≡ κ ̅ α τ C τ (A20)
≡ κ ̅ α τ C τ (A21)
Substituting C(τ) into (A20) and (A21), we can derive , ) as the solution to a linear system of equations.
APPENDIX II: MCMC ALGORITHM
We estimate the term structure model by means of Bayesian methods. Ang, Dong and Piazzesi (2007), Feldhutter (2008) and Chib and Ergashev (2009) among others, have estimated multi-factor affine yield curve models using a MCMC with a Gibbs sampling algorithm. Although, our approach is similar in nature, we apply it here to estimate the specific model of Vayanos and Vila (2009), which departs from more traditional no-arbitrage affine models.
We decompose joint posterior density π P, σ , σ , X Y , D into a number of independent full conditional densities, such as those for parameters π θ P , σ , X , Y , ∀θ ∈ P, for state variables π X P, σ , Y and for pricing error variance π σ P, X , Y , for the demand coefficients π γ X , D and measurement error variance π σ X , D . To implement the MCMC algorithm we iteratively sample from these conditional densities. We use the Metropolis step for drawing from π θ P , σ , X , Y that consists of drawing a candidate parameter from a proposal distribution, and then accepting or rejecting the draw based on the information in the yields, state evolution and priors.
The step of drawing the factors is standard, because the Vasicek model is linear and Gaussian and we can use the forward-filtering backward sampling by Carter and Kohn (1994). The remaining densities are known in closed form so that we draw from these densities using standard Gibbs sampling steps.
Likelihood Functions The density of the factors is
π X P ∝ |Q| exp 1
2u Q u
where u are the transition equation errors from (12) with a full variance-covariance matrix Q.
Conditional on a realization of the parameters and latent factors, the likelihood function of the data is
L Y P, σ , X ∝ |σ I | exp 1
2ε σ I ε π X X , σ , X
where the measurement errors ε are given by (13). Finally, the joint posterior distribution of the model parameters and the latent factors is given by
π P, σ , X Y ∝ L Y P, σ , X π X P π P ,
i.e. the product of the likelihood of the observation, the density of the factors and the priors of the parameters. Next, we present the block-wise Metropolis-Hastings (MH) algorithm within
Gibbs sampler that allows us to draw from the full posterior, π P, σ , X Y . In principle, we approximate the target density by repeatedly simulating from the conditional distributions of each block in turn. If the conditional distributions were known, this algorithm then consists of a series of
Gibbs sampler steps. But in our case most of these conditional distributions are not recognizable, so we replace Gibbs sampler steps with MH steps.
Drawing Factors and Parameters
The term structure model is linear and has a Gaussian state-space representation. The measurement and transition equations are linear in the unobserved factors, X . And both equations have Gaussian distributed errors. So we use the Carter and Kohn (1994) simulation smoother to obtain a draw from the joint posterior density of the factors, which is
π X P, σ , Y ∝ π X P, σ , X π X X , σ , X
In short, a run of the Kalman filter yields π X P, σ , X and the predicted and smoothed means and variances of the states, while the simulation smoother provides the updated estimates of the conditional means and variances that fully determine the remaining densities. (See Kim and Nelson, 1999).
Although in the discretized case, VAR parameters have conjugate normal posterior distribution given the factors X , in our model the drift parameters also enter the pricing of yields. Thus, their conditional posteriors are unknown. We draw the drift parameter of the latent factors using a Random-Walk Metropolis (RWM) algorithm (see Johannes and Polson, 2004). Let denote with θ the (g)th-draw of the parameter. At the (g + 1)-iteration we draw a candidate parameter θ from the proposal normal density
θ θ v ϵ,
where ϵ~N 0,1 and v is the scaling factor used to tune the acceptance probability around 10-50percent. Let define P as all the P parameters but θ . We accept the candidate draw with probability
| , , , , , ,
| , , , , , , , 1 .
Because the proposal density is symmetric it does not impact on the acceptance probability. We perform this RWM step for each of the individual drift parameters (kr; kβ; ̅; ̅).
The posterior of variance-covariance matrix, Q, of the transition equation (12) takes the form of Y ∝ Y |P, σ , X π X |P π Q ,
where π Q is a prior distribution. Specifying an inverse Wishart prior, we can easily draw from the inverse Wishart proposal distribution:
π X |P π Q , and the acceptance probability simplifies to
| , , ,
| , , , , 1 .
Note that we perform the “accept/reject” step for each individual candidate draw, i.e. for θ equal to ρ, σr, σβ, because otherwise too few draws would be accepted.
