A Geometric Approach to Differential Forms
This text presents differential forms from a geometric perspective accessible at the undergraduate level. It begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The subject is approached with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. The book contains excellent motivation, numerous illustrations and solutions to selected problems.


The Numbers GamePosted: April 10, 2008
Guess I should have listened to my math teacher.
A group of number crunchers at George Tech used two mathematical processes to correctly pick this year’s Final Four….and 30 of the 36 Final Four teams in the past nine years, according to InfoWorld. Those results are more accurate than the tournament seeding system and polls, which have picked 23, and RPI, which ID’d 21 of the Final Four participants.
The Georgia Tech system, uses two mathematical processes — logical regression and Markov chain, aka LRMC — to make selections. This year LRMC not only picked the Final Four — which was four #1 seeds — but had Kansas beating Memphis for the championship.
Don’t ask me to explain how it works. But it does work way better than my system.
Having said that, bet LRMC didn’t pick George Mason a few years back.


Moreover, the increased volatility may be predictive of volatility going forward. Volatility may then return to levels resembling that of pre-crisis levels or be more uniform going forward. A simple regression model does not account for this variation in volatility exhibited in financial markets. It is not representative of the “black swan” events that occur more often than predicted. Since the original introduction, many variations of GARCH have emerged. These include Nonlinear (NGARCH), which addresses correlation and observed “volatility clustering” of returns, and Integrated GARCH (IGARCH), which restricts the volatility parameter.
We assume that this is due to the worse predictive power of the GARCH model in turbulent periods. In addition, the specific form of the conditional variance does not affect the QML in the above form. This opens up the possibility of using much more complicated nonlinear forms, such as NN (Goodfellow et al., 2016).
- This is rather undesirable behavior due to the use of a non-linear approach.
- We used a rolling-window estimation approach (Zanin & Marra, 2012).
- We tested a framework where the model was fully reset (random weights fully initialized) less often than with each timestep forecast.
- We look at volatility clustering, and some aspects of modeling it with a univariate GARCH(1,1) model.
- However, there exist investments which have considerably more risk and create opportunities for profit through that risk.
GARCH models are viewed to provide better gauges of risk than can be obtained through tracking standard deviation alone. These properties of the ARCH(1) model match many of the stylized facts
of daily asset returns. The half-life is log(0.5)/log(alpha1 + beta1), where the units will be the frequency of the returns.
In other words, it has conditional heteroskedasticity, and the reason for the heteroskedasticity is that the error term is following an autoregressive moving average pattern. This means that it is a function of an average of its own past values. Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) is a statistical model used in analyzing time-series data where the variance error is believed to be serially autocorrelated.
It gave us tools in the EDA phase to recognize when to use the ARCH and GARCH models. We fit models and built forecasts that could be done on a rolling basis to predict the volatility in the most practical manner. Lastly, we ended on an exploration of what can we do to improve our model by subtly changing our assumptions. We then generate autocorrelation (ACF) and partial autocorrelation (PACF) plots to visualize how previous returns with different lags are correlated with one another. Recall that because of properties of stationarity, the correlation and covariance at different points in time should be the same for the same gap — also know as lag h — regardless of the exact point in time t. That is, ACF and PACF should be constant across the same lag in time and should depend only on the lag h and not directly on t.
Autoregressive conditional heteroskedasticity
However, in this case, we specify that there is actually heteroskedasticity (non-constant variance). Since volatility is conceptually linked with variance, garch we are interested in modeling how the variance changes over time. In the stock market, volatility refers to the amount and frequency of price fluctuations.
2.1 Statistical Properties of the GARCH(1, Model
Below, we see how well this rolling volatility prediction is able to capture the volatility in our unseen test data. The model we will use for this is called auto-regressive conditional heteroskedastic model of order P, ARCH(P). The efficient market hypothesis states that the price of a stock is reflective of an efficient market — that is, all information about the company is reflected in the current value of the stock. While controversial, this view is at the basis of a lot of economic theory. It implies that consistently beating the market is impossible based on information about the price of a security up until time t, since all information is available publicly through time t. In the previous post, we concluded that the closing price of the S&P 500 could be modeled adequately by a random walk model.
This is rather undesirable behavior due to the use of a non-linear approach. The GARCHNet model with a normal distribution appears to have the lowest ABLLF cost function value among the GARCHNet models and can usually compete with the same cost function calculated for its GARCH family counterpart. In summary, based on the cost function results, we assume that GARCHNet at this stage is a relatively conservative model. The results converge across the index tested—with noticeable differences, but these are due to the distribution of the data rather than the model specification. One of the most influential drivers of risk is variance, particularly its changing temporal structure or tendency to cluster (Cont, 2002).
Where n is the size of the data, S is the sample skewness (which measures the symmetry of the data relative to the mean) and K is the sample kurtosis (which measures the shape of distribution, especially the tails). We implement the process by using a training and testing data split of 80%/20%, with more recent historical data serving as the test set. The above ACF and PACF plots do not indicate that there are ARCH and GARCH effects present.
The standardized residuals from a model assuming a normal distribution will be closer to normally distributed than the residuals from a model on the same data assuming a t distribution. The in-sample https://1investing.in/ estimates of volatility will look quite similar no matter what the parameter estimates are. Figures 1 and 3 would not change much if we changed the parameter estimates for the respective models.
We use arch_model() from the arch package and specify that the data is of mean zero and modeled with a GARCH process. We specify the arguments of P,Q as 1,1 and choose not to standardize (“rescale”) the data. After the above transformation to absolute returns, there is now a pattern indicative of ARCH and GARCH effects being present. In the PACF, there is not a big drop until about lag 4, which might indicate that we should have an order of P,Q up to 4. For simplicity and illustration we will use a GARCH(1,1) model before assessing and testing different specifications.
3.1 The Likelihood Function
The assumption of the distribution when fitting the model does have an influence even when using the empirical distribution. The persistence of the model is a key driver of the predictions — it determines how fast the predictions go to the unconditional volatility. If there really is a lot of persistence in the volatility and your model accurately captures the persistence, then you will get good predictions far ahead. The persistence is estimated by seeing how fast the decay looks during the in-sample period. If there is a trend in the volatility during the in-sample period, then the estimator “thinks” that it never sees a full decay. The shorter the sample period, the more likely there’s a trend that will fool the estimation.
Uncertainty in financial markets has been a main point of risk-related research for decades (Segal et al., 2015; Vorbrink, 2014). The market standard, which was established more than 30 years ago for use as a measure of risk, is Value-at-Risk (VaR) (Duffie & Pan, 1997). It is the simplest way to express potential losses over a target time horizon with a specified statistical certainty.
The aforementioned studies, however, do not specifically focus on the implementation of NNs for conditional variance alone. For example, studies by Kristjanpoller and Minutolo (2015, 2016) use GARCH estimates of variability as inputs to the NN model, while Kim and Won (2018) build NNs with covariates that are parameters of artificially generated GARCH models. Our approach leans toward estimating conditional moments of an assumed distribution using NNs, such as in Rothfuss et al. (2019). The first to propose such approach were Nikolaev et al. (2011), who investigated an approach with recursive NNs (RNN) to represent conditional variance and found that incorporating nonlinear methods (RNN-GARCH) reduces model uncertainty. Further on, Liu and So (2020) consider using the LSTM NN to model conditional variance directly through the maximum likelihood approach of the density function of the assumed distribution. They showed that this method can successfully determine both the standard deviation and variance of financial returns.
Specifically, the name of such a test is the Traffic Light Test (Costanzino & Curran, 2018). In the case of VaR at 2.5% significance level and 250 testing instances the ’safe’ (green) zone ends with 10 exceptions (95% cumulative probability) and yellow (warning zone) ends with 16 exceptions (99.99% cumulative probability). Financial professionals often prefer the GARCH process because it provides a more real-world context than other models when trying to predict the prices and rates of financial instruments.


Have you tried our mortgage offset calculator. This is a powerful tool that shows how extra repayments and money held in an offset account helps pay your mortgage quicker.
Here are a few examples;
1. Jim and Mary have a $300,000 loan over 30 years at an interest rate at 7% per annum. They place $10,000 in their offset account over the life of the loan and they save $63,043 in interest and take 3 years off their loan duration.
2. Tom has a $350,000 loan over 30 years at an interest rate at 7.25%. He pays $300 extra into his loan each month above the minimum repayments. Over the life of the loan he saves $165,257 on interest and reduces his loan term by nearly 9 years
3. In the above example with Jim and Mary and they make extra repayments of $200 per month plus have $10,000 in their offset account they would save $153,056 on interest and pay out the loan in 21 years and 2 months.


So um I take notes in class and typing them out helps me remember. So um im going to type them out to the box. Maybe you guys can learn stuff from me or from others so post your school notes!
A complete circle is 360 degrees.
A half circle is 180 degrees.
Fourth of a circle is 90 degrees and also called a quadrant.
There are two circle directions they are clockwise and counter clock wise.
A part of a circle is called an arc.
A horizontal line is level straight across .
A vertical line is level up and down.
A horizontal line connected to a vertical line forms a 90 degree angle or an perpendicular angle.
A 90 degree angle is also known as a right angle.
Parallel lines are lines moving in the same direction and are a constant distance apart.
Shapings are made into the hair.
A wave is two or more connected shapings moving in the opposite direction.
example of a wave or shaping in hair.


