MKT, and expectancy and value in using MKT as a resource for teaching. The project uses mixed methods to provide empirically-based accounts of how factors of quality of instruction and of instructional resources impact prospective teachers’ MKT.
* Note on terminology: We acknowledge that statistics and mathematics education are disciplines with both overlapping and distinct components. For simplicity of language in this paragraph, we use mathematical knowledge for teaching and MKT in the very broad sense to include both mathematical and statistical knowledge for teaching (SKT).


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- Pronunciation Practice Activities (5). Quizzes, Questionnaires and Puzzles (5). Reading Extra (4). Reflective Teaching in Second Language Classrooms (5). - 'PET for Schools' Answer Key and Recording Scripts Practice Test (Complete PET). PDF, 109kb. |
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- Ask two or three students around the class to describe the picture and identify what the unit appears to be about. Help them with vocabulary if necessary. Repeat this procedure for photographs 2-5, trying to ensure that as many students as possible are given the chance to contribute. |
- Learn seventh grade math for free—proportions, algebra basics, arithmetic with negative numbers, probability, circles, and more. Full curriculum of exercises and videos.
Homework: Diﬁerence Quotient Practice Page 1 This worksheet is homework to be included in your homework notebook. [Odd-Numbered Answers on Back] 1. Given f(x) = 4x2, ﬂnd the following and simplify. (a). f(x+h) (b). f(x+h)¡f(x) (c). f(x+h)¡f(x) h (d). If you let h = 0, what do you get from your answer to part (c)? 2. Need extra practice resources for students who are struggling with grammar?Looking for further consolidation of grammatical skills? We've looked at all the tricky areas of grammar in the New Inspiration syllabus and have created some worksheets for extra grammar practice for your students.
Passing exams with a good score is the key learning outcome of Wider World. Through the Students' Books and Workbooks, the course provides extensive exam support, as well as extra Practice booklets for Cambridge and PTE exams.lesson 3 extra practice angles of triangles answer key, What is the third angle of a right triangle if one of the angles measures 51°? 12. ALGEBRA Find m∠A in ABC if m∠B = 38° and m∠C = 38°, 13.
Chapter 5 Extra Practice Answers Lesson 5-1 1. 3 8 2. 3 7 3. 2 3 4. 2 9 5. 2 5 6. 5 1 7. 7 1 8. 35 3 9. 107 32 10. 1 4 11. 1 3; For every striped tie, there are 3 solid ties. 12. 7 15; For every 7 crayons, there are 15 markers. 13. 2 7; For every 2 roses, there are 7 other types of flowers. 14. 3 5; For every 3 golden retrievers, there are 5 poodles. Lesson 5-2 1. 38.2 mi/gal 2. $1.59/lb Practice your Addition, Subtraction, Mutliplication and Division - self-marking Online exercises and worksheets with answers, you create and print from your browser. Set your level of difficulty. Math Worksheet Generator - Free - Practice Math with these Worksheets
Big Ideas MATH: A Common Core Curriculum for Middle School and High School Mathematics Written by Ron Larson and Laurie Boswell. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key. Absolute Value Equations; Complex Numbers. Simplify Imaginary Numbers Adding and Subtracting Complex Numbers Multiplying Complex Numbers Dividing Complex Numbers
(Note: These are mostly multiple choice, for extra practice. Your Final Exam will NOT have any multiple choice!) Part A - Multiple Choice Indicate the best choice for each question in the indicated space. Each correct answer is worth 2 marks. There is no penalty for incorrect answers. Select only one answer per question, or your unwanted extra blank pages in Excel 2010 Sometimes, when I'm working in an Excel 2010 document, I'll click File | Print just to see the preview of what the document will look like (in terms of its division into pages) when it's printed.
unwanted extra blank pages in Excel 2010 Sometimes, when I'm working in an Excel 2010 document, I'll click File | Print just to see the preview of what the document will look like (in terms of its division into pages) when it's printed.
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Example: A group of 3 vfd's of the same rating 1 feeder/branch ckt to supply all 3 How is ampacity determined? Assume pu vfd input current = 1 430.24: 1.25 x 1 + 1 + 1 = 3.25 or 430.122: 1.25 x (1 + 1 + 1) = 3.75
Thank you, Arthur Brower
Hey Arthur thanks for your question. Parts I through IX of Article 430 apply unless something in Part X changes those rules. Your first example is correct, but only if the input current rating is 1 ampere.
Conductors supplying three adjustable speed drives would be sized at 125% of the largest plus 100% of the other two as stated in 430.24 using the drives rated input current as stated in 430.122(A). So if the adjustable speed drives are all the same size and have input current rating of 10 the minimum feeder conductor ampacity would be 32.5 amperes [10 x 1.25 = 12.5 + 10 + 10 = 32.5]


