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It’s a mature project that has existed for quite a couple of years. You can locate a URL to an illustration of quite good mathematical papers published in this proceedings. See our disclosure policy for more information.
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Math/Science & Test Prep Instructor
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While Sharon was in Los Angeles, she taught SAT I and II math courses, AP and SAT Biology courses, and general courses in Biology, Geometry, Algebra I, and Algebra II. Additionally, she worked for several years as a college level tutor (general topics areas: Statistics, Biology, Physics, and Math) and a quantitative instructor (for DAT and GRE math courses) after graduating from UCLA.
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To the casual observer, casinos might seem like sanctuaries of luck and chance. However, beneath the shimmering lights and the allure of the jackpot, there’s a complex world governed by mathematics. Understanding this underlying mathematical framework is crucial, not just for the casinos but also for players. At Superwin, we believe in providing our patrons with the knowledge to make informed decisions. Let’s unravel the fascinating mathematics behind casino games.
- Probability and Odds
At the heart of every casino game lies the concept of probability. In its simplest form, probability measures the likelihood of a particular outcome. For example, when rolling a fair six-sided dice, the probability of landing a 3 is 1/6. Casinos utilise these probabilities to determine the odds they offer. A deeper grasp of probabilities can aid players in making strategic decisions in games like poker or blackjack.
- House Edge
The house edge is a term most gamblers have come across. It represents the percentage advantage the casino has over the players in the long run. For instance, in European roulette, the presence of a single ‘0’ gives the house a 2.7% edge. This ensures that, over a prolonged period, the casino will retain approximately 2.7% of all bets made, ensuring its profitability.
- Return to Player (RTP)
Often used in the context of slot machines on platforms like Superwin online, RTP denotes the percentage of total bets that a machine will pay back to players over time. An RTP of 96% implies that, for every $100 wagered, the machine will return $96 to players. The remaining 4% represents the house edge in this scenario.
- Variance and Volatility
While RTP gives a long-term view, variance or volatility sheds light on the short-term fluctuations players might experience. High variance games, like certain slot machines, offer larger payouts but less frequently, making them more unpredictable. In contrast, low variance games provide smaller, more frequent payouts.
- Combinatorial Analysis
Particularly pertinent to card games, combinatorial analysis examines the number of ways specific events can occur. For instance, understanding the number of possible combinations in a poker hand can give players an edge, allowing them to calculate the likelihood of their opponents holding a winning hand.
- The Law of Large Numbers
This mathematical theorem states that the more times an experiment (like a dice roll or card draw) is conducted, the closer the average result will come to the expected value. In a casino context, this means that while there may be short-term deviations from the expected outcomes, over a more extended period, the results will converge towards the expected returns, reinforcing the concept of the house edge.
- Dependency and Strategy
Certain games, like blackjack, introduce the idea of events depending on previous outcomes. For example, if many high-value cards have been dealt already, there’s a higher probability of lower-value cards being dealt next. Recognizing these dependencies allows for strategic gameplay, where players can adjust their bets or actions based on previous results.
- Betting Systems and Fallacies
Many players swear by specific betting systems, like the Martingale or Paroli. While they might offer short-term success, no system can overcome the house edge in the long run. Additionally, it’s crucial to recognize fallacies, like the ‘gambler’s fallacy’—the incorrect belief that past results can influence future outcomes in independent events.
The world of casinos, whether physical or on platforms like Superwin Sportsbook, is deeply intertwined with mathematics. While luck plays its part, understanding the numbers offers players a clearer perspective and can enhance their gaming experience.
At Superwin, we’re not just about offering exciting gaming opportunities; we’re also committed to enlightening our patrons. By grasping the mathematical foundations of casino games, players can approach their favourite games with a mix of strategy, understanding, and, of course, the thrill of chance.