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For students who are in high school all through to college, having the right kind of calculator often makes the difference whether you are learning a new topic or doing a test. The best algebra calculators are designed with mathematical functions that students will find useful for solving scientific and mathematical functions. They also feature long-lasting, replaceable or rechargeable batteries for efficient and reliable use.
- Table of the top 10 best algebra calculators:
- 10. Texas Instruments TI-84 Plus Graphics Calculator, Black
- 9. Calculated Industries Scale Master Pro XE Advanced Digital Plan Measure
- 8. EL-W516XBSL Scientific Calculator
- 7. Canon 12-Digit Scientific Calc.
- 6. TEXAS TI-30XIIS 2-LINE SCIENTIFIC TEXAS INSTRUMENT OEM Calculators
- 5. Sharp Scientific Calculator Office Machines/Calculators & Counters
- 4. Orion T136X Talking Scientific Calculator
- 3. Texas Instruments TI-89 Titanium Programmable Graphing Calculator
- 2. Casio fx-9860GII Graphing Calculator
- 1. Texas Instruments TI-84 Plus CE Graphing Calculator, White
Table of the top 10 best algebra calculators:
10. Texas Instruments TI-84 Plus Graphics Calculator, Black
This calculator comes pre-loaded with software including the interactive geometry software Cabri Jr. It also comes with 10 graphing functions that have been defined, graphed, analyzed and saved. The advanced functions in this calculator are accessed through a pull-down display menu with a vertical or horizontal split screen. Furthermore, this calculator is equipped with Input/output port that enables communication with other TI products. It is powered by 4 AAA batteries and comes with lithium battery as the backup to protect the calculator’s RAM memory when changing the batteries.
9. Calculated Industries Scale Master Pro XE Advanced Digital Plan Measure
Designed to be used for everyday construction calculations, this calculator comes in Armadillo protective gear, CR2025 long life battery, and a pocket-size. It is equipped with solutions for over 140 building calculation costs and quantities of about eight commonly used construction materials. Additionally, this calculator is bundled with a Scale Master Pro XE digital plan measure.
8. EL-W516XBSL Scientific Calculator
This scientific calculator comes with over 535 advanced mathematical functions for trigonometric, statistical and other applications. Featuring WriteView technology, it is able to display formulas exactly as they are written in textbooks. Moreover, it also has Direct Algebraic Logic (DAL) which makes it easier to enter formulas into the calculator. It also has a multi-line playback function that enables easy editing of formulas.
7. Canon 12-Digit Scientific Calc.
This 12 digits scientific calculator comes with over 139 functions that solve trigonometric, statistical and other mathematical problems. It also has a graphing function with several graphing functions defined, graphed, analyzed and saved. Moreover, it has a multi-line playback function that is used when one needs to edit the formulas. It uses AAA batteries and is equipped with a backup lithium battery that protects RAM memory when the primary batteries are being replaced.
6. TEXAS TI-30XIIS 2-LINE SCIENTIFIC TEXAS INSTRUMENT OEM Calculators
This scientific calculator offers a 2-line display that enables you to see your formulas and results. It comes with several trigonometric, statistical and other mathematical functions including graphing function. With a premium build, this calculator is powered by triple A batteries. Moreover, like all TI calculators, it comes with a backup lithium battery which helps preserve RAM memory when the other batteries are being changed.
5. Sharp Scientific Calculator Office Machines/Calculators & Counters
This 12-digit, 2-line LCD display scientific calculator performs over 270 advanced functions. It has multi-line playback which makes it easier to edit formulas and view earlier formulas. This calculator is PTA endorsed for use in algebra, trigonometry, geometry, biology, chemistry and other general science calculations. It is also permitted for use in AP Physics, AP Chemistry, PSAT/NMSQT, SAT I/II, Math IC/IIC tests. It is powered by triple A batteries and has a backup lithium battery.
4. Orion T136X Talking Scientific Calculator
This scientific calculator is ergonomically designed to be used by visually impaired people. It is ideal for students tackling high school algebra all the way to college calculus. It offers clear and high-quality natural speech and teachers can easily help along. When a key is pressed, it is spoken as well and there is a repeat function that speaks out the display at any given time. There is an option for verbose or quiet operation and a unique learning model that allows for key identification without interrupting the operation. Moreover, it comes with over 127 statistical and trigonometric functions and formula correction function. It also comes with a rechargeable battery that lasts can last over 6 hours of continuous operation.
3. Texas Instruments TI-89 Titanium Programmable Graphing Calculator
This graphing calculator comes with pre-installed software with up to date functions. There are 16 pre-loaded graphing functions for engineering and mathematics coursework. Moreover, it has memory management and is able to create folders for specific subjects or applications. There is also an icon desktop that allows easy access to editors and applications. This calculator is permitted for use in standardized and state tests.
2. Casio fx-9860GII Graphing Calculator
This scientific calculator comes pre-loaded functions for solving trigonometric, statistical and graphing functions. It also has 1500 KB of Flash ROM memory which enables quick and efficient operation. Moreover, it has a built-in spreadsheet, eActivity applications and can be connected directly to Casio projectors for large screen display. With an icon-based menu, it is easy to access functions and applications and makes overall usage of this calculator easy and simple.
1. Texas Instruments TI-84 Plus CE Graphing Calculator, White
With a newer sleek design, this calculator offers the great TI-84 functionality in a crisp LCD screen. With several trigonometric, statistical and other mathematical functions, this calculator makes learning science and math easier and quicker. Moreover, it has a built-in MathPrint feature which displays formulas exactly as they appear in textbooks. It is ideal for use starting from middle school through college. It also comes with a rechargeable battery.
Whether you are preparing for a state/standardized test or learning a new math or science topic, these top best algebra calculators will make it easier for you. With multiple commonly used functions and clear LCD displays, you get to enter and edit your formulas easily. Moreover, these calculators are permitted for use in most state and standardized tests and can be used with a wide range of students in several levels. More: Best glass computer desks