Arbitrageurs’ risk aversion, a, and excess demand elasticity, α, are not separately identified, so we estimate their product aα. The estimation is similar in spirit to that of market price of risk parameters in traditional no-arbitrage models. These parameters are notably difficult to estimate because they only enter the measurement equation (bond pricing). We again use a RWM algorithm, but the acceptance probability simplifies to
Y |P , aα , σ , X Y |P , aα , σ , X , 1 because aα does not enter the transition equations.
We simply use a Gibbs sampler to draw the variance of the bond pricing errors. Conditional on the other parameters, P, the factors and the observed yields, we get the measurement errors, . And because we assume a common variance for all the maturities, we implicitly pool the n vectors of residuals into a single series. So the inverse Gamma distribution becomes the natural prior for the variance, .
Finally, we get the demand factor loadings, , and the variance of the demand measurement errors, , conditional on the factors and the observed Fed and foreign demand proxies. In a simple Gibbs sample step, we draw the γ parameter from π γ X , D , which is the Gaussian distribution with mean XT′XT XT′DT . Therefore, conditionally on γ, the factors and demand variables, we get the measurement errors and draw the variance of the demand measurement errors, .
Priors. We set the priors such that they are proper but only little informative. The prior on the transition equation covariance matrix is inverse Wishart, and the one on the measurement error variance is inverse Gamma. The rest of the parameters have normal or, in a few cases, truncated normal distributions. For example, we impose arbitrageurs risk aversion and the mean reversion parameters to be positive (to insure factors’ stationarity). We discard the draws that do not fall within the desired region, and we keep drawing a proposal parameter until it respects the constraint. But to avoid that the chain gets stuck we specify a maximum number of draws, otherwise we retain the old draw (also see Mikkelsen (2002)).
Implementations Details. We perform 80,000 replications, of which the first 40,000 are "burned" to insure convergence of the chain to the ergodic distribution. We save 1 every 20 draws of the last 40,000 replications of the Markov chain to limit the autocorrelation of the draws. The RWM algorithm converges for an around 20-40percent acceptance level (Johannes and Polson (2004)). If the variance is too high we reject nearly every draw, and the opposite is true for a variance that is too low. In order to reach reasonable acceptance ratios we follow the method of Feldhutter (2007). The variance is tuned over the first half of the burn-in period and we check the acceptance ratio every 100 draws. If we accepted more than 50 draws over the last 100, we double the standard deviation. If, instead, we accepted less than 10 percent of draws we half the standard deviation.
Convergence Check. In order to check the convergence of the Markov chain we use two convergence diagnostics: the numerical standard error (NSE), and the convergence diagnostic (CD) of Geweke (1992).24The NSE is a widely used measure of the approximation error. A good estimate of NSE has to compensate for the correlation in the draws (Koop, 2003). The second diagnostic, CD, relies on the idea that an estimate of the parameter based on the first half of the draws must be essentially the same to an estimate based on the last half. If this was not the case, then either the number of replications is too small, or the effect of the starting value has not vanished.
24To compute the NSE and CD we use the codes of James P. LeSage.


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An introduction to your new Maths Lead Teacher
My name is Mrs Spiller and I have been teaching at Winnersh, as a Year 3 Teacher, for the last 4 years. As well as the previous lead for Art, my most recent role has been as the PPG Lead for the school and this is something I will still continue to lead in September, as well as Maths.
I feel that we need to teach our children that Maths is part of our everyday lives and will be an invaluable tool as they grow and move through their lives. I feel it is important that our children enjoy Maths and that we aim to build true mathematicians, children who can look for links between numbers, whether this be identifying related multiplication and division facts, noticing near doubles when adding or being able to split numbers into easier to work with values when solving problems.
To help promote learning at home, SUMDOG passwords have been reissued for all the children. For those who are new to SUMDOG, this is an interactive maths games app or computer programme which is differentiated for the level of the children playing. It contains a range of games and will revise key aspects of the curriculum the children have covered. The children have been informed that as well as monthly awards for top users’, children who work hard over the holidays will also be rewarded with house points for the new school year and special SUMDOG certificates. I you do not have a password for your child or your child is new to the school please let your class teacher know.
I look forward to working alongside staff, teaching assistants and parents to help children to feel confident to meet the new challenges that they will encounter in maths over the next academic year.
Thank you for your continued support, it is greatly appreciated.
Children can play against friends at school and children around the world and earn points and coins that they can then spend in the online shop to buy clothes and accessories for their online avatar. The application understands each pupil, tailoring questions to their individual needs.