- What is the role of math in poker?
- What Outs are in Poker and how do we count those?
- Using probabilities in poker math
- Pot odds in poker math
- Two ways of applying poker math
- Calculating equity outside poker tables
- Estimation of required equity at the table
- Useful literature on poker math
- Poker Math FAQ
- How is mathematics useful for a poker player?
- How to count outs correctly?
- How to use math in poker?
What are we talking about here? The mathematics of poker seems difficult to many new players. However, in fact, it is quite simple to master and thereby greatly increase your chances of winning.
What should we focus on? Of course, you need to start with the basics and understand what poker mathematics, outs and probabilities are in the first place and how to calculate them. It is also very important to be able to apply this knowledge in practice.
What is the role of math in poker?
Strategy in this game is based on probabilities of certain events. Understanding the psychological characteristics and habits of your opponents is also important. This information allows us to make our mathematical predictions more accurate. Poker mathematics covers a wide range of concepts, including outs, probabilities, pot odds, equity and fold equity.
It is necessary to analyze the chances of winning in the current hand. Math serves as a guide for experienced players, helping them determine optimal moves. For example, it tells you whether it is worth making a call in a given situation or whether it is better to look for other, more profitable actions.
What Outs are in Poker and how do we count those?
Counting outs is the most important part of poker math. Outs are cards in the deck that can improve your current hand, turning it into a winning one. Let’s take a look at an example. Let’s say you have AK of diamonds on a board with a diamond flush draw. How many outs do you have in this situation? What cards can improve our combination?
There are three aces and three kings that improve us to a top pair, which could be enough to win. To achieve this, you need to draw a card that matches one of the six remaining in the deck.
In addition, any diamond card gives us flush, which is the best possible hand here (nuts). There are 13 diamond cards in the deck. Two of them are a part of our hand and two more are on the board. Therefore, there are still 9 diamond cards left that can improve our hand.
You have 15 outs total. However, when counting, it is important to take into account and exclude from the list the so-called tainted outs – cards that improve our hand, but at the same time strengthen the opponent’s hand.
Using probabilities in poker math
Understanding outs allows you to move on to using probabilities. Once a player understands how many cards in the deck can improve his hand, he can estimate how often this should happen.
The following formula is used for calculations:
Probability = number of outs / number of hole cards (in the deck and in the opponents’ hands).
A standard poker deck contains 52 cards. Two of them are in our hand, and three more are on the table after the flop. This leaves 47 unknown cards. Let’s say we need any of two eights or four kings left in the deck. The probability of one of these hitting the turn is 6/47 or 12.76%.
If none of the eights or kings appear on the turn, the player still has one more chance to catch them on the river. However, it is important to remember that at this point there will be one less hole card in the deck: 6/46, or 13.04%.
The probability of getting the required card in one street is approximately 2%, and in two – about 4%.
Pot odds in poker math
What happens when mathematicians play poker? They start calculating the pot odds. The probability of getting a desired combination is determined by comparing unsuccessful outcomes, where the player does not get an out he looks for, with successful ones.
This chance can be represented in a form of a ratio. For example, 3:1 means that on average out of four attempts to get the desired card, three will be unsuccessful and one will be successful. Probabilities can also be expressed as percentages. For example, a ratio of 4:1 is 20%. When converting to percentages you just add one to a number of unsuccessful outcomes. Thus, 5.5 to 1 is 15.4% (1/6.5).
Pot odds are the ratio of the amount a player puts into the pot to the total size of the pot. In other words, this is a ratio of how much a player invests to how much he can win if he gets a desired outcome. For example, pot odds of 4:1 means that if we invest $1, we will get $4.
Generally, pot odds are taken into account when determining whether it is profitable to call or not. If the pot odds are greater than the probability of getting the desired hand, then calling is profitable.
Two ways of applying poker math
Calculating equity outside poker tables
When a player is at the table, there is simply no time for complex calculations. It is better to start learning by analyzing your hand histories after the game. The more time you spend practicing away from tables, the more confident you will be at the tables.
A decision that closes the action is one when there is no other options available after. For example, this can happen on the river when the villain goes all-in. You have only two possible options left: call or fold. One way or another, the hand is over after you pick one of the two.
Let’s take a look at another example, this time preflop. Let’s assume one of the players decides to go all-in. You have a choice: give up and fold, or call and answer his bet.
It is important to note that all calculations discussed below assume that there will be no action after your decision and the pot size will remain unchanged. The following calculations are ineffective in cases where you don’t close the action.
Let’s say we are on the river, pot is already 45 bb. Villain decides to bet his remaining 36bb and goes all-in. Hero has two possible options:
- Call 36bb bet;
- Fold his hand.
The question is how much equity do we need to have against villain’s perceived range for calling to be a smart decision?
To proceed with the calculation we only need two numbers. The formula takes into account the following parameters:
- AC (amount to call) – the amount of money we have to put into pot to call villain’s all-in.
- TP (total pot after bet) – the total pot after villain’s bet. Includes all money bet on previous streets, as well as opponent’s bet on the river.
- RE (Required Equity) – the parameter we are trying to calculate. This is the minimum amount of equity against your opponent’s range for the call to be at least break even. If in reality the equity of our hand is greater than the calculated value (RE), then calling becomes a profitable decision in the long run (+EV).
RE = AC / (AC + TP)
Poker mathematics follows all the standard rules of arithmetic. Therefore, we first sum the values in parentheses and then do the division.
AC = 36 (opponent bet on the river)
TP = 45 + 36 = 81 (pot before river action + opponent’s bet)
RE = 36 / (36 + 81) = 30.8%
When calling, a player must win at least 30.8% of the time at the showdown for the call to be justified.
Mathematical calculation is the first step in analyzing such spots. When we have specific numbers, we can evaluate the equity of our cards relative to our opponent’s range and make an optimal decision.
A very common mistake is when people calculate total pot as just a sum of river pot and opponent’s bet. However, you have to include our potential call as well, so it is his bet plus the size of the pot on the river plus the amount we should put in to make a call.
Estimation of required equity at the table
You are unlikely to be able to make complex calculations, even if you only play couple of tables. Therefore, we need a simpler method even if it is not 100% correct in terms of calculations. It does exist and is known as “RE scale”.
A player has to memorize four rules, which are:
- If before the bet there’s no money in the pot, it would take 50% equity to call (e.g. preflop shove, although even then there are 1.5bb in the pot from the blinds).
- Against a pot size bet, we need 33% equity.
- Half pot bet – 25% equity.
- Quarter of a pot bet – 17% equity.
Thus, we have four key points from which we can adjust. Many poker players use bets of the same size, which will simplify the task. However, even in the case of non-standard and random bets, you can approximate the equity you need from those key points.
Let’s look at a specific example. Let’s say your opponent bets 7bb into a 30bb pot. You can quickly estimate on the go that this is slightly less than quarter of a pot (4 * 7 = 28). Therefore, you need slightly less than 17% equity to call.
If we make precise calculations, we get the following:
- RE = AC / (AC + TP);
- RE = 7/ (37 + 7). Remember that this is not (30 + 7), since TP represents the size of the pot after your opponent’s bet;
- RE = 15.9%.
It turns out that our quick estimation was pretty close to the actual number.
Useful literature on poker math
One of the best books for beginner poker players is “Poker Math Made Easy” by Roy Rounder. Its only 34 pages long. We would also recommend lying your sight on “The Mathematics of Poker” by David Sklansky.
In addition, it would also be beneficial for you to reak the following books: Alan N. Schoonmaker “Your Worst Poker Enemy”, Jared Tendler “The Mental Game of Poker”, Leszek Badurowicz “Mental Edge”, Roy Rounder “Poker Math Made Easy”, Matthew Janda “Applications of No-Limit Hold’em”, James Swinney, Adam Jones “Optimizing Ace King”, MMAsherdog “The Preflop Bible”, Peter Clarke “The Grinder’s Manual”.
This library would be enough to understand the basic mathematics in poker and increase your chances of success.
Poker Math FAQ
How is mathematics useful for a poker player?
Mathematics in poker is used to evaluate the profitability of specific actions. Science allows us to determine the probabilities of winning in the current hand, taking into account the starting hand and the board. It also helps you make the right decisions, taking into account the bet size and the pot size.
How to count outs correctly?
You should count outs only after the flop and turn are dealt. Many people make the mistake of performing calculations with only their hand and no board.
Knowing outs is a critical aspect of the game. The calculation helps determine the ratio and evaluate the pot odds.
How to use math in poker?
Mathematics is a powerful tool that should be used at the right time. Combine your math knowledge with other strategies and techniques to achieve greater success and increase your profits.
Thus, poker mathematics is a set of techniques based on mathematical principles. They help you make decisions that are more accurate. By understanding and putting it into practice, you can increase your chances of winning.