Already in the not too distant past , was used to calculate the amount of fat you step on the bathroom scale BMI formula Us more obese , as a country , BMI calculator formula we looked for other ways to find the right scale It can not be a book , right? After all ,
some people are big boned , some people are short, some tall.Sure, BMI calculator formula you 're supposed to BMI formula calculate your weight in relation to height and same age, and while the bathroom scale has become a technologically advanced tool that allows you to store various figures of weight in his report speaks well you and optionally washed the dirty dishes , it does not take other important factors into BMI formula consideration.
At some point , the body mass index became BMI calculator formula the new scientific way to determine whether you were rich or poor. The formula BMI formula works like this:
BMI = weight in kilograms / (height in inches x height in inches ) x 703.
( 703 is to convert the original metric of the rate formula . )
BMI formula The Center for Disease Control said that a BMI of 18.5 to 24.9 is ideal.
Less than 18.5 is insufficient BMI formula, 25.0-29.9 means you are overweight, and 30.0 and that means you are obese .A mathematician named RAJ Lambert Queenliest came up with the formula for BMI in the nth century BMI calculator formula as a means of measuring the degree BMI formula of obesity in the general population.
He stated that BMI should not be used to indicate the level of fat in an individual. U.S. scientists have ignored the warning BMI formula .If used separately , BMI calculator formula you end up with some of those most physically fit on the planet that falls into the category of "overweight" - people like BMI formula Michael Jordan, Brad Pitt, Arnold Schwarzenegger (30 is considered " obese ! ") and Tom Cruise are on this list BMI calculator formula.
These celebrities can BMI formula weigh more than the average because they have denser muscle tissue , which is heavier than the "normal" muscle. In addition, strong bones are denser than fat in bulk, which could add points for body mass index .
Formula , meanwhile , assumes low muscle mass and high relative fat BMI formula.Thus, it seems science , the formula is FALSE and misleading , and for many people , quite inaccurate .
One thing that is not considered when looking at the body mass index is the waistline Dr. Oz, who appeared in the show BMI formula " Oprah " several times and is ready to launch his own TV show in the fall, spoke at length about the importance is the size of the size, it is the general health Site Oprah.com :
"Instead of focusing BMI formula on the number on the scale , Dr. Oz says to focus on the number on the ideal size for women size BMI calculator formula is 32 inches and 35 ½ inches for a man - . a goal that is achievable by everyone , regardless of age . "
Even a statement that seems to be very one size BMI formula fits all. Thirty-two and a half inches for all women ? And 35 inches for men?With a performance of " Oprah ," Dr. Oz has a BMI formula couple of copies of a body called momentum , which plays a key role in making us fat . The body stores fat around the stomach ,
where the body has an easy to access when or if necessary . But fat creates an inflammatory process that can endanger clogged BMI calculator formula arteries BMI formula.Ideally, the momentum BMI formula should be thin and lacy, not thick and greasy . This should result in a fine size instead of a " beer belly " .
Body fat can be measured more BMI formula accurately , perhaps , by a couple of other tests in addition to the BMI . You can use a fat caliper body to enter a fold of skin at certain points on the body and measure its thickness . BMI formula Then there is the art of " immersion " This method can based BMI calculation formula on the idea that lean body mass ( muscle, bone, etc) tends to sink in water, while fat floats .
A person is sitting in a chair that is in the balance. The person in the chair is lowered into a pool of water until completely BMI formula submerged , and the weight of the person ( immersion ) is recorded. The bigger you are, the more it tends BMI calculator formula to float and reduce your weight will be overwhelmed - the people muscle weighs more than fat people in immersion. This method is very accurate, but requires a lot of equipment BMI formula.One thing is certain : The body mass index is not particularly BMI formula useful as a tool to assess whether you are fit .
BMI formula You might as well return to the scale of the bathroom ; find to talk carefully to you and give you a little encouragement when the numbers are too high.Or completely abandon BMI formula the numbers ... a three-digit number has nothing to do , really, with its value.Roshana Ariel is the author of "I hate diet book ",
which is no longer in print. She is an editor of a newspaper, and is interested in BMI formula telepathy and other methods to make our lives better BMI calculation formula , including weight loss and a healthy diet. More information can be found
Body mass index (BMI) is a way to assess a person 's weight in relation to height . The body mass index is sometimes called BMI formula the Queenliest index with the Belgian pioneer Adolph Queenliest name Jacques Lambert has made withe system during the nineteenth century. Obesity or obesity level gravity of a BMI calculator formula person is BMI calculation formula determined primarily by their BMI using a BMI of formulas ( formulas BMI). BMI can also determine other levels of weight, BMI formula such as underweight, normal weight or overweight.
BMI does not measure actual body fat of a person.A predominant formula used to BMI formula calculate body mass index and obesity of gravity is in the International System of Units , commonly known as SI or metric units. This is the weight of a person in kilograms divided by height in meters squared , BMI formula presented as a BMI = kg / m². The " ²" simply means that you multiply the number twice . For example 1.5m ² yield a figure of 2.25 its only 1.5 x 1.5 .
BMI formula The rest of the formulas used to calculate the severity of obesity is based BMI calculation formula on the system of imperial units. The two methods will be mentioned BMI are similar to the above formula BMI calculator formula with a few twists BMI formula .The primary function of imperial units is calculated using weight BMI calculation formula in pounds ( lb ) multiplied by 703 ( seven hundred and three ) BMI formula and then divided by your height in inches squared .
This can be demonstrated that BMI = ( pounds x 703) / in ². What is in the parenthesis is what needs to be solved first, and then divided by the square inch.BMI formula Another way to calculate this is BMI = ( lb / in ² ) and multiplying by 703 to get.
BMI formula Second, based on imperial units is calculated using weight in pounds ( lb) , but this time multiplied by 4.88 BMI calculator formula ( four point eight eight) , and then divided by their height in meters squared. BMI calculation formula This can be demonstrated that BMI = ( 4.88 x pounds) / ft ². Another way to calculate the BMI = ( lb / ft ² )
and multiplying what BMI formula they get for 4.88 .One of three formulas BMI above , if the final result is 30 or higher, the person tested is obese BMI calculation formula and has a level of obesity associated BMI formula with body mass index . A body mass index ranging from 25 to 29.99 is classified as overweight. BMI less than 18.5 is considered underweight . The optimal BMI for individuals is 18.5 to 24.99 , which is where it should be BMI formula.
There are different classes that describe BMI formula the severity of obesity based on the BMI of a person. BMI calculation formula Class 1 is slightly obese obesity and BMI ranges from 30 to 34.99. Class 2 obesity is moderately obese with a BMI of 35 to 39.99 . BMI formula Class 3 is morbidly obese is above 40 BMI . More class severity of obesity, which are more vulnerable to weight related health problems and the most obvious BMI calculation formula and dangerous they become.To determine the severity of obesity in children BMI formula and adolescents, a BMI above formulas are used , but the result is then compared to what is normal for this BMI calculation formula age and sex boy in particular and gives a percentile .BMI formula You can see the graphics samples provided by the U.S. Center for Disease Control percentile for children with this link and this link for girls .
Under the fifth percentile means that the child is underweight BMI formula . Of 85 , just below the nth percentile means that the child is overweight. Above the nth percentile means that the child is obese. Between 5 and 85 percentile is the optimal level for children and adolescents.
BMI formula Having the ideal weight is very important not only BMI calculation formula for growth but also for your overall health. This is the most important consideration for you to be able to say that you are in fact healthy factor . The problem with most people these days is that they are so aware of the implications of BMI calculation formula going under or over their ideal weight BMI formula.They may lack knowledge BMI calculation formula about the importance of ideal body weight . And because of this lack of knowledge that many people think and feel they are in good health, even if they are overweight or underweight .
For this reason , it is very important that you learn how to measure the weight of your body and also whether your current weight BMI formula is ideal, if not, then you need to do something BMI calculation formula about it. Health , after all, is the most important factor one must possess .
How can you determine the body mass index in which you own ?
BMI calculation formula It is very simple to determine your own BMI and you just have to follow a simple formula that you can master in a short time BMI calculation formula without effort. It is this way.
A . The first formula you can use to check your BMI is :
X 703 BMI = weight (kg ) / height ( in ) - The number 703 is a constant number you multiply your current weight in pounds divided by its current height in inches squared .
Two . The following formula can be used to determine your body mass index is ;
4.88 X BMI = weight (kg ) / height ( M's) - In addition, the 4.88 value is constant in the formula
for you to be able to multiply your current weight in pounds and then divided into existing meters height square BMI calculation formula .
Three . The last formula can be used to determine your body mass index is :
BMI calculation formula BMI = weight (kg ) / height ( M's) - Similarly, you can determine your BMI by dividing your current weight in kilograms to height in meters squared .
Let's take an example to better illustrate how you can use one of the above formulas to BMI calculation formula determine your BMI.
Weight = 50 kg ( converted into pounds : 50 * 2.2 = 110 kg )
Height = 5.2 m ( convert to inches : 5.2 * 12 cm / 1 m = 62.4 * 62.4 = 3893.76)
X 703 BMI = weight (kg ) / height ( in )
BMI = 703 X 110 lb / 3,893.76
BMI = 19.68
Thereafter, the value obtained will determine if your BMI is normal or not . To do this, you need a simple chart that will tell you exactly which category fall BMI calculation formula.If the BMI value obtained was 18.5 and below which means that you are underweight or malnourished . If the BMI value BMI calculation formula is obtained from 18.5 to 24.9 , then your BMI and weight is normal , if the value is 25.0 to 29.9 then you are considered overweight .
On the other hand values walkover 30.0 to 40.0 is considered overweight while a little morbidly obese . The only reason you need to determine the body mass index is to BMI calculation formula know which category you fall into this way to learn how you can manage your weight problem. Also ask if you need to gain weight or lose weight this way you can see your body mass index close to stay BMI calculation formula healthy .


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The Difference Between Math and Arithmetic: How to Be Great in Mathematics
What exactly is the distinction between mathematics and arithmetic? Both math and arithmetic are crucial in everyone’s daily lives, but recognizing their distinctions will help you succeed at mathematics and handle math issues successfully and easily. Take some time to learn about how math and arithmetic differ, how they relate to each other, and how you can use this information in your life as well as your education. You’ll be glad that you did when you see the results!
Math and arithmetic are both crucial components of understanding our surroundings, but they are not the same thing. Math is a broad field of study that covers numerical problem-solving, abstract reasoning, and the analysis of patterns and relationships. Arithmetic, on the other hand, is the practice of manipulating numbers to solve problems. It focuses on fundamental mathematical computations such as addition, subtraction, multiplication, and division. Math encompasses a wide variety of topics, including algebra, calculus, geometry, and statistics. While these topics all involve numerical computations, they also involve higher-level reasoning and analysis. See, this website has all the info you need to learn about this amazing product.
Arithmetic, on the other hand, is much more focused on numerical calculations. The basic operations like addition, subtraction, multiplication, and division are all included. The main goal of arithmetic is to be able to quickly solve problems involving numbers without having to reason or analyze complicated equations. This makes it simpler for kids to comprehend fundamental math ideas before going on to more challenging ones. One needs a strong foundation in both math and arithmetic to be a smart mathematician. Understanding how and why certain equations work is just as important as memorization of formulas and equations. With a strong foundation in both math and arithmetic, you can begin to tackle more difficult topics such as calculus or statistics. You can read more on the subject here!
Math application in our daily lives is a valuable weapon to have in your armory. If you want to master math and arithmetic, finding a tutor can be a great way to go. With a tutor, you’ll have someone who can guide you through difficult topics and provide feedback on your work. A tutor can also help you with test prep, problem solving, and other areas of math and arithmetic. When looking for a tutor, it’s critical to choose someone with experience in the subject matter. This could include teachers, professors, or even tutoring companies that specialize in mathematics. It’s important to choose someone who is able to explain concepts in an easy-to-understand manner and will be patient as you learn. Additionally, be sure they offer a variety of services that fit your needs and budget.
Finally, remember that finding a tutor isn’t a one-time thing. Make sure you keep up with your studies by scheduling regular sessions with your tutor. Making the most of your tutor’s expertise can help you stay on top of your academics. This page has all the info you need.