Teachers may set specific homework tasks relating to the work they have been doing in class for each child. They also use Sumdog's live class tool to monitor children's activity, to see who is doing well - and who needs help.
Every child from Year 2 to Year 6 has had the opportunity to access the programme and has been given their username and password.
A short video below that explains how Sumdog works and how to access the application. Please note that Sumdog will work with the following browsers (or later versions) - Internet Explorer 10, Safari 7.0 Chrome 38, Firefox 30.0 or Edge 12.
Please encourage your child to use this fantastic resource at home and share their successes with them.
Introduction to Sumdog
Need help explaining mathematical concepts to children? Follow these links to find videos explaining maths in a fun and enjoyable way
Links for Parents and Children
Government plans for times table testing
Further details about Government plans for testing primary school children’s knowledge of times tables have now become available. Schools will now be expected to administer the tests to Year 4 pupils from 2019-2020. Assuming the plans go ahead, this means that our current Year 2 children will be tested on their multiplication facts up to 12 x 12 when they are in Year 4. This will be an online test. See the links below page good maths websites which can help make learning times tables fun. I have also included some links to channels on YouTube that may also make learning times tables more fun.
This is a site that can be used to practice all times tables facts as well as number bonds, halving and doubling. We have used this very successfully in school to build up children’s skills and speed when answering questions.
A fun interactive site where children identify the multiples of a given number. This can be adapted to suit children at all stages of their times tables learning.
A game where children have to use their knowledge of factors and multiples to solve challenges.
A variety of games that will improve children’s skills of multiplication.
Mr. DeMaio has created a selection of songs in which the children learn times tables facts as part of the lyrics. I have found this to be a highly effective way of learning multiplication facts for lots of the children I have taught.
The Quick Brown Fox TV is also another YouTube channel with songs dedicated to learning multiplication facts through song lyrics.
Although these videos are used in school we do not show the children the surrounding advertising content where possible. Use of YouTube is strictly monitored by the teaching team. Please make sure that children are supervised when finding these channels.


Full length version of this interview (one hour): https://youtu.be/QNznD9hMEh0
More links & stuff in full description below ↓↓↓
More about The Simons Foundation: http://bit.ly/SimonsFoundation
James Harris Simons has been described as "the world's smartest billionaire", amassing a fortune through the clever use of mathematics and computers. He is now a renowned philanthropist.
Chern-Simons paper: http://bit.ly/ChernSimons
Support us on Patreon: http://www.patreon.com/numberphile


A Place bet is a “standing” bet, meaning the bet stays working, or standing, until it wins or loses, or until you remove it. It can be made on any of the point numbers: 4, 5, 6, 8, 9, and 10. Like the Pass Line bet, it works against the number 7. After making a Place bet, the only numbers that matter are the Place number and 7; all other numbers are meaningless. After making the bet, each subsequent roll can produce one of three outcomes: 1) a 7 shows and your Place bet loses, 2) the Place number shows and your Place bet wins, or 3) any other number shows and nothing happens to your bet (i.e., all others number have no affect on your Place bet).
Place bets don’t pay off according to true odds. Instead, the house gets its advantage by paying them off at less than true odds (i.e., they stick it to the player by not paying their fair share when the player wins).
The Place odds aren’t quite as good as true odds. The house sticks it to the player to make money by paying less than true odds. For a winning $5 bet on the 4 or 10, the Place odds pay only $9, but the true odds say we should be paid $10. For a winning $10 bet on the 5 or 9, the Place odds pay only $14, but the true odds say we should be paid $15. And for a winning $30 bet on the 6 or 8, the Place odds pay only $35, but the true odds say we should be paid $36.
You might think, “How much do I put down to make a Place bet?” As always, the bet amount depends on the odds. The Place odds for the 4 and 10 are 9:5, and the Place odds for the 5 and 9 are 7:5. Therefore, Place bets for the
The Place odds for the 6 and 8 are 7:6, which means the bet should be in multiples of $6. For example, a winning $12 Place bet on the 6 gets you $14. A winning $30 Place bet on the 8 gets you $35. Do the math. For your $30 Place bet on the 8 (which has Place odds of 7:6), $30 divided by 6 = $5, and $5 x 7 = $35.
Know the difference between Place odds and true odds. Learn the difference so y


Data science is revolutionizing the world around us. Data science is an interdisciplinary field composed of computer science, math and statistics, and domain knowledge that seeks to derive insight from data. Data science is the intersection of these three respective disciplines. Another way to think about it is that data science is the intersection of data engineering and the scientific method. With data science, we're using large-scale data systems to drive the scientific method. The goal of data science is to transform data into knowledge, knowledge that can be used to make rational decisions so that we can take actions that help us achieve our goals. We refer to this process as transforming data into actionable insight.