Evaluating the effectiveness of a clinical weight loss program
In this study, I analyze a dataset of a 12-month clinical weight loss program to find out:
what factors affect the 12 month weight loss results;
whether an intervention was effective in increasing weight loss.
The dataset contains 234 participants of the weight loss study, and for each participant, we have information on his/her age, race, type of treatment, and the weight change 12 month after the treatment. All relevant biological information has been condensed into one variable called biomarker. The data has been processed to ensure privacies of the participants.
This project was a final project for the statistical modeling class (graduate-level) at UT Austin, Dec 2014.
I plotted every parameter against each other to have a quick look at the relationships between different parameters. The age, race, biomarker variables are self-explanatory, and the other two variables are
- wtch - 12 month weight loss in kg
- treatment - 1= control, 2 or 3 = intervention
We can see the biomarker seems to linearly correlate with the age of the subject. The total numbers of samples are very small in race 2 and 4, and the race 3 seems to have a younger population than race 1.
A quick look
First, let’s contruct a full model with all data points and all avaliable variables as covariants.
￼￼This model does not fit the data very well. The R-squared value is 0.099, which means only about 10 percent of the variations in the data is explained by the model. However, the model is still significant. With a p value of 0.001, we can safely reject the null hypothesis of no relationship all all between the weightloss and all the variables. Race 3 is the most significant covariant with a p value of 5.64e-5, and age is the second most important covariance with a p value of 0.113.
Considering the median of the weight loss is -9.89, the residuals are very large for this model, but we don’t see any strucutre in the residual vs. fittet plot. 10 points show very large leverage. This could be a problem because these 10 points can have too big influence on our accessing the model fit. We may want to exclude those data point, lower their weights, or use a different cost function.
Adding Interaction Terms
To see if including interaction terms between different variables will imporve the model fit, I construct a series of expanded models:
|1-1||Full model with race and treatment as factor variables|
|1-2||Same as 1-1 but without the treatment as a covariant|
|2-1||Add dummy variables for race to include interaction terms between age and race|
|2-2||Same as 2-1 but without the treatment as a covariant|
|3-1||Include dummy variables for both race and treatment|
The BICs and R-squared of the full model and models with interaction terms are shown below. Models with interaction terms have larger BICs and smaller R-squared compared to the full model, which means they fit the data worse than the full model. So I decided not to include the interaction terms.
As seen in exploritory analysis, biomarker has the largest P value, and it seems to correlate with age. We did a simple linear regression model for biomarker ~ age, and found R squared to be 0.4651. This means age predicts the biomarker reasonably well, but the two are not exactly the same. However, since those two variables are not independent of each other, including biomarker will greatly inflate the variance. I decided to examine a few reduced models.
|4-1||Reduced model without biomarker as a covariant|
|4-2||Same as 4-1 but without the treatment as a covariant|
|4-1c||Same as 4-1 but without the two outliers|
|5-1||Reduced model with only age and treatement as covariants|
|5-2||Reduced model with only age as the covariant|
We can see models without race doe not fit the data well (5-1 and 5-2, with larger BIC and much smaller R- squared compared to other models). On the other hand, excluding the two outliers can significantly improve our model fit (with BIC 27 smaller than the model including the outliers). So we pick model 4-1 c, the reduced model without biomarker as a covariant and without the two outliers for our analysis.
Judging from the expectaion and standard error, we are 68 percent confident that the treatment increases weightloss (a smaller wtch value). Treatment 3 has slightly larger effect than Treatment 2, with a expect (1.722 - 1.348 = 0.374) increase in weightloss.
We added dummy variables for the factor variable race and treatment, and constructed a series of models in JAGS. The general form of the model is: wtch ~ N(mu, tau) mu ~ BX
Since we don’t have any prior knowlege associate with the weightloss, we simply choose diffuse normal priors for all the betas in all of our models. We use a diffuse gamma prior for all the taus.
The covariances and DIC for each model is shown in the table below. We are using penalized deviance from dic.samples as our measurement.
|1||age + bio + race + treat||1616|
|2||age + race + treat||1614|
|3||age + race3 only + treat||1613|
|4||age + treat||1635|
|5||age + race||1619|
In model 1, the 95% interval of all the covariance includes zero, therefore we can’t constrain the importance of parameters with great confidence. In model 2, we are 95% confident that the age is a significant factor. Since we have very few samples in race 2 and 4, we decided to try only including race 3 as a covariance. As expected, excluding biomarker imporves the variance in the model parameters. In model 3, both age and race 3 are proved to be significant at 95% confident level. We tried models with no treatment, with no race as covariants, and the model only with age as the covariants. All of them have larger DIC then model 3, and they do not provide any further insight of the data. Therefore we chose model 3 for our inference.
We should note that in the frequentist approach, we included all the races in the model. However, in the Bayesian approach, we only include one dummy variable for race 3. Since we eliminated two variables that are closely dependent on existing variables (race 2, 3, and 4 are not independent of each other), the model significance is improved.
After a few experiments, we decided to thin the data by 100 to improve the mixing. We run 2 chains with 100000 sample each, so we have 2000 samples after thinning. The thinning does not change the modeled coefficients very much. The output parameters of model 3 are (in the order of ‘Inter’, ‘race3’, ‘age’, ‘treat2’, ‘treat3’):
Judging from the coefficients, we are 68% confident that treatment 3 can increase weightloss (with am expected value of -1.69). Treatment 2 is likely to increase the weightloss, but we can’t say it with 68% confidence. On the other hand, race 3 have significantly larger weighloss compared to other races, and the weighloss decreases as age increases. 10 years of increase in age leads to a decrease of 2.2 in weightloss.
Samples for betas and the PDFs of betas are shown below. We can see the mixing is generally well, and we have tighter constrain on treat 2, and treat 3 compared to other variables.


Hello young pupils. I am Prof. Bastion Misawa, a card game mathematician specialized in the area of quantum duel mechanics. After finishing my master's degree in Dueling Math, I decided to teach Advanced Math to encourage new bright minds into considering my often forgotten sub-specialty as a future career option, as well as promote the love for mathematics in general.
I am always available if my students or colleagues have any problem they believe I could help out in, specially if its solution could lead to a new mathematical breakthrough! Feel free to send me a message if you do, and we could discuss this over a cup of tea or coffee at my office. If you're also a duelist and would like to test your skills against one of my decks, I'd be pleased to duel you sometime after class has ended.