This value bet calculator is a nice too to pick the right the match and place a bet. All you need to do is to estimate how much is the probability for the tip to outcome.
For example you estimate that the probability of a match is about 60% to be right and the odds are 2.2. Now when calculating if a value bet exists we have 2.2*0.6=1.32, so
we see that a value of 32% exists and this tip is a value bet. On long run you will a 32% profit of your betting amount by placing such matches.
In this value bet calculator simple put the odds and the estimated probability and the value bet calculator will show you if a value bet exists and how much is the profit.


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


I studied Data Analysis and Machine Learning at the French Engineering School Centrale Lille. This specialization was significantly focussed on statistics modeling, prediction and computer science. As is the case in many of the top French engineering schools, this study track also provided courses in general engineering, project management and entrepreneurship throughout the duration. Alongside this, I completed a Bachelor’s degree in Mathematics by correspondence from UPMC in Paris. I then completed a Master’s degree in Applied Mathematics with Lille 1.
I started working for the financial industry, developing decision-making tools for the hedging division of Axa. I thought at the time that Mathematics was only applicable in Finance. I learned that I was wrong when I worked for the startup Nitrogram (now acquired by Stripe), implementing a machine learning algorithm on their Instagram data analysis platform.
Following this experience, I went to Laos to work with the Ministry of Transport, for the company Phoenix-ISI, where we developed aroad maintenance monitoring platform. This two-year project was where my interest in geographic information systems and web maps began.
At the end of the project, I started my own company - GeoQuants and focused on creating applications around geo-spatial data and data visualisation. A full list of my projects can be found here.


No one liked algebra in high school. 3xy - 400x + 904 = 43x(18xy -12)? That is an actual algebra problem I have no idea how to solve.
Now high schoolers don’t need to figure out how to answer those equations thanks to a new app called PhotoMath. Users of PhotoMath can scan their math problems on their smartphone and the app will show them the solution and how to get there. The video below shows how it works:
The creators say this helps cut down on homework time since students don’t have to flip to the back of their textbook on particularly tricky questions. And by showing the process, the app acts as a teaching aid that can help people struggling to understand certain concepts.
That’s a noble intention, but we all know how it will really be used: Cheating. Kids these days grew up on cell phones and know all the tricks to using them in class without being noticed. Unless they own an iPhone 6 or Galaxy Note 4 or any other unwieldy phablet, high schoolers will be using this app to cheat.
PhotoMath is currently available on iOS and Windows Phone. Don’t worry Android users, the app will be released for you in 2015. At least you’ll have time to think up a plausible explanation for how your C+ in Algebra turned into an A overnight.
Joseph Misulonas is an intern at Playboy.com. He was originally accepted into the advanced mathematics program at Northwestern University, but decided to study journalism instead. When not regretting his life choices, he can be found on Twitter at @jmisulonas.