Is Problem Solving an Art or Science?
Problem solving is as much a science as an art.
Problem solving is taught and deemed as numbers game.
The general practice is to fire-up Microsoft Excel (or its equivalent), list parameters to evaluate, give weightage to these parameters and then list out numbers in favor or against each of the options. Every problem solving method is a derivative of SODAS methodology.
But does this cover all aspects of problem solving?
Let us take a simple example of buying a tablet. For simplicity, let us limit the choices between Apple iPad and Samsung Tablet.
Factor | iPad | Samsung Tab


7 Simulated concentration-time profiles for escalating IV bolus doses (0.1, 0.02, 0.002 nmol/kg) of darbepoetin (DA). TMDD model for just two drugs contending for the same receptor continues to be suggested. The explicit answers to the equilibrium equations make use of complicated numbers, which can’t be solved by pharmacokinetic software conveniently. Numerical bisection algorithm and differential representation were established to resolve the functional system rather than obtaining an explicit solution. The numerical solutions had been validated by MATLAB 7.2 solver for polynomial root base. The applicability of the algorithms was showed by simulating concentration-time profiles caused by exogenous and endogenous IgG contending for the neonatal Fc receptor (FcRn), and darbepoetin contending with endogenous erythropoietin for the erythropoietin receptor. These versions were applied in Phoenix WinNonlin 6.0 and ADAPT 5, respectively. and (RB) or (QSS) [11, 12]. Therefore, the concentration from the drug-target complex could be expressed being a function of free medication concentration  explicitly. When two molecular types bind towards the same focus on competitively, the focus of drug-target complexes from two types can be portrayed with regards to free of charge Lp-PLA2 -IN-1 medication concentrations through the Gaddum equations . The computation of free of charge medication focus for the RB or QSS TMDD model for just two molecular species contending for the same focus on is mathematically complicated. When a one medication binds to its focus on, the free of charge medication concentration could be computed by resolving a quadratic formula and explicitly portrayed under RB Rabbit polyclonal to PBX3 or QSS assumption . Nevertheless, when two types of substances compete for the same focus on, the free concentrations of the two species are answers to a operational system of two quadratic equations with two variables. This provokes a problem in applying such a model, in PK software especially. In the next sections we presented the speedy binding TMDD model explaining two medication species contending for the same focus on. Our main objective was to propose different methods solving the operational system equations that may be emulated in PK software. The use of these procedures was showed through two case research regarding erythropoiesis-stimulating agent contending using the endogenous EPO for erythropoietin receptor and a monoclonal antibody contending using the endogenous IgG for FcRn. Theoretical The TMDD model for just two drugs contending for the same receptor is normally proven in Fig. 1. The notations and symbols of the super model tiffany livingston act like the overall TMDD super model tiffany livingston with one medication . As proven in Fig. 1, the main element feature of the model is normally that two molecular types (and and and and and and and and and and and so are dissociation equilibrium constants for medications A and B, respectively. Upon presenting the total medication plasma concentrations: +?RCand +?RC+?RC+?RCand could be calculated from Eq. 19 through free of charge and total medication concentrations, or equivalently, from Eq. 18 simply because functions of free of charge medication concentrations and and so are the just solutions from the equilibrium equations Eq. 18 rewritten the following: (+?=?+?=?will be the baseline plasma concentrations for total medication A, total medication B, and total receptors, respectively. For the entire model, the receptor synthesis price can be computed from Eq. 26: and denote the baseline beliefs from the drug-receptor complicated concentrations that may be computed in the Gaddum equations: and one must solve the equilibrium circumstances Eqs. 27 and 28 at baseline beliefs for and +?and =?0 (43) where =??(2 +?+?+?+?=?1 +?2+?2+?+?+?+?+?=??(+?+?+?= resolving the equilibrium equations Eqs. 27 and 28 could be reduced to locating a reason behind a quadratic formula: =?0 (57) where =??(1 +?+?+?=?+?= resolving the speedy binding TMDD model using the explicit alternative is supplied in the supplementary materials. Pc simulations of TMDD model with two medications contending for same receptor had been performed using the MATLAB m-function = and = = = and so are similar, Lp-PLA2 -IN-1 resembling the TMDD model with one medication. When = 1 and = 0.1, weighed against the simulation using = = 1, the (free of charge medication concentration for medication B in = 0) decreased instantaneously, whereas the (free of charge medication concentration for medication A in = Lp-PLA2 -IN-1 0) increased instantaneously. That is because of the more powerful receptor binding affinity of following the equilibrium. Since much less receptor are for sale to medication A, boosts. The Lp-PLA2 -IN-1 difference between and.


A school performs the necessary function in any child’s life by providing him or her with the educational basis that will guide the student in his or her career later in life. However, more often than not the students face serious problems in grasping what is being taught at school. This can happen due to a number of factors. For instance, a child may not clearly understand the instructional style of the teacher or may have problems with the subject matter itself. In such cases, a Darwin private tutor can help the student with the lessons in school and aid him or her in securing better grades.
The English language is the most widely spoken language in the world and therefore it is necessary that a student develops proficiency in its written as well as spoken forms. If a student is not being able to bring good grades with English, it is important that the child’s parents get in touch with an experienced English tutor Darwin who can guide the student in the best way possible regarding the various simple and advanced aspects of the English language studies. This can help the student to do well both in the current standard as well as in advanced classes.
Likewise, a Darwin math tutor can be just what a student needs if he or she has been facing problems with mathematics. The subject of mathematics is dreaded by many since they imagine it to be incredibly complex. However, like with any other subject, one can develop proficiency with mathematics by practising it regularly. Developing a finer understanding of mathematics can also help those students who are looking to take up science and related subjects in the advanced classes. The best mathematics teachers in Darwin can also help the students to do well in younger classes.


What will this cost me?
There are two aspects of gaming in how many trophies you got from my perspective. One being time and the second money/budget. Currently I don’t have as much time to play as before but on the other hand I have money to buy games, which was not the case when I was younger. But when I was younger I had all the time the world, but no as much money.
As I posted before I made a rough calculation on how long it would take for me to become Top 100 on PSNProfiles. So I have “cost” in time.
But how much will this cost me in money?
So let’s play with some numbers again!
I need to complete 184 Games to get to Top 100 (everything else equal)
I have Playstation Plus Extra, which gives me a free backlog of roughly 40 Games.
Then we are down to 144 Games needed.
But let’s guess that each monthly game from PSN+ is a new one for me.
This means 12 games a year. (times 8 years), will give me 96 games.
Then we are down to 48 games that I need to buy.
To make the calculation somewhat simple, each game will cost retail price of $70 (I would however look for deals, so the real number should be smaller)
So 48 Games x $70 = $3 360 is what this goal will cost me.
As with the timeline, I will use this cost as a baseline and see how I can improve it.
So in 8 Years for the cost of $3 360 I will becoming Top 100 in Sweden on PSNProfiles!


There are many different ways that will make your math learning a lot of fun, and it is important to kids because it’s sometimes tough to keep the attention focused at one thing for longer time frame. For the young kids the best method is either buying the math cards or to make them yourself.
How to make it fun?
You will make learning math a lot of fun just by turning the math problems in the game. You can make the game like using various items to represent different types of the currency amounts in the country. Have extra items of the currency and see which will become richest, just by answering some problem correctly. Lots of children do better whenever they will do some actual hands on the math problems; thus the best way to make the learning more fun is using the tape measure or ruler or has this search through house for the spoons that are same size. Or you can take help of the o level a math tuition that will use tape measure for teaching them the fractions too.
One good idea for to make math learning fun; is in case your kid has an access to computer, then you may go on internet and find a few math games that can come very close to the math potential. It won’t just help them to have fun learning the math problems but help them with the computer skills too.


CBSE New Rule For Class 10 Maths Examination 2020 Basic and Standard Mathematics Full Details Summary : From academic year 2019-2020 cbse is going to introduce two types of Mathematics examination for class 10, cbse named it as Mathematics Basic & Mathematics Standard. We are going to tell you more about this new circular of cbse in detail.
What Is Mathematics Basic & Mathematics Standard ?
Cbse said that there will be two types of maths question papers in examination of class 10 from 2020, one will be paper of Mathematics Standard and other will be paper of Mathematics Basic. Question level for the Mathematics standard will be same as of now and the question level for Mathematics basic will be easier as compared to Mathematics Standard.
A student can choose any one of the two papers while filling the registration and examination form which is usually done in the months of October and November. The mathematics basic is introduced for those who are not will to take maths in their class 11.
Below is the table that will clear all the doubt and confusion regarding the Cbse new Rule for Class 10 Maths Examination.
|Level Of Examination
|Same Of Now
Table above show the clear picture and difference between these both level of examinations.
What if Any Student gets Compart or Fail ?
Cbse has plan for those who are not able to get passing marks in any of math examination, those who opted for Mathematics – Standard can choose any one of two papers i.e bsic and standard for their compart examination and those who choose mathematics basic can only choose basic paper for compartmental examination.
Table below will make it more clear :
|In case of failure in
|Math - Basic
|Math - Basic
|Math - Standard
|Math - Basic Or Math - Standard
We Provide study materials for Class 1 to Graduation in Our education section, have a look at our EDUCATION SECTION, it will help you alot.Hope we have clear your confusion , if you still have any doubts, please ask us in comment section .


Belmont Hill welcomes Mr. Hargrove as a new math teacher and sports coach. He teachers Algebra I, Algebra II, and Geometry, and he will be the assistant coach for the third hockey and lacrosse teams. Mr. Hargrove joins us from the Salisbury School in Connecticut, where he taught for five years. He attended Strath Haven High School in the Philadelphia suburbs, where he played soccer, hockey, and lacrosse.
Then he went to Williams College for his undergraduate studies and to Tuck Graduate School of Business at Dartmouth, where he played lacrosse. Volunteering on the third lacrosse team in 2017, Mr. Hargrove really enjoyed being a part of the Belmont Hill culture and community, and those experiences led him to join the school. He has no pets but hopes to get one when his kids are a little older.
In his spare time off campus, he likes to spend time with his wife and three daughters Leah (7), Gracie (5), and Emma (3).