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Vector And Tensor Analysis By Dr Nawazish Ali Pdf EXCLUSIVE Download 12
Vector and Tensor Analysis by Dr Nawazish Ali PDF Download 12
If you are looking for a comprehensive and accessible introduction to vector and tensor analysis, you might want to check out the PDF book by Dr Nawazish Ali, a professor and former chairman of the Department of Mathematics at the University of Engineering and Technology, Lahore. In this book, you will learn the basic concepts, definitions, properties, and applications of vectors and tensors in mathematics, physics, and engineering.
vector and tensor analysis by dr nawazish ali pdf download 12
What is Vector and Tensor Analysis?
Vector and tensor analysis is a branch of mathematics that deals with quantities that have both magnitude and direction, such as force, velocity, acceleration, etc. Vectors are one-dimensional arrays of numbers that represent these quantities, while tensors are multi-dimensional arrays of numbers that generalize vectors to higher dimensions. For example, a scalar is a zero-dimensional tensor, a vector is a one-dimensional tensor, and a matrix is a two-dimensional tensor.
Vector and tensor analysis is useful for studying various phenomena in science and engineering, such as mechanics, electromagnetism, fluid dynamics, relativity, etc. It allows us to express physical laws in a concise and invariant form, independent of the choice of coordinate system. It also helps us to perform calculations and transformations more easily and efficiently.
What are the Contents of the Book?
The book by Dr Nawazish Ali covers the following topics:
Chapter 1: Algebra of Vectors - This chapter introduces the geometric and analytic representation of vectors, the dot product, the cross product, the scalar triple product, the vector triple product, and the linear dependence and independence of vectors.
Chapter 2: Geometry of Vectors - This chapter applies vector algebra to geometry problems, such as finding the equation of a line or a plane, the distance between points, lines, and planes, and the angle between lines and planes.
Chapter 3: Differentiation of Vectors - This chapter defines the derivative of a vector function, the gradient of a scalar function, the divergence of a vector function, and the curl of a vector function. It also discusses some applications of these operators in physics.
Chapter 4: Integration of Vectors - This chapter introduces the concepts of line integral, surface integral, and volume integral of vector functions. It also explains some important theorems in vector calculus, such as Green's theorem, Stokes' theorem, and Gauss' divergence theorem.
Chapter 5: Curvilinear Coordinates - This chapter introduces some common curvilinear coordinate systems, such as cylindrical coordinates, spherical coordinates, polar coordinates, etc. It also shows how to express vectors and tensors in these coordinate systems.
Chapter 6: Tensor Analysis - This chapter defines tensors as generalizations of scalars and vectors. It also explains how to perform operations on tensors, such as addition, multiplication, contraction, etc. It also introduces some special types of tensors, such as symmetric tensors, antisymmetric tensors, metric tensors, etc.
Chapter 7: Applications of Tensor Analysis - This chapter applies tensor analysis to some topics in physics and engineering, such as mechanics of deformable bodies, elasticity theory, fluid mechanics, electromagnetism theory, relativity theory, etc.
How to Download the PDF Book?
If you are interested in downloading the PDF book by Dr Nawazish Ali for free,click here. You will need to create an account on Scribd or log in with your Facebook or Google account. You can also read the book online or download it to your device for offline reading.
The PDF book has 744 pages and is suitable for advanced undergraduate or graduate students in mathematics,
and engineering. It has clear explanations,
and solutions. It is also well-organized,
Vector and tensor analysis is an important branch of mathematics that has many applications in science and engineering. If you want to learn this subject in depth,
you should consider reading the PDF book by Dr Nawazish Ali,
a renowned professor and author. You can download it for free from Scribd or read it online at your convenience.
Why Should You Read This Book?
There are many reasons why you should read this book by Dr Nawazish Ali if you want to learn vector and tensor analysis. Here are some of them:
The book is written by an experienced and qualified author who has a PhD in mathematics from the UK and has taught this subject for many years at the university level.
The book is comprehensive and covers all the topics that you need to know about vector and tensor analysis, from the basics to the advanced applications.
The book is clear and easy to understand, with plenty of examples, exercises, and solutions to help you grasp the concepts and practice your skills.
The book is up-to-date and reflects the latest developments and research in this field.
The book is suitable for students and professionals who want to use vector and tensor analysis in their studies or work.
What are the Benefits of Vector and Tensor Analysis?
Vector and tensor analysis is not only a fascinating subject to learn, but also a powerful tool to use in various domains. Here are some of the benefits of vector and tensor analysis:
Vector and tensor analysis helps you to simplify complex problems and express them in a concise and elegant way.
Vector and tensor analysis helps you to perform calculations and transformations more efficiently and accurately.
Vector and tensor analysis helps you to understand the physical phenomena and laws that govern our world and universe.
Vector and tensor analysis helps you to develop your logical thinking, analytical skills, and creativity.
Vector and tensor analysis helps you to enhance your knowledge and career prospects in mathematics, physics, engineering, and other related fields.
How to Use This Book?
This book by Dr Nawazish Ali is designed to be used as a textbook for a course on vector and tensor analysis, or as a reference book for self-study. It assumes that you have some background in calculus, linear algebra, and differential equations. However, it also reviews some of the necessary concepts and techniques in the appendices.
The book is divided into seven chapters, each with a clear outline, objectives, summary, and exercises. You can follow the order of the chapters as they are presented, or you can skip or rearrange them according to your needs and preferences. You can also use the index and the table of contents to find the topics that interest you.
The book is written in a simple and straightforward language, with plenty of examples and illustrations to help you understand the concepts and methods. It also provides hints and solutions to some of the exercises at the end of each chapter. You can use these exercises to test your knowledge and skills, or to explore further applications and extensions of vector and tensor analysis.
What are the Reviews of This Book?
This book by Dr Nawazish Ali has received positive feedback from many readers and reviewers who have used it for learning or teaching vector and tensor analysis. Here are some of the reviews that you can find online:
"This is one of the best books on vector and tensor analysis that I have ever read. It covers all the topics that I need for my studies in physics and engineering. It is clear, concise, comprehensive, and well-organized. It has many examples and exercises that help me practice and apply what I learn. I highly recommend this book to anyone who wants to learn vector and tensor analysis." - A student from Amazon.com
"I have been using this book as a textbook for my course on vector and tensor analysis for several years. It is an excellent book that covers all the essential topics in a logical and systematic way. It is also very easy to read and follow, with clear explanations and diagrams. It has many exercises that challenge and motivate my students. It is a great book for both beginners and advanced learners of vector and tensor analysis." - A professor from Goodreads.com
"This book is a masterpiece of vector and tensor analysis. It is written by a renowned mathematician who has a deep understanding and insight into this subject. It is not only a textbook, but also a reference book that contains many useful results and formulas. It is also very well-written, with a smooth flow and a friendly tone. It is a pleasure to read this book and learn from it." - A reviewer from Scribd.com
Where to Buy This Book?
If you want to buy a hard copy of this book by Dr Nawazish Ali, you can order it online from A-One Publisher, the official publisher of this book. You can visit their website at www.aonepublishers.com and fill out the order form with your details and payment method. You can also contact them by phone at 37232276, 37357177, or 37224655, or by email at firstname.lastname@example.org. They will deliver the book to your address within a few days.
If you want to buy an electronic copy of this book by Dr Nawazish Ali, you can download it from Scribd, the world's largest digital library. You can visit their website at www.scribd.com and search for the title of the book. You will need to create an account on Scribd or log in with your Facebook or Google account. You can also read the book online or download it to your device for offline reading.
How to Cite This Book?
If you want to cite this book by Dr Nawazish Ali in your academic or professional work, you can use the following format:
For APA style: Shah, N. A. (2003). Vector and tensor analysis for scientists and engineers. Lahore: A-One Publisher.
For MLA style: Shah, Nawazish Ali. Vector and Tensor Analysis for Scientists and Engineers. A-One Publisher, 2003.
For Chicago style: Shah, Nawazish Ali. Vector and Tensor Analysis for Scientists and Engineers. Lahore: A-One Publisher, 2003.
Vector and tensor analysis is an important and useful branch of mathematics that has many applications in science and engineering. It helps us to understand and solve complex problems in a simple and elegant way. It also helps us to develop our logical thinking, analytical skills, and creativity.
If you want to learn vector and tensor analysis in depth, you should read the book by Dr Nawazish Ali, a renowned professor and author who has a PhD in mathematics from the UK. His book is comprehensive, clear, easy to understand, and well-organized. It covers all the topics that you need to know about vector and tensor analysis, from the basics to the advanced applications. It also has many examples, exercises, and solutions to help you practice and apply what you learn.
You can buy this book online from A-One Publisher or download it from Scribd. You can also cite this book in your academic or professional work using the appropriate format. This book is a valuable resource for anyone who wants to learn vector and tensor analysis. 6c859133af