As many of you know we use the Math-U-See curriculum and LOVE IT! Our preschooler always wants to “do school” just like her big sister and that means that she wants to use her big sisters math manipulatives too! So I created some pages for her to use to build shapes with her Math-U-See blocks!
These pages can be printed out and laminated or can be placed in a page protector and the student can just build right onto the sheet. I tried to use the size block that corresponds with the number of sides on the shape. For example the triangle uses the pink-3 block and the hexagon uses the blue-5 blocks. Some of these shapes may require more than one set of blocks. But if you only have one set of blocks, this is the perfect time to teach your toddler addition without them even knowing it! Show them (or have them show you) how to use other blocks together to reach the desired block length. You can print out the colored outlines or black & white outlines to increase difficulty. You can also have your preschooler color in the blocks before they build.
This is a great way for younger students to become familiar with the blocks while they are learning their shapes!
Other Math-U-See Helpers:
Let us know what resources you would like to see, in the comments below!Tweet


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Sunday, March 13, 2005
You remember the Great Summers Squall, in which my aggravation at frail and wilting biologists was matched only by my irritation at those manly men who knew all along that womens warn't no good at figurin' and such. Not like a man, nowise.
Toward the end of this post I mentioned something about writing a paper about shapes rotating in 3D. When I was studying these things in the first place, it was true that I (as befitting a girlie) found visualizing some of the geometry a little difficult.
So I taught myself how to use Mathematica to render and rotate the shapes, and figure out some angles I was having trouble with, and I was good to go.
Another woman with a similar difficulty (shared, in this instance, by many men), took a somewhat different approach. She crocheted some hyperbolic surfaces. The beauty of this approach, besides creating attractive and durable models, is that you can demonstrate such concepts as geodesics, parallel lines, and non-Euclidean triangles.
Not only that, but I think that models such as this and this would make nifty Christmas ornaments, if rendered in metallic yarn or thread.
(In this interview with married mathematicians David Henderson and Daina Taimina -- Taimina was the one who thought to crochet the surfaces -- has this disturbing note:
I didn't realize that there was such a thing as mathematics that wasn't expressed by equations. Oh, I knew that there were calculations that were much easier when done by computational methods but I always assumed there was a big, hairy equation lurking in the back somewhere. It was too much trouble to work out, but it was there. I'm very suspicious of things labelled "intuitionist"; it's too often a synonym for "bullshit".)
Anyway, here are some instructions, if you want to try it yourself. I learned to crochet when I was little, but I find these instructions confusing. Perhaps if I had the yarn in my hand, they'd be clearer. (We -- my sister and I -- pretty much limited ourselves to Euclidean planes when we were learning.)
On the topic of mathematical yarncraft, here's a model of the Lorenz manifold, and the instructions (in the form of a PDF file) to go with it. The paper says that this is much harder than the hyperbolic surfaces. (The Lorenz manifold was found via this crocheting blog.)
And, finally, a Klein bottle woolly hat (sold out for now). Made -- not by, but to the specifications of -- the guy who also sells glass Klein bottles. The guy happens to be Cliff Stoll, who wrote The Cuckoo's Egg, and Silicon Snake Oil. The former book led to him portraying himself in a NOVA, in which he was shown showering with his girlfriend, and then scampering out naked in order to answer the phone. Niles and I agreed that, should we ever be subjects of a NOVA episode, naked scampering will be Right Out. The world will thank us.


Text by Kamana Regmi.
The second day of the training by OLE Nepal had a session to give the participants a touch of how the Xo can be integrated in the class. An activity on developing the concept of addition was done for about one and a half hour.The activity was about developing the concept of counting and adding. the first thing they did was colour pictures on picture cards. Then they used the same cards to play a winning and losing game by drawing a card, counting the number, counting the same number of beans, adding the beans, writing the mathematical sentence. The player who got the larger number won all the beans. Then came the role of the XO where a game with the same concept was played and students got to assess themselves through the score that was displayed in the screen.
Participants were really excited about the whole activity and said it raised their confidence on the use of Xo in their calss. Today, the third day of the training, the participants are making their own lesson plans and conducting sample classes. Hope this will raise their confidence even more.