Fact Family Turkey Craft. Their goal was to make the turkey with the most plumage. • 1st and 2nd grade addition and subtraction fact families
Included are a variety of feathers that can be printed on. Check out our math fact families selection for the very best in unique or custom, handmade pieces from our shops. • 1st and 2nd grade addition and subtraction fact families
• 3Rd And 4Th Grade Multiplication And Division Fact Families.
To add variety to everyone’s turkey, run the feathers off on lots of different colors and give your students a choice. They teach basic addition and subtraction. Roll the dice and create a turkey fact family craft using the numbers.
Touch Device Users, Explore By Touch Or With Swipe Gestures.
Scribd is the world's largest social reading and publishing site. After, they will write the 4 number sentences that belong in. Add a real feather to the turkey’s head for that finishing touch.
After, They Will Write The 4 Number Sentences That Belong In.
Allow students to demonstrate their fact family skills by creating more and more and more feathers for our friend tom the turkey! Feel free to use the black and. There is a page for addition/subtraction fact family and a page for multiplication/division fact family.
Then They Will Write The 3 Numbers That Are In The Fact Family On The Turkey.
Well, a fact family is made of 3 numbers. Why are fact families important? Then they will write the 3 numbers that are in the fact family on the turkey.
To Add Variety To Everyone’s Turkey, Run The Feathers Off On Lots Of Different Colors And Give Your Students A Choice.
Plus, it makes for a cute bulletin board! Allow students to demonstrate their fact family skills by creating more and more and more feathers for our friend tom the turkey! Their goal was to make the turkey with the most plumage.


Alma Mater: Villanova University
Hometown: Acton, MA
I took several classes in college on the mathematics behind financial derivatives and option pricing models. I knew that after college I wanted to apply my math skills to financial markets. The Operations Analyst Program especially is a great transition from college to a career because of the emphasis on learning and education both from classroom sessions and from other SIG employees. With each rotation, I was able to shadow other group members to learn about different products, functions, technical skills, and day-to-day work.
Through the rotational program, I was able to work with people in offices around the world and I’ve been able to leverage those relationships in my current role. I’ve also worked on a lot of rewarding projects that I was able to see through from start to finish. Being able to take a problem or a process that was entirely manual, come up with a solution, and write all of the code in an efficient and logical manner gave me a great sense of accomplishment.
Copyright © 2024 SIG Susquehanna. All rights reserved. Susquehanna Financial Group, LLLP (SFG), an affiliate of SIG, is a member of FINRA.
Copyright © 2024 SIG Susquehanna. All rights reserved.


I think that I have gotten to the point where I have become skilled in drawing Isometrics, but I have yet to fully understand the "triangle method of drawing circles". I just plot out the square and the midpoints of each segment, skip the triangles, and then draw the isometric circle using 2 different circles. Or just find a template that matches. But I have been trying to work on my line weight to try to make it as consistent as possible. This was my first check plus and I was so happy when I found out!


Jaya Mathew Senior Data Scientist, Microsoft
Jaya Mathew is a Senior data scientist at Microsoft where she is part of the Artificial Intelligence and Research team. Her work focuses on the deployment of AI and ML solutions to solve real business problems for customers in multiple domains. Prior to joining Microsoft, she has worked with Nokia and Hewlett-Packard on various analytics and machine learning use cases. She holds an undergraduate as well as a graduate degree from the University of Texas at Austin in Mathematics and Statistics respectively.


Exhaust pipe diameter calculator. There are many opinions on this matter some choosing a 10-15 increase from. Or Philip A Smiths Scientific Design of Exhaust and Intake Systems. Typical exhaust pipes are 18ga. Heres the formula to figure out your primary header tube diameter. Then divide Figure by Figure B and you have pipe length in inches. Enter the diameter of your chimney pipe to find out how long it should be.
Too large might not provide any benefits. When youre taking your. In the base model C6 NPP mufflers were 25 in diameter vs 3 in the Z06. The following calculator can be used to calculate the header dimensions best suited to your combination. The formula to calculate pipe size is. Som e exhaust systems use different diameter tubing at different points along the path. A flexible exhaust connection has three primary functions. You just have to do the steps in the correct order.
Determine the correct diameter of exhaust pipe you should use on your car by consulting a general guide for pipe sizing. Or Philip A Smiths Scientific Design of Exhaust and Intake Systems. Notes 2 A reduction in primary pipe Diameter of 18 of an inch will move the torque peak down by 650-800 RPM. Looking into the the exhaust diameter would you recommend the 225 or bump up for the 25.
Looking into the the exhaust diameter would you recommend the 225 or bump up for the 25. For exhaust systems with a diameter of 200 mm 8 in and larger. An 18 change in primary diameter will move it 650-800rpm up or down. Exhaust pipe diameter has a significant impact on engine characteristics. Exhaust fumes flow in one direction from your engine back to the tailpipe. Som e exhaust systems use different diameter tubing at different points along the path. 2 primary and 2 secondary.


An online personal loan EMI calculator helps in calculating your EMIs in a hassle-free manner. This online device with its simple and well-designed algorithms allows you to know the exact calculations of your monthly installments, interest outgo and the total amount payable. This tool comes with a lot of features like accuracy, time-saver, easy comparisons, adjustable, etc. The tool starts functioning the moment you feed-loan amount, tenure and interest rate into the same. And, gives the correct values in the least time frame. Thus, it helps in taking a calculative decision as far as availing a personal loan is concerned.
What is a Personal Loan EMI Calculator
An online tool that can be used to calculate EMIs, Interest Outgo and Total Amount Payable towards a personal loan. It works on a well-designed algorithm wherein it simplifies the values and shows the results quickly. The device starts to function the moment you feed:
Yes, soon after feeding the credentials above, the device churns out the accurate values, making your personal loan a hassle-free experience.
Components of EMI Calculator
Principal Paid:This is the portion of your monthly payment that is applied towards the loan principal. And, this portion keeps on increasing each month as the loan matures.
Interest Paid:This is the interest portion of your monthly payment that is applied towards interest. This portion will keep on decreasing every month as the loan matures.
Total Payment:It is a total sum of principal and interest paid.
Outstanding Loan Balance:The remaining loan balance at any given period corresponds to the principal amount that is owed to the lender at the end of that period.
What EMI Stands for
Equated Monthly Installment or commonly known as EMI is the amount that you pay every month to the lender to pay off your loan.
EMI Consist of Interest on the loan along with the principal amount.
However, the total sum of the principal amount and interest is further divided by the tenure of the loan.
How Personal Loan EMI Calculator Works
An online EMI calculator for personal loan works on a simple phenomenon and lets you know the values faster. The smart tool basically works on two basic formulae.
The first one is used to calculate the monthly interest rate while on the other hand the second one is used to calculate the monthly installments.
Let’s get into the detail and understand the functioning of both.
How to Calculate Monthly Interest
Whenever you apply for a personal loan online, it would not be wrong to say that interest rate is the first and foremost thing that grabs the attention of all. So, before you start using the EMI calculator, you need to convert your annual interest rate into the monthly interest rate. And, in order to convert that, below formula is used:
r=annual interest rate/12
For example- If a bank offers you 18% of interest rate annually, your monthly interest rate will be:
Thus, 1.5% will be your monthly interest rate for personal loan
How to Calculate EMI
Calculating the EMI is a bit difficult as compared to calculating the monthly interest rate. To calculate your EMI, you need to use the below formula:
E = P . r . (1+r)^n/((1+r)^n – 1)
P=Principal Loan Amount
r=Monthly Interest Rate
n=Monthly Loan Tenure
For example- If you have borrowed a sum of Rs. 10,00,000 from a bank at an annual interest rate of 10.5% (10.5%/12=0.875 monthly) for 10 years ( 10×12=120 months), your EMI will be:
Rs.10,00,000 *0.00875 * (1 + 0.875)120 / ((1 + 0.875)120 – 1) = Rs. 13,493
Now, you will pay Rs. 13, 493 for 120 months to repay your personal loan amount. Thus, your total payable=13, 493×120=16,19,220.
Features of Personal Loan EMI Calculator
Accuracy: This device is so accurate that it will perform the calculations in a few seconds.
Time Saving: It does complex calculations in just a few seconds thus saves a lot of time of the user.
Easy Comparisons: EMI calculator allows you to have easy comparisons. Yes, you can compare different lenders easily, and know different EMIs offered by them.
Adjustable: An EMI calculator can be used endlessly which means you can use it many times. Also, you can re-adjust the settings, and can get the results endlessly.
Tells You More Than Just the EMI: The calculator helps you gain more information about your loan. Yes, apart from calculating your monthly EMIs, the device showcases interesting graphic representations, pie charts ,and tables.
Factors Affecting Personal Loan EMI
Loan Amount: The higher loan amount you demand, the higher EMIs you need to pay. However, the maximum loan amount is decided by the lender based on your personal loan eligibility factors.
Interest Rate: The rate of interest is directly linked to the EMIs. If the personal loan interest rates are higher, your EMI amount is also higher. Well, your interest rate is determined by several factors like income, cibil score, credit history, repayment capacity, etc.
Loan Tenure: The personal loan tenure that you choose is inversely proportional to the EMI. The shorter the tenure, the higher the EMI amount. A personal loan tenure ranges between 1 year to 5 years.
Ways to Reduce Personal Loan EMI
- Go for a Low Interest Rate: The lower interest rate you have, the better as it keeps your EMI pocket-friendly. Both of them are directly proportional so if your rates are lower, your monthly installments are also reduced.
- Opt a Suitable Tenure: If you opt for a longer tenure, EMI payable would also be reduced. Yes, both of them are inversely proportional. So, expect to have a reduced EMI burden with longer personal loan tenure.
- Category of the Employer:In case you work in a leading MNC or Fortune 100/500 company, it would be easy for you to get a personal loan at attractive interest rates. And, by getting a loan at competitive interest rates, you can expect to have pocket-friendly personal loan EMI.
- Get Balance Transfer: Opt for personal loan balance transfer and enjoy affordable EMIs. Simply transfer your existing personal loan to a lender offering a lower interest rate and enjoy a reduced monthly installment.
Frequently Asked Questions (FAQs)
Q.Should I enter the monthly interest rate or yearly interest rate in the EMI calculator?
A.You have to enter the yearly interest rate in the online EMI calculator to get the precise EMI amount.
Q.How long does it take to calculate the personal loan EMI?
A. One cannot describe the exact timing as each personal loan calculator is different. But a calculator roughly takes a few seconds to calculate the EMIs on time.