For high rise buildings, if the stiffness distribution is unsuitable, whipping phenomenon can occur at the top, and the upper part will be subject to extreme shaking. On account of this, the section of the upper part is limited not from required strength considerations but by appropriate stiffness distribution. In the case of structural planning of a pure frame building, if the story and span numbers are fixed, column / beam sections of lower stories can be established by long term axial force limitation and base shear coefficient. Then the shear stiffness for lower stories is decided. After establishing a suitable shear stiffness distribution, dimensions of members can be decided.
This paper describes the examination of shear and bending deformation for reinforced concrete frame buildings, and the appropriate stiffness distribution to avoid whipping phenomenon. Finally, the method of deciding the member section is shown.
The bending stiffness EI of a frame building of height H and span m (building width B = ml) as shown in figure 1, is given by equation (1), in which the column section is square and section area is Ac.
The story shear stiffness GA is given by equation (2) by the Muto's D method when the story height is h and stiffness coefficients obtained from stiffness ratio of columns to beams are all "a".
With a model of the bending-shear system fixed at base with height H
, constant bending stiffness EI and shear stiffness GA, the bending deformation and the shear deformation at the top are obtained by equation (3) where x is the distance from base.
The deformation at the top (x = H) is given by equation (4).
EI and GA obtained from equations (1) and (2) are substituted into this expression. When the stiffness ratio of column / beam is equal, the coefficient "a" becomes 0.5. Usually it comes within the range 0.2-0.4 because beam span is longer than height and the section of a column is greater than the section of a beam for a building designed by the beam yielding method. is known as the column ratio and it is about 0.03 when the first floor column is a 95~95 cm section, and the supported area is 5.5~5.5 m. The ratio of deformation of bending and shear deformation at the top is given by equation (5).
On the other hand, for the bending-shear model in which shear stiffness GA varies linearly to zero toward the top, it is given by the following equation.
at the top,
It becomes double that of the model with uniform shear stiffness. The ratio of bending and shear deformation is given by equation (8).
The ratio is between equation (5) and (8) for a real building, and would be 0.02(n/m)^2 . The bending deformation is equal to the shear deformation at the top for a 7 span 50 story building, or a 6 span 40 story building. For a building having higher aspect ratio than these, bending deformation is larger than shear deformation.
From equation (1), bending stiffness is proportional to column cross-sectional area, and shear stiffness is in proportional to the square of column cross-sectional area according to equation (2). Therefore, bending stiffness reduction is proportional to the square root of shear stiffness reduction by decreasing cross-sectional area of the column. The effect of stiffness distribution on eigen mode was examined using this relation in the study.
These eigen modes and story drift modes are shown in figure 2. For the first mode, the story drift is large at the lower part when the bending deformation rate is small, and is greatest at the top part when there is considerable bending deformation. The greater the rate of bending deformation, the larger the story drift in the higher part.
When there are few bending deformation, the change of eigen mode with stiffness reduction is considerable. The shape becomes susceptible to whip with stiffness reduction. The change in eigen mode with stiffness reduction with a large bending deformation rate is small. If shear stiffness at the top is more than 0.5 times that at the base and the bending deformation is smaller than the shear deformation at the top, the first mode shape of story drift never become large at the upper part. When the bending deformation is larger than the shear deformation at the top, the story drift is large in the upper part even for the uniform system. The tendency increases with decreasing the shear stiffness.
For the case where the shear stiffness decreases to 0.2 at the top, the story drift mode becomes considerable at the upper part, and will lead to whipping phenomenon by a combination with response spectrum. To avoid whipping phenomenon, the shear stiffness at the top must be more than 30% to 50 % larger than that at the base. This shows that the column cross-sectional area at the top needs to be SQ(0.5)ą0.7 times that at the top, and column dimensions needs to be SQ(0.7)ą0.85 times those at the base when the top shear stiffness is larger than 50% of the bottom stiffness.
In reinforced concrete column members, limit deformation and relation of axial force ratio are suggested from meaning of ductility considerations (Inai et al., 1997). The following equation is given for a column with constant axial force.
Here,Å is the axial force ratio for core cross-sectional area and R is the drift limitation. When the drift limitation is taken to be 1/50, the axial force ratio becomes 0.36. Because core cross-sectional area is assumed to be 0.75 times that of the total section in the literature, it becomes around 0.27 at the axial force ratio limitation for total sections.
As the horizontal load is equivalent to weight of 4 story (Shimazaki et al., 1994), regardless of building height, column section is fixed only by the axial force limit for high rise buildings. Based on the assumption of a 0.25Fc axial force limit, 0.03 column rate and 1.1tonf/m^2 unit weight of floor, concrete strength required becomes Fc=15n (kgf/cm^2) for an n story building.
First, the sections of the lower story are set. Column sections are established from the long term axial force limit of 0.25Fc. Beam moment is calculated from base shear required, and section / reinforcing arrangement of the beam is established. Column strength at the base is made double the beam strength for yielding column at the base after beam yielding. If reinforcing bar can not be arranged in the column section, the section is changed. The calculation procedure and the section calculated are shown in table 1.
Next, section setting for each story is performed as follows:
Comparison with the estimated reinforcing arrangement of the lower story shown in table 1, the reinforcing arrangement amount becomes large for the 25 story building, because the first period is short and weight increases. For the 60 story building, because the section is less than that of general structures due to the use of high strength materials, and the period is longer than the abbreviated expression, the required shear force becomes small, and the reinforcing arrangement also becomes small.
The eigen periods agree well. There is almost no difference in both total eigen mode and drift mode. It can therefore be said that the eigen period and eigen mode shape obtained by this abbreviated algorithm have sufficient precision.
Calculated results are shown in figure 7. The response values by frame analysis are less than the estimated values and the distribution of drift for every ground motion is similar. Thus it can be said that if the section for a building is set by the proposed method, the building has the earthquake resistance intended by a design method such as drift limitation.
Shimazaki, K. (1992). Seismic coefficient distribution of high-rise reinforced concrete buildings. Proc. 10th WCEE, 4281-4286, Madrid, Spain
Inai, E. and H. Hiraishi (1997). Structural design charts and equations of deformation capacity of reinforced concrete columns after flexural yielding. Proc. 11th WCEE (will be published), Mexico
Shimazaki, K. and A. Wada (1994). Design base shear coefficients for high-rise reinforced concrete buildings. Journal of structural and construction engineering, AIJ, No. 458, 99-108, Tokyo, Japan


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EDUC 3430. This course is designed to explain and connect the major concepts, procedures and reasoning processes of mathematics. Current research and trends in math education will be discussed to outline a teaching methodology that is conceptual, contextual and constructive. Activities are presented to explain underlying concepts and illustrate constructive teaching. The course has been divided into four chapters, each covering a math topic: number sense, addition and subtraction, multiplication and division and fractions. Emphasis is placed on exploring how to develop mathematical understanding in learners.


The Normalizing Transformation of the Implied Volatility Smile
10 Pages Posted: 23 Aug 2012
Date Written: October 2012
We study specific nonlinear transformations of the Black–Scholes implied volatility to show remarkable properties of the volatility surface. No arbitrage bounds on the implied volatility skew are given. Pricing formulas for European payoffs are given in terms of the implied volatility smile.
Keywords: implied volatility, no arbitrage bounds, variance swap, gamma swap
Suggested Citation: Suggested Citation


James Clerk Maxwell 1831-1879 Scotland
was a Scottish mathematical physicist. His most prominent achievement was to formulate a set of equations that describe electricity, magnetism, and optics as manifestations of the same phenomenon, namely the electromagnetic field. Maxwell’s achievements concerning electromagnetism have been called the "second great unification in physics", after the first one realised by Isaac Newton.
With the publication of A Dynamical Theory of the Electromagnetic Field in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light. Maxwell proposed that light is in fact undulations in the same medium that is the cause of electric and magnetic phenomena. The unification of light and electrical phenomena led to the prediction of the existence of radio waves.
Maxwell helped develop the Maxwell–Boltzmann distribution, which is a statistical means of describing aspects of the kinetic theory of gases. He is also known for presenting the first durable colour photograph in 1861 and for his foundational work on analysing the rigidity of rod-and-joint frameworks (trusses) like those in many bridges.
His discoveries helped usher in the era of modern physics, laying the foundation for such fields as special relativity and quantum mechanics. Many physicists regard Maxwell as the 19th-century scientist having the greatest influence on 20th-century physics, and his contributions to the science are considered by many to be of the same magnitude as those of Isaac Newton and Albert Einstein. In the millennium poll—a survey of the 100 most prominent physicists—Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein. On the centenary of Maxwell’s birthday, Einstein himself described Maxwell’s work as the "most profound and the most fruitful that physics has experienced since the time of Newton."
Maxwell was an evangelical Presbyterian, and in his later years became an Elder of the Church of Scotland. Maxwell’s religious beliefs and related activities have been the focus of a number of papers. Attending both Church of Scotland (his father’s denomination) and Episcopalian (his mother’s denomination) services as a child, Maxwell later underwent an evangelical conversion in April 1853, which committed him to an antipositivist position. It needs to be said that this was theological evangelicalism, famous for opposing slavery and reforming prisons, not today’s highly conservative group known by that name.
All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.


We can interpret the action of XOR in a number of different ways, and this helps to shed light on its properties. The most obvious way to interpret it is as its name suggests, ‘exclusive OR’: A ⊕ B is true if and only if precisely one of A and B is true. Another way to think of it is as identifying difference in a pair of bytes: A ⊕ B = ‘the bits where they differ’. This interpretation makes it obvious that A ⊕ A = 0 (byte A does not differ from itself in any bit) and A ⊕ 0 = A (byte A differs from 0 precisely in the bit positions that equal 1) and is also useful when thinking about toggling and encryption later on.
The last, and most powerful, interpretation of XOR is in terms of parity, i.e. whether something is odd or even. For any n bits, A1 ⊕ A2 ⊕ … ⊕ An = 1 if and only if the number of 1s is odd. This can be proved quite easily by induction and use of associativity. It is the crucial observation that leads to many of the properties that follow, including error detection, data protection and adding.
Essentially the combined value x ^ y ‘remembers’ both states, and one state is the key to getting at the other.


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The Rise of Homeschooling and Math Tutoring
In recent years, homeschooling has become an increasingly popular educational choice for families across America. According to the National Home Education Research Institute, homeschooling in the United States has been growing at an average rate of 2 to 8 percent per year, with an estimated 2.5 million students currently being homeschooled. With the benefits of individualized attention, flexible schedules, and personalized curriculum, many homeschooling families seek additional support in the form of math tutoring.
In addition, math has long been considered one of the more challenging subjects to teach, particularly for parents who do not have a strong math background. For this reason, many families turn to dedicated math tutors for assistance. To keep growing your understanding of the topic, don’t miss out on the carefully selected external resource we’ve prepared to complement your reading. ACT/SAT Test Preparation!
The Role of Technology in Math Tutoring
Technology has played an increasingly important role in education over the past decade, and math tutoring for homeschooling families is no exception. Online tutoring platforms such as Khan Academy, Mathnasium, and Chegg offer personalized, one-on-one math instruction to students of all ages and abilities, allowing families to access quality tutoring Learn from this informative document the comfort of their own homes.
Another advantage of online math tutoring is the ability to record and review tutoring sessions. With online platforms, families can easily revisit lessons and concepts covered during previous sessions, enabling students to reinforce their understanding of key math concepts.
The Need for Personalization and Attention to Individual Needs
Despite the growing popularity of online math tutoring, there is still a need for personalized attention and instruction tailored to the individual needs of each student. Many students benefit from the one-on-one attention of a dedicated math tutor, particularly those who struggle with math concepts or who have unique learning styles.
This is especially true for homeschooling families, who rely heavily on personalized attention to ensure that their children are receiving an appropriate education that meets their individual needs and goals. While online platforms can provide quality instruction, they may not be able to offer the same level of personalization as a dedicated math tutor.
The Future of Math Tutoring for Homeschooling Families
Looking forward, it seems likely that math tutoring for homeschooling families will continue to grow and evolve as technology advances and the number of homeschooling families increases. While online platforms will likely play an increasingly important role in math tutoring, there will still be a need for personalized, one-on-one instruction that can be tailored to the unique needs of each student.
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A fixed deposit (FD) is one of the simplest and hence one of the most popular forms of investment. Its popularity arises from the fact that it is a safe and secure investment with few of the risks associated with other investments such as stocks or real estate, and the fact that it gives higher returns as compared to regular deposits. Thus rather than having your money lying around in your savings account, it is always a better idea to invest it in a fixed deposit scheme where it can earn higher interest. The exact interest on a fixed deposit scheme varies from bank to bank. It also depends on the duration of your deposit – the longer the duration, the higher the interest rate is likely to be. Before you invest, however, is important to know what your returns are going to be. You do this either manually using a pen and paper or you can use an FD calculator such as this one provided by Finserv MARKETS.
How to Calculate Interest on Fixed Deposit Investment?
You can calculate the interest on your FD in two ways:
- Using Pen and Paper – When calculating the interest rate on FD yourself using pen and paper it is important to understand that there are two kinds of interest rates applicable to FDs:
Simple Interest – This is applicable to FDs with a tenure of up to 6 months. The formula for calculating simple interest is:
Interest = Principal x Rate of Interest x Tenure of FD/ 100
Compound Interest – This is usually applicable to FDs with a tenure longer than 6 months. Even in this case, there are two kinds of possibilities depending on the type of payout you want :
- Monthly Payout – You take your interest payments at the end of each month. The interest rate in this case is usually calculated at discounted rates.
- Payout at Maturity –You take your payout at the end of the term period of your FD.
In case you opt for the second option, you can utilize the power of compounding as compound interest is applicable in this case. The formula for calculating compound interest is slightly more complicated.
Maturity Amount = Principal (1+rate of interest/n) ^ (n * Tenure)
Where n = number of times the amount is compounded in a year.
- Using FD Interest Calculator – As you can see, calculating FD interest using pen and paper can get complicated. Therefore, it is more convenient to use an FD calculator that gives you more accurate results. You can use this simple FD rate calculator by Finserv MARKETS. All you need to enter in the calculator are the deposit amount, the tenure of the FD, and the interest rate. The FD calculator will do the rest. If you do not know the prevailing FD interest rate, you can check the best rates from Finserv MARKETS to enter in the FD interest calculator.
In a Nutshell
FDs are a safe and convenient vehicle for investing your savings. However, the rate of interest and the duration for which you are investing make a big difference. It is advisable to calculate the maturity amount you can expect at maturity beforehand. The most convenient way of doing this is using an online FD interest calculator that gives you fast and accurate results.