Maths at Lordship Lane
Mathematics is, of course, a fundamental part of every primary school's curriculum because of the profound importance of the subject. At Lordship Lane, we aim to develop children's maths skills and understanding so that they are comfortably ready for the next stage in their education. This includes developing a high level of arithmetic fluency in addition to developing strong problem-solving skills.
Lordship Lane Primary Schools aims to prepare our children for their future success, providing them with the essential knowledge required to encompass a Cultural Capital – in the world of work, in relationships forged throughout life and as a valued contributor to society. Many children begin their primary education with a range of limited experiences, which is often evident during their time in EYFS. Therefore, our curriculum for mathematics aims to provide our children with the opportunity to link their learning to real-world problem solving, whilst also ensuring that they are equipped with the relevant mathematical vocabulary.
We ensure that contextual learning takes place during enrichment days with focus on cross-curricular links, such as P.E, Science, History and Geography, which enables our children to form direct links to their wider experiences outside of the school environment.
At Lordship Lane, we use the Maths-No Problem! scheme for teaching mathematics. This is inspired by the maths approaches used in Singapore - one of the highest performing countries in mathematics in the world! We have chosen this mastery approach because we believe it is very well-organised and offers children the range of maths experiences that are so essential for them to make good progress.
All children have a maths lesson every day across the school. These maths lessons make extensive use of concrete resources, not only in younger year groups but across the school. This is because we understand their importance in the development of children's conceptual understanding. All maths lessons also contain elements of direct teaching, guided practice (when children learn by working together with others or the teacher) and independent practice (when children practice on their own).


Students in the Scientific Computing and Numerical Analysis group receive PhD degrees from the Department of Computer Science at the University of Illinois at Urbana-Champaign.
Are you interested in numerical algorithms? Topics like finite elements, multigrid solvers, integral equations, spectral element methods, fast algorithms, large-scale applications, or high-performance parallell algorithms?
Students in our group pursue research in areas ranging from computation to mathematical theory. We collaborate closely with our colleauges in the numerous excellent engineering departments at Illinois, as well as with the Physical and Social Sciences.
The PhD requires 64 or 96 hours of credit, depending on whether you enter with a masters or a bachelors degree. The specific Program of Study (POS) is felxible, but generally our students take Numerical Analysis (CS450), the 500-level courses in our area, and some courses in parallel computing, theory, and engineering. In addition, the Computational Science and Engineering (CSE) Certificate Option is often pursued.
We typically have around 20 graduate students in Scientific Computing (check out our alumni over the last several decades!) Our goal is a fun and positive atmosphere in Scientific Computing to help you take classes, study for the qualifying exam, and collaborate on research toward your PhD.
We maintain close connections with major national and international research centers in numerical analysis and computing. For example, many of our students find it useful to collaborate with scientists at Department of Energy national laboratories such as Lawrence Livermore, Argonne, Sandia, and Los Alamos.