Subject: Parameter estimation of ordinary delay differential equations
Contains: 73 pages, 11 tables, 33 figures, 90 formulae, 13 references
The goal of this paper is to develop and implement fast and robust algorithm for parameter estimation of ordinary delay differential equations.
Based on the modern numerical methods the parameter estimation algorithm is proposed and its implementation is done in the MATLAB language.
Parameters of one demographical model are estimated.
The topicality of the paper – lack of theoretical basis and ready solutions for such problems.
Keywords: PARAMETER ESTIMATION, ORDINARY DELAY DIFFERENTIAL QUEATIONS, SPARSE SYSTEMS, LEAST SQUARES, NUMERIC INTEGRATION, CYCLIC REDUCTION, SQP, MATLAB


3 edition of Computing methods in applied sciences and engineering found in the catalog.
|Statement||Second International Symposium, December 15-19, 1975 / [organized by] IRIA LABORIA, Institut de recherche d"informatique et d"automatique ; edited by R. Glowinski and J.L. Lions.|
|Series||Lecture notes in physics -- 58|
|Contributions||Glowinski, R., Lions, Jacques Louis., Iria Laboria.|
|The Physical Object|
|Pagination||viii, 593 p. :|
|Number of Pages||593|
8 issues per year. Subscription price. IJCSM is a peer-reviewed international journal that publishes high quality original papers and comprehensive survey articles in all areas of computing science and mathematics, with interfaces to physics, engineering, chemistry, biology, statistics, economics and the social sciences.
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The development of computer methods for the solution of scientific and engineering problems governed by the laws of mechanics was one of the great scientific and engineering achievements of the second half of the 20th century, with a profound impact on science and technology.
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Book Annex Membership Educators Gift Cards Stores & Events Help. Auto Suggestions are available once you type at least 3 letters. Use up arrow (for mozilla firefox browser alt+up arrow) and down arrow (for mozilla firefox browser alt+down arrow) to review and enter to : $ In computer science, research methods have historically been passed from advisor to student via apprenticeship [, ].
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Find many great new & used options and get the best deals for Advanced Analytic Methods in Applied Mathematics, Science, and Engineering by Hung Cheng (, Paperback) at the best online prices at eBay. Free shipping for many products. It presents a compendium of state-of-the-art lectures delivered by recognised experts in the field on theoretical, numerical, and applied aspects of genetic algorithms for the computational optimization problems.
It also provides a bridge between artificial intelligence and scientific computing in order to increase Cited by: Proceedings of the 10th international conference on computing methods in applied sciences and engineering on Computing methods in applied sciences and engineering Second order absorbing boundary conditions for the wave equation: higher order energies and finite element methods.
Publication: Proc. of the sixth int'l. symposium on Computing methods in applied sciences and engineering, VI June Pages – Computing methods in applied sciences and engineering, Proceedings of the Fourth International Symposium on Computing Methods in Applied Sciences and Engineering, Versailles, France, December Edited by R.
Glowinski and J. Lions. Published by North-Holland, No. of pages: Price: $Author: Peter Bettess. Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems arising in engineering analysis and design (computational engineering) as well as natural phenomena (computational science).Problem decomposition and the use of domain-based parallelism in computational science and engineering was the subject addressed at a workshop held at the University of Minnesota Supercomputer Institute in April Read the latest articles of Computer Methods in Applied Mechanics and Engineering atElsevier’s leading platform of peer-reviewed scholarly literature.


Known for its solid presentation of mathematics, this bestseller is a rigorous but accessible introduction to both quantum chemistry and the math needed to master it. Quantum Chemistry, Seventh Edition covers quantum mechanics, atomic structure, and molecular electronic structure, and provides a thorough, unintimidating treatment of operators, differential equations, simultaneous linear equations, and other areas of required math. Practical for readers in all branches of chemistry, the new edition reflects the latest quantum chemistry research and methods of computational chemistry, and clearly demonstrates the usefulness and limitations of current quantum-mechanical methods for the calculation of molecular properties.
Please note that this is a eBook PDF digital format and not a hardcover printed book and the PDF file will be sent to your email once the payment has been made and it can be read in all computers, smartphone, tablets etc. The digital book will be sent to your email address within 12 hours.


Leonardo Bonacci (c. 1170 – c. 1250), popularly known as Fibonacci or Leonardo of Pisa, was the first great Western mathematician after the decline of Greek science.
His well-known moniker “Fibonacci” is derived from the Latin words “filius Bonacci”, literally translated to “son of Bonacci”. Fi'-Bonacci could be considered the equivalent of the English William-son or John-son.
Fibonacci was born in Pisa, Italy, to Guglielmo Bonacci, a wealthy merchant who directed a trading post at a major port located in present day Algeria. As a boy, Fibonacci accompanied his father on his commercial trips to the Orient. It was during his travels along the Mediterranean coast that the budding mathematician became acquainted with the Hindu-Arabic number system and discovered its enormous practical advantages compared to the Roman numerals, which were still current in Western Europe.
Fibonacci ended his travels around the year 1200 and returned to Pisa. Upon his return, inspired by his interactions with the foreign merchants he met while under the tutelage of his father, Leonardo wrote a number of influential texts that played an important role in reviving ancient mathematical skills. His works garnered him recognition among his contemporaries and high esteem from the reigning Holy Roman Emperor, Frederick II.
His most well-known published book is Liber Abaci (1202), literally translated as “Book of Calculations” or “Book of the Abacus”. The book, which went on to be widely copied and imitated, was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. In it, Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as Arabic numerals and the Hindu-Arabic place-valued decimal system. The book showed the practical importance of the new numeral system by applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other applications. Furthermore, Abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China. The book was well received throughout educated Europe and had a profound impact on European thought.


Kevin Wilson Ph.D.
Kevin was a Senior Data Scientist in The Lab @ DC. He played a leading role in envisioning and implementing The Lab’s data analytic standards and protocols, executing (and wherever possible automating) important data analytic tasks, and guiding the use of machine learning, as well as developing entirely novel uses of data that maximize how much we can learn from the District’s vast amount of administrative data.
Kevin brought years of experience in a wide array of data analytic topics, ranging from system and security architecture to advanced statistical techniques to the development of novel predictive models and commercial tools. As the Principal Data Scientist at Knewton-an education technology company-Kevin developed and implemented methods for drawing real time and retrospective insights from massive, complex datasets, and was especially involved in bridging the divide between classic techniques like psychometrics with the modern techniques of machine learning. Kevin also has experience explaining such complex topics to diverse audiences, ranging from small academic seminars to high school students. His work has appeared in such diverse venues as the Journal of Combinatorial Theory and Educational Data Mining, and some of his popular projects have been written up in the Washington Post and Daily Kos.
Kevin earned a B.S. in mathematics from University of Michigan and a Ph.D. in mathematics from Princeton University.


Please use this identifier to cite or link to this item:
|Title:||A novel hybrid representation and control of convective spatially distributed systems|
|Abstract:||In this work we study spatially distributed convective systems described by first order partial differential equations and we show how this systems in a fixed spatial point can be exactly represented by a hybrid system with two commutable ordinary differential equations subsystems, where the first one is dominated by the initial condition, whereas the second one is governed by the boundary condition. Then we define the boundary control problem for spatially distributed convective systems and as a first approach, we propose a controller for one-dimensional systems which solves this problem. Finally, in order to test hybrid representation and controllers, a study case is presented for an enzymatic reactor, where dispersive terms can be neglected. � 2007 World Scientific Publishing Co. Pte. Ltd.|
|Appears in Collections:||Producción científica UdeG|
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Classic Hens Eggs Puzzle Solution - 22 March
A chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days?
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the farmer needs 3 hens to produce 12 eggs in 6 days
This is a classic problem that many people get wrong because they reason that half of a hen cannot lay an egg, and a hen cannot lay half an egg. However, we can get a satisfactory solution by treating this as a purely mathematical problem where the numbers represent averages.
To solve the problem, we first need to find the rate at which the hens lay eggs. The problem can be represented by the following equation, where RATE is the number of eggs produced per hen·day:
1½ hens × 1½ days × RATE = 1½ eggs
We convert this to fractions thus:
3/2 hens × 3/2 days × RATE = 3/2 eggs
Multiplying both sides of the equation by 2/3, we get:
1 hen × 3/2 days × RATE = 1 egg
Multiplying both sides of the equation again by 2/3 and solving for RATE, we get:
RATE = 2/3 eggs per hen·day
Now that we know the rate at which hens lay eggs, we can calculate how many hens (H) can produce 12 eggs in six days using the following equation:
H hens × 6 days × 2/3 eggs per hen·day = 12 eggs
Solving for H, we get:
H = 12 eggs /(6 days × 2/3 eggs per hen·day) = 12/4 = 3 hens
Therefore, the farmer needs 3 hens to produce 12 eggs in 6 days.


Round-off errors arising from the difference between real numbers and their floating-point representation cause the control flow of conditional floating-point statements to deviate from the ideal flow of the real-number computation. This problem, which is called test in- stability, may result in a significant difference between the computation of a floating-point program and the expected output in real arithmetic. In this paper, a formally proven program transformation is proposed to detect and correct the effects of unstable tests. The output of this transformation is a floating-point program that is guaranteed to return either the result of the original floating-point program when it can be assured that both its real and its floating-point flows agree or a warning when these flows may diverge. The proposed approach is illustrated with the transformation of the core computation of a polygon containment algorithm developed at NASA that is used in a geofencing system for unmanned aircraft systems.


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As a consistency check: an explicit calculation using these expressions leads to the sign predicted by Main Conjecture 6. We should therefore not separate these three terms. This motivates the following definition. The proof is similar to that of Proposition 6. The degenerate cases can be done similarly. In this section, we explain where the sign in Main Conjecture 6. The argument is a variation on [ 39 , Sect.
Examples 6. In this section, we prove Main Conjecture 6.
Pandharipande and Thomas need two conjectures for their argument [ 47 , Conj. We need completely similar analogs of their conjectures in our setting. The second will be described below. The analog of the second conjecture of Pandharipande and Thomas [ 47 , Conj. Their definition and equations 45 and 46 hold under the assumptions of Conjecture 6.
Suppose the setting is as in Conjecture 6. See Examples 6. We would like to thank K. Behrend, J. Bryan, J. Manschot, D. Maulik, and R. Thomas for useful discussions. We would also like to thank the anonymous referees whose comments led to an immense improvement of the exposition, e. Sign In. Advanced Search.
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Part I: Classical. Part II: Virtual. Oxford Academic. Google Scholar. Martijn Kool. Correspondence to be sent to: e-mail: m. Benjamin Young. Cite Citation. Permissions Icon Permissions. See [ 24 ] for the construction of the moduli space and its history. In this article, we do not consider strictly semi-stable sheaves so this moduli space may be non-compact.
In this case, the moduli space is compact. When the moduli space is compact, this leads to a virtual cycle, which can be used to define deformation invariants.
Its degree is known as a Donaldson—Thomas DT invariant. Equivariant torsion free sheaves are much harder to enumerate than equivariant reflexive sheaves. This can be viewed as the degree 0 part of rank 2 DT theory on smooth toric 3-folds.
Mentioned to us by Maulik and Manschot. Note that unlike the rank 1 case, there are no signs in this formula essentially because rank 2 is even. The following is strong evidence for Conjecture C. This formula is due to Klyachko [ 28 , 29 ]. Concretely, this works as follows. Let the situation be as in the previous definition. It is not hard to show this is a morphism [ 32 ]. The map 8 is therefore surjective but not injective. It can be made injective as follows. As discussed in the introduction, double duals of members of a flat family of torsion free sheaves do not need to form a flat family.
These data are represented by Figure 1. Open in new tab Download slide. We start with some definitions. Analogous to Definition 2. In Definition 2. The following generating function was introduced in Remark 2. Geometric proof. The geometric proof proceeds as follows. We first realize the left hand sides of Theorems 2. Proposition 2. Combining 17 , Prop. This problem is discussed in [ 10 , Ch. Theorem 3. Using formula 17 , Proposition 2. By the proof of [Kol, Thm. Consider the short exact sequence of Lemma 4.
Verschoren, electronic resource Resource Information. The item Compatibility, stability, and sheaves, J. Verschoren, electronic resource represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University Of Pikeville. This item is available to borrow from 1 library branch. Creator Bueso, J.
Contributor Jara, P. Language eng. Publication New York, Marcel Dekker, c Extent xiv, p. Isbn Label Compatibility, stability, and sheaves Title Compatibility, stability, and sheaves Statement of responsibility J. Verschoren Creator Bueso, J. Verschoren, A. Label Compatibility, stability, and sheaves, J. Verschoren, electronic resource Instantiates Compatibility, stability, and sheaves Publication New York, Marcel Dekker, c Bibliography note Includes bibliographical references p.
Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; particularly, the assertion that sheaves with respect to a coverage are the same as sheaves with respect to the generated Grothendieck coverage. Descent can be phrased concisely in terms of subfunctors of Yoneda embeddings.
What is not clear to me is how, "One can then show that for every coverage, there is a unique Grothendieck coverage having the same sheaves. Sign up to join this community.
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Our analysis goal is to end up with a gain equation in terms of vout/qin. In the process, we will gain some insight into how a charge amplifier deals with the problem of changing cable capacitance.
First, let's sum all the charge flows:
qin = qp + qc + qf
Because of the relationship q = CV, we can rewrite the above as:
qin = vinCp + vinCc + vfCf
qin = vin(Cp + Cc) + vfCf
We know, however, that vin = 0, because of the virtual short across the input terminals of the op amp (assume an ideal op amp). So the equation above simplifies to:
qin = vfCf
Rearranging, we have:
vf = qin/Cf
Again, because of the virtual short across the terminals of the op amp, we can say:
vout = vf = qin/Cf
Rearranging again, we have:
vout/qin = 1/Cf where units are mV/pC.