This will give you a brief overview of calculus. This will not replace your calculus textbook. This will help you understand what you are learning in class, and why. When I first learned calculus we focused on the mechanics of solving problems. I wish we’d spent more time discussing why we were learning these things and what they meant.
You need to remember some algebra to understand this document. Here is a quick refresher of the relevant facts. You will need a lot more algebra when you are ready to actually solve calculus problems.
You should be familiar with graphs of functions. Graphs are the easiest way to explain and understand calculus.
2 - x2Click on any graph for an interactive version.
The dark horizontal line is called “the x axis.” The numbers along this line are the inputs to the function. These are often things that you can change.
The dark vertical line is called “the y axis.” the numbers along
this line are the outputs from the function. If you give your computer or
calculator an input to a function, it can compute the output of that
function. Sometimes we write
f(x) (and say say “F of X”)
instead of “y.”
Slope is defined as “rise over run.”
f(x) = 2x ÷ 3
In graph above you can see that the function moves
2 units up every time it moves
3 units to the right. So the slope is
2 ÷ 3, or ⅔.
f(x) = 2x ÷ 3
You could also say that the function moves
4 units up every time it moves
6 units to the right. And it moves
6 units up each time it moves
9 units to the right.
2 ÷ 3,
4 ÷ 6, and
6 ÷ 9 all give
you the same answer, ⅔.
In the next graph notice that
g(x) = 2x ÷ 3 + 1
has different values than
f(x) = 2x ÷ 3,
but they both have the same slope. In fact there are an
infinite number of lines with that
f(x) = 2x ÷ 3,
g(x) = 2x ÷ 3 + 1, and
h(x) = 2x ÷ 3 + ⁇
The next graph shows a line going down, so it has a negative slope. You
can see that the function moves
2 units down
every time it moves
3 units to the right. So the slope is
(-2) ÷ 3, or -⅔.
f(x) = -2x ÷ 3
A horizontal line has a slope of 0.
f(x) = 1.5
In this case the “rise” will always be 0. The “run” can be any number I
choose, let’s say 42.
0 ÷ 42 = 0 So the slope is 0.
If the line is vertical, its slope is undefined.
x = -1.6
In this case the “run” is always 0. And anything ÷ 0 is undefined. (Or is it? Keep reading!)
Limits are a way to divide by zero and otherwise make infinity work for you. In calculus class you will learn a more precise definition of limits, and a lot of rules to help you compute them. You might have to use limits directly to solve some problems. Mostly you learn about limits so you can understand the definitions of derivatives and integrals, described below.
Calculus focuses on smooth functions. There is a formal definition of smoothness, but your intuition is right. Let’s look at some examples.
2 - x2 (2 minus x squared)
is smooth everywhere.
2 - x2
|x| (the absolute value of x) is smooth everywhere except at
x = 0.
n! (n factorial) is not continuous, so it can’t be smooth.
If you zoom into a smooth function enough, it will eventually look like a straight line. That makes the function very easy to predict.
Smoothness is the opposite of a fractal. If a function looks like the Mandelbrot set, you probably can’t apply calculus to that function. No matter how much you zoom into the Mandelbrot set you’ll always find places where it is not smooth.
The same goes for the natural fractals, like the coastline of Britain. No matter how much you zoom in to a natural coastline, you will always find more details. It will never look like a straight line. It will never be smooth.
Let’s look at the function
f(x) = x3 ÷ x. Naively you
might simplify that to
x2. That’s almost right, but remember that you are not allowed to divide by
zero. So you should really say
x3 ÷ x = x2, but only when
x ≠ 0.
x3 ÷ x is
undefined when x = 0.
x3 ÷ xNotice the open circle at x=0.
In simpler terms,
x3 ÷ x looks a lot like
x3 ÷ x has a hole it
If you only had the first graph, you could probably guess what the second graph looked like! You could fill in that hole with anything you want, but only one value would make the function smooth. Congratulations, you just computed a limit! Calculus class will teach you when it is okay to use the complete version of the function, rather than the original version with the hole.
If you see delta and epsilon in a problem, it is probably talking about limits. These are written with lowercase Greek letters: 𝛿 and 𝜺.
A derivate is a rate of change. For example, let’s say
f(x) tells us the total number of cookies in your house at a
given time. Then the derivate of
f(x) tells us how quickly
you are eating those cookies or baking new ones.
The graph above shows that the number of cookies goes down over time. That
means someone is eating your cookies! The graph would go up if you baked
or bought more cookies. Also notice that
continuous and smooth. That means that someone is eating a
little bit of a cookie at a time, continuously. Alternatively, what if
whole cookies were disappearing at once? Then the graph might look like
This function is not smooth everywhere. We can only talk about the derivative where it is smooth. Where it is not smooth, we say the derivate is not defined.
Our cookie example is a straight line. The derivative of a line is the slope of the line. In this case someone is eating one cookie per minute. The rate of change is -1 cookies per minute.
The following example is a little more complicated. Instead of a single line, this function is made of several different line segments. You could say that the average rate of change for the first 5 minutes is -1, just like the previous graph. That’s correct, but it’s leaving out a lot of details.
Let’s examine the graph in pieces:
The following graph shows the derivative of the previous graph. Notice that the derivative is only defined where the original function is (a) defined, (b) continuous, and (c) smooth.
That last example was a little too contrived. Let’s go back to a parabola. What’s the derivative of x2 ÷ 2?
x2 ÷ 2
I’ll start by drawing some line segments to create a simple approximation of this parabola. I’ll draw one point at each integer value of x.
x2 ÷ 2in red, approximation in green
You know how to take the derivate of a series of line segments! Just like in the previous section, you need to plot the slope of each line segment, and draw an empty circle anywhere the slope changes suddenly. That will give you a graph like this for the derivative x2 ÷ 2.
x2 ÷ 2
And you can break up the parabola any way you want. What if you asked for twice as many line segments? Then the answers would look like the next two graphs.
x2 ÷ 2with a better approximation
x2 ÷ 2
Is it reasonable to approximate a parabola with a lot of line segments? Yes! Remember what you saw in the zooming in demo, above. As we zoom in, any one part of a smooth function will look more and more like a straight line.
As you look at smaller and smaller sections, you see more and more detail. How small can we make the sections?
x2 ÷ 2covered by an even better approximation
x2 ÷ 2
As the line segments get smaller and
smaller, our results get more and more precise. What if set the length of
the blue lines to zero? Every time
x moves 0 units to the right, x2 ÷ 2
moves 0 units up. And
0 ÷ 0 = what?
The orange line, above, shows the actual derivative of the function, and the blue lines show our approximation of the derivative. No surprise: our approximations were getting closer and closer to the right answer. Also notice that we have more open circles (unknown values) as the approximations get more precise.
Can we really jump from an
infinite number of open circles to a
That may look intimidating now, but by the time you get to this you will
already be a master of limits. When you work out the problem in
class you’ll eventually learn that the derivative of
x2 ÷ 2 is
matching the graph. Along the way to
x you’d see something
(0 ÷ 0) + x + 0. Because you will have the power of
limits you can simplify that to
x without being afraid to
divide by 0.
This first example highlights the position, velocity, and acceleration of a pendulum. As you can see, they are clearly related.
Velocity is the derivative of position. The velocity arrow points in the direction that the pendulum is moving. That arrow disappears, and the velocity becomes 0, each time the pendulum changes direction. The position of the pendulum is changing fastest when the pendulum is in the middle, so the velocity arrow is the biggest at that time. That's the definition of a derivative!
Acceleration is the derivative of velocity. Notice how the acceleration arrow acts similar to the velocity arrow, but the acceleration arrow is slightly in the lead. And notice how the velocity arrow is similar to the position arrow, but the velocity arrow is slightly in the lead.
Let me plot those arrows on the graph below. The vertical line shows the current time. Up 👆 on the graph corresponds to an arrow pointing right 👉 on the image.
The graph shows that all three functions are actually sine waves. The only difference between them is that they are offset from one another. In the proper terminology, they are out of phase from each other. Or, as I said above, they act similar but each is in the lead of the next.
Let’s turn that last physics experiment on its side.
A mass on a spring acts very similar to a pendulum. Once again we use a sine wave as a very good approximation of the position of the mass. So the math will be the same. A mass on a spring problem is just a pendulum problem in disguise.
The red line shows the position of the mass. Notice how the mass moves up when the slope of that graph is positive. When the mass changes direction, the slope of that graph is 0. When the mass is moving down, the slope is negative. Rate of change = slope of graph.
Pay attention when your teacher explains the calculus of sine waves. This subject is important and it comes up a lot in math and physics. Here's one more example.
In this case you are watching an electron bouncing around a circuit. But the math is the same. We talk about charge instead of position. And the derivative of charge is current, instead of velocity. Charge, current, and the derivative of current will all be sine waves, as in the previous examples.
I should remind you that these are simple approximations. This is as far as you will get in high school calculus. If you want to learn about friction and resistance you’ll need to wait for differential equations class. Until then you live in a safe and magical world with perpetual motion.
Interest is money the bank puts into your bank account. Your account balance will change over time because the bank is adding money.
f(x) represents your bank balance, then the derivate of
f(x) says how quickly the bank is adding money.
Bankers often quote interest rates in different ways. They might quote a rate for a year. But they add money more than once a year, and the rate constantly changes.
Sometimes they talk about interest “compounding” yearly, quarterly, monthly, daily or continuously. “APR” is banker-speak for “continuously.” That’s the same approximation that we did above. The smaller the time span you look at, the closer you get to the ideal or continuous value.
There are more details, of course. I’ll leave those to your teacher.
Pay particular attention when you get to
The derivative of
f(x) is often written as
f′(x), pronounced “F prime of X.”
The derivate of y is often written as
dy / dx, pronounced
“the derivate of y with respect to x.” It can also be written as
You may hear the term “tangent line.” A tangent line is a slight variation on what I discussed in Really Small Pieces, above. Start from any one of those really small line segments. Extend it forever in both directions so it’s easy to see the slope of that line segment. The graph below shows multiple tangent lines. Notice how each tangent line touches the parabola at exactly one point. That point is where the original tiny line segment is. The derivative at any point on a curve is the slope of the tangent line at that point.
2 - x2with tangent lines
When you see the ∫ symbol, that means to take the integral of a function.
Integration and differentiation are inverses of each other. If g(x) is an integral of f(x) then f(x) is the derivative of g(x).
That applies to all of the examples above. E.g. Position is the integral of speed.
How did ships know their position before GPS was invented? One way is called dead reckoning. From time to time they would measure their speed. They would multiply each speed measurement by the amount of time until the next measurement, then add all the values together. That’s the inverse of what we did in the derivative examples, above.
The x axis represents time. The y axis represents distance. Slope represents speed. If you know the speed and time, you can easily compute the distance.
Each pair of arrows represents one measurement. The position of the pair represents the time the measurement was taken. The direction each arrow points shows the result of the measurement, i.e. the speed of the boat. Notice:
The more often the sailors measured their speed, the more accurate their estimate would be. If they could measure their speed constantly, they would know exactly where they were.
But that’s tricky. If they measure the speed constantly, they will have an
infinite number of distances to add up. And each distance is a speed times
a time, and the times are all 0. If you’re not careful you’ll get
∞ ⨉ 0. But by this time you’ll be an expert at limits, so you
will know how to get a real answer.
How do you calculate an integral? One method is to look at all of the
rules for computing derivatives, and to invert each of them. For example,
we saw above that the derivative of
x2 ÷ 2 is
any time you have to integrate
x, you can look that up in
your list and say an answer is
x2 ÷ 2.
An answer. Remember, a derivative is just a fancy version of a slope. And as we saw above, more than one line can have the same slope. So more than one function can have the same derivative. So each function will have multiple anti-derivatives.
The short version is that you need to write
+ C at the end of
your answer. So the integral of
x2 ÷ 2 + C. Your teacher
will take off a lot of points of you forget the
C is any number. That means that
x2 ÷ 2,
x2 ÷ 2 + 2,
x2 ÷ 2 + 3,
x2 ÷ 2
x2 ÷ 2 + π,
x2 ÷ 2 - 27, etc., are all valid integrals of
This method is nice because it gives you a formula for an exact answer. However, this method can be difficult in some cases and impossible in others.
The previous section begs an important question. Are you expecting the answer to be a number or should it be a formula?
If you see a problem like
∫ 2x dx, that’s called an
indefinite integral. The answer should look like
x2 + C, as described above.
What if you wanted an exact value? You can’t just plug in a number for
C can be any number, so
x2 + C can be any number.
That’s not very helpful!
Back to our boat example: What if we add up all of our measurements and
got a total of 10 miles? That means the boat moved 10 miles. But
that doesn’t tell us the position of the boat. There could be two
or more boats all going the same speed, but if they start at different
positions they will end in different positions. Crudely speaking,
C corresponds to the initial position of the boat.
In practice, if you want an actual number from an integral, you will use what’s called a definite integral. A definite integral looks just like an indefinite integral, except that the problem will specify a start value and an end value.
In our boat example, you might ask how far did the boat move between 10am
and 10:05. In our
2x example you might ask what’s the
x = 3 and
x = 7. Those problems are both definite integrals.
If you know the indefinite integral of a function, then it is
trivial to compute a definite integral of the same function. You
compute the indefinite integral at the start and end values, and
you subtract the second from the first. What’s the integral of
x = 3 and
x = 7? Start
x2 + C. For the end
value we get
72 + C and for
the start value we get
32 + C. So the answer will
have to be
(72 + C) - (32
+ C). The
C’s cancel out to give us
72 - 32
49 - 9 or
A different way to think about an integral is the area under the curve.
Same problem: What’s the integral of
x = 3 and
x = 7?
2x and marked
x = 3 and
x = 7 on
the graph. The area I’ve highlighted in
is the answer we want. You could notice that it’s a trapezoid, and Google
“area of a trapezoid” if you don’t remember that formula. Or break it into
a triangle and a rectangle if that’s easier. Or
count the number of squares. (Each
little square has area of ¼.) In each case you’ll get
40, just like in the previous section.
What if the function we are integrating is more complicated than a line? Break it into smaller pieces. Compute each piece separately then add them up. Like we’ve been doing!
What if the function we are trying to integrate is a curve, not a set of line segments? Break it into small pieces then add them up. How many pieces? How small? It depends how accurate you want your estimate to be.
But this method is not just for estimates. If you take the limit as the pieces get infinitely small, that will be the exact answer.
Let’s take another look at the
dead reckoning example. I cheated the
first time I presented that problem. I started by plotting
the answer in red! It was
satisfying to watch the estimate approach the final answer. But this time
let’s solve the problem honestly. Let’s call that original
Those black and
silver arrows show the
f(x) at various values of
x. When you
take an infinite number of these measurements, we call this
f′(x), the derivate of
f(x). The next animation
f′(x) in red just
like any other function.
Notice that these two animations show the same data. In the previous
animation I plotted
f(x) as a
red line, and I showed
f′(x) as a series of black and
silver arrows. In this
animation I plotted
f′(x) as a
red line, and the
area under the curve shows the
rectangles represent negative numbers.) In both animations we are
multiplying the derivative at a place by the
distance to the next place and adding up all of the results.
When we solve an integral we start with
f′(x) and use that to
recreate the original
f(x). We use the area under the curve
f(x). We use a series of thinner and thinner
rectangles to approximate the area under the curve. The limit as we
approach an infinite number of zero-width rectangles will be
f(x) + C.
The formal definition of an integral involves an infinite sum. And that’s exactly what we did with both versions of this dead reckoning example. You can also use a finite sum as an approximation.
You will learn other ways to solve an integral. Some of them will look a bit mysterious. They are all different ways of approaching the same problem I demonstrated here. Any time you have to solve an integral, or your teacher says “the anti-derivative exists,” remember the animations in this section. You are calculating the area under the curve. And that is the inverse of a derivative.
That was a quick tour though a full year’s worth of calculus! There’s no need to memorize anything. But keep this handy while you study calculus.
Enjoy your studies. I’m just scratching the surface of what you will learn in class. I am Philip Smolen writing this for the Summer of Math Exposition 3 contest. Check #SoME3 for other entries. Check behind the scenes of my project for even more interesting math.
Click on the images for attribution and other details.
As you decide who is worthy of receiving a golden pi creature, you should know that I have experience in the care, display, and maintenance of soft, math-related sundries.
I taught myself to crochet for the sole purpose of touching the hyperbolic plane. I've also made Möbius strips and other interesting shapes from crochet. I can provide a loving and nurturing home for a golden pi creature.


You don’t need a degree in Science but it helps to know a few simple formulas and conversions. More details under each section.
Do not forget that adding seconds is a tricky business. There are some special watches, calculators or apps that add time accurately. You’ll need one because when you need to add the times in seconds for 381 scenes you’re bound to make mistakes and lose a minute on the way somewhere.
32″+45″ = 1’17 rather than 77″
a second of video is 25 frames
a second of film is 24 frames
The formula I use the most is the proportions formula. It helps me calculate feet of film stock to time.
if X feet = Y minutes then Z feet = how many minutes?
answer = Z feet x Y minutes / X feet
Reverse formula for finding out feet.
if X feet = Y minutes then how many feet = Z minutes?
answer = Z minutes x X feet / Y minutes


A friend's Christmas present to himself (and indirectly his family and friends) was a MakerBot Replicator 3-D printer. Down at Bristol Hackspace we have several repraps in various stages of completion, but none yet ready for novice use. Marc's printer seems to have worked out of the box, so I was able to get my first test piece send via Dropbox to London, printed appropriately enough, since it involved hearts, in a pretty pink plastic and posted to Bristol in a couple of days.
The key to my personal progress was the discovery of OpenSCAD. I've looked at using GUI-based 3-D tools like Blender but OpenSCAD is a programming environment, just my kind of 3-D tool. I soon found a weath of freely avaliable tutorials, reference material and scripts:
OpenSCAD uses two modes of construction: Constructive Solid Geometry (CSG) using the Computational Geometry Algorithms Library (CGAL); and extrusion(two modes) of a 2-D shape.
Some lessons learnt:
The challenge for the budding designer is to learn how to compose a desired shape by the application of the operators to the primitives. In the physical world, we typically construct objects in a 'carpentry' mode, glueing non-intersectiing objects (union) and removing waste (difference). Also our range of primitives is vast and I generally have no idea how, say, a nut is formed. CSG allows us to union() overlapping objects, get the intersection of objects and use the less-known operators minkowski sum and hull. To see what experienced openSCAD programmers use, I analysed the collection of 22 library files for basic shapes, gears and threads etc. There were a total of 10000 lines including comments. I used grep to do the counting of patterns such as 'rotate\s*(' to get counts of each construct in the language.
|union||576||Severe underestimate since union() is the default for a sequence of objects|
|intersection_for||2|| replacement for loop within intersection which doesnt work
This data seems to indicate that designers are mainly using operators which correspond to the physical operations (carpentry mode) and are generally not using the advanced operators. This is not surprising because even intersection can yield surprising results. An example in the wikibook shows the construction of a dodecahedron from the intersection of 6 boxes:
but I havent been able to trace the source of this construction. It's tempting to look for ways of constructing the other regular solids using intersection and indeed playing with this code, I found I could make an octahedron:
What other solids could I make this way? One feature of OpenSCAD is very useful here - the ability to display a parameterized sequence of objects as an animation. The system variable $t changes from 0 to 1 in increments determined by the total number frames entered in the animation panel, so to see what solids are generated as the dihedral angle is changed :
and its fun to watch the shapes morph, becoming dodecahedrons at several angles and curious asymmetric faceted shapes in between. Moreover you can edit the script as it runs to change other parameters such as the number of intersecting boxes. You can also create a video. Each frame is saved as a .png file. I used ffmpeg installed on Ubuntu:
> ffmpeg -f image2 -i frame%05d.png -r 12 dodec.avi
to create this little video


#92 A comparison diagram of the gravitational field strength of typical inverse square gravity versus hyperbolic 1/kr gravity illustrating Birkhoff's Theorem that states gravity must be asymptotically flat, that is, it must display no asymptotes. This means that, for a perfectly spherically symmetric gravitational field, no asymptotes are possible. That is, it must follow an inverse square law and cannot be hyperbolic. No real black hole is anywhere near spherically symmetric, however. And, they are always grossly distorted by external gravitational fields. So, the theorems do not apply and are good only for approximate computations, not for "precision cosmology".


We've all probably heard that we need to think about retirement, even from a young age.
We're told that we need to get a retirement account, and start saving – the earlier the better!
Have you ever wondered why you're better off if you start saving for your future earlier? It's the wonders of compound interest!
Compound interest is the eighth wonder of the world. He who understands it, earns it.. he who doesn't.. pays it.
– Albert Einstein
What Is Compound Interest?
Everyone knows that earning interest on your money is a great thing – but what are the different kinds of interest you can earn?
- Simple interest: Simple interest charges interest only based on principle value; Let's say you lend $1000 to someone you know. If you lend it to them on a 5% rate per year on simple interest – that friend would owe you $50 each year before they pay you back.
- Compound interest: Compounding interest charges interest on the principle value as well as the value of any interest previously accrued. So in the same example, if you charged your friend a 5% yearly compounding rate, he'd owe you $50 the first year, then 5% of $1050 the second year – $52.50 – (if no payments were made), and so on. The end result is that your friend would owe you increasingly more under compounding interest the longer it took for him to pay you back.
Here is the formula for calculating compound interest – from Wikipedia:
- P = principal amount (initial investment)
- r = annual nominal interest rate (as a decimal)
- n = number of times the interest is compounded per year
- t = number of years
- A = amount after time t
So here's an example:
An amount of $1500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years.
A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6:
So, the balance after 6 years is approximately $1,938.op.
Why Is Compound Interest So Great?
The reason why compound interest is so great is because not only do you earn interest on your principal balance, or the money you started with – but you also earn interest on your interest! If you are looking to grow wealth over a long period of time, compounding will definitely work in your favor and help you to achieve some exponential gains.
In fact, if you have a long enough time horizon and depending upon the interest rate, the money earned off of the interest alone can be greater than the principal invested at the start! See this chart for details:
Chart via thetaoofmakingmoney
Remember, however, that compound interest can work against you as well, so make sure that you're the one holding the reins, and like Einstein said, that you're earning it and not paying it.
What do you think about compound interest? Are you earning it or paying it? Tell us your thoughts in the comments!
Want to find out more about compound interest? Check out this great post and infographic about compound interest: Compound Interest Infographic.


Another month has passed and it is time to reveal how a programming challenge, codenamed Mu, can be solved. You can still give it a try, although no certificate will be granted. If you want to get certified, a new Codility Certificate task has been prepared for you.
The task was to count the total number of zeroes in a decimal representation of numbers 0, 1, 2, ..., N. The important aspect is that N could be as large as 1010,000. Not all programming languages support such large integers. Therefore, two technical simplifications have been applied:
- number N is given as a string containing its decimal representation,
- it suffices to calculate the result modulo 1,410,000,017.
1,410,000,017 has appeared in Codility Certificates before, and there is a good reason: it is a prime number which is relatively large but which fits easily into 32-bit signed integers. Therefore, all arithmetical operations can be performed modulo 1,410,000,017 and intermediate values fit into 64-bit arithmetic. That simplifies the technical details a lot.
The simplest, and the least efficient, solution enumerates all integers from 0 to N and counts all the zeros that appear in their representations. Here is its implementation in Python:
def number_of_zeros(S): N = int(S) a = 0 while N >= 0: a = a + str(N).count('0') N = N - 1return a % 1410000017
Obviously, such a solution works only for small values of N, since it runs in O(N * log N) time. Even if it ran for a year (on a typical PC), it wouldn't calculate the result for N = 1016. Instead of counting zeros one by one, we must count many of them at once.
Let us assume that we have calculated F = number_of_zeros(N). What is number_of_zeros(10*N+9)? Each counted zero (except the one in number 0) is repeated ten times. Additionally, there are N+1 zeros at the least significant positions. Hence:
How about number_of_zeros(10*N+C) for 0 ≤ C ≤ 9? If Z is the number of zeros in a decimal representation of N, then there are (9−C)*Z fewer zeros. Therefore:
Additionally, for N ≤ 9 we have number_of_zeros(N) = 1. This leads to the following algorithm:
def number_of_zeros(S): P = 1410000017 Z = 0 N = 0 F = 0 for j in xrange(len(S)): F = (10*F + N + P - Z*(9-int(S[j])) ) % P if S[j] == '0': Z += 1 N = (10*N + int(S[j])) % P return ((1 + F) % P)
The invariant of the for-loop is crucial to show that this algorithm really implements the above recursive equations. The invariant consists of the following conditions:
- Z = number of zeros in S[:j],
- N = number represented by S[:j] (modulo P),
- F = number_of_zeros(S[:j])-1 (modulo P).
It is important that we process the digits of N from most to least significant. This way, we can maintain variables N and Z, and at each step of the iteration update their values in constant time. Otherwise, we would have to calculate their values from the beginning at each step of the iteration. Since that requires scanning S[:j], it yields O((log N)2) overall time complexity. For N=1010,000 it would mean a minute or two instead of a fraction of a second.