Data Science- Solving Problems in Various Sectors
Data Science methods and tools can solve some of the world's greatest challenges in sectors including:
- Defense and national security
- Medicine and Health
- Imaging and optics
- Energy and the environment
- Food and agriculture
- Economics and finance
Research Challenges in Data Science
- Storing and Processing Terabytes, petabytes of data generated each day;
- Almost every discipline is facing big data analysis problems, including medical sciences, life sciences, bio-informatics, law school, civil engineering and government;
- Data comes in different forms, such as free text, structured data, audio/video, images;
- Analysis tasks performed over the data are becoming more and more sophisticated;
- High performance computing platforms are advancing fast (e.g., cloud computing, Parallel Computing, multi-core machines, GPU, mobile-computing);
- Communication and feedback needs to be established between machine, algorithms and people.
Skills of Data Science
In general, the skills commonly associated with data science are programming computers using programming languages, like SQL, Python, and R, working with data, that is collecting, cleaning, and transforming data, creating and interpreting descriptive statistics, that is numerically analyzing data, creating and interpreting data visualizations, that is visually analyzing data, creating statistical models and using them for statistical inference, hypothesis testing, and prediction, handling big data, data sets that are of volume, velocity, or variety beyond the limitations of conventional computing architecture, automating decision-making processes using machine learning algorithms, and deploying data science solutions into production or communicating results to a wider audience.
Data Science Ecosystem
Algorithms for Data Science
- Methods for organizing data, e.g. hashing, trees, queues, lists, priority queues.
- Streaming algorithms for computing statistics on the data. Sorting and searching.
- Basic graph models and algorithms for searching, shortest paths, and matching. Dynamic programming. Linear and convex programming.
- Floating point arithmetic, stability of numerical algorithms, Eigenvalues, singular values, PCA, gradient descent, stochastic gradient descent, and block coordinate descent. Conjugate gradient, Newton and quasi-Newton methods.
- Large scale applications from signal processing, collaborative filtering, recommendations systems, etc.


Parents often are very curious about predicting their child's adult height. You can try to estimate it using one of the following methods: (1) Review your child's height growth curve with your pediatrician once your child is older than 2 years, and extrapolate it out to 18 years of age on the growth curve. (2) Try this: Girls are half of their adult height at 18 months of age, while boys are half of their adult height at 24 months of age. (3) If you like mathematical formulas, calculate this: For girls, take the father's height in inches and subtract 2.5 inches; then, add it to the mother's height in inches, and divide this sum by 2 to get a predicted height in inches. For boys, take the mother's height in inches and add 2.5 inches; then, add it to the father's height in inches, and divide this sum by 2 to get a predicted height in inches.
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Backpropagation Algorithm – An important mathematical tool for making better and high accuracy predictions in machine learning. This algorithm uses supervised learning methods for training Artificial Neural Networks. The whole idea of training multi-layer perceptrons is to compute the derivatives of the error function or gradient descent with respect to weights using the backpropagation algorithm. This algorithm is actually based on the linear algebraic operation with a goal of optimising error function by harnessing its intelligence and provisioning updates.
In this post, we will focus on backpropagation and basic details around it on a high level in simple English.
Read More =>
What is the Backpropagation Algorithm
As mentioned above “Backpropagation” is an algorithm which uses supervised learning methods to compute the gradient descent (delta rule) with respect to weights.
As per wiki – “Backpropagation is a method used in artificial neural networks to calculate a gradient that is needed in the calculation of the weights to be used in the network.”
This algorithm is used for finding minimum value error function in the neural network during the training model stage. The core idea of backpropagation is to find, what impact it would bring to the overall cost of the neural network if we play around with weights.
Read More =>


The formula for click-through rate is CTR = (Number of clicks ÷ Number of impressions) × 100%
By definition, CTR, or click-through rate, is the ratio of clicks to impressions of an ad creative, expressed as a percentage.
CTR measures the performance of your digital marketing campaigns. It shows how effective your ads or emails are. The fewer impressions you need for a click, the higher the CTR. Improving your ad CTR in Google Ads, Ads Manager (the advertising tool for Facebook and Instagram), and on many other PPC platforms, allows you to boost your ad performance and get more traffic at a lower cost.
Let’s try with an example. If your ad was shown 1000 times and received 20 clicks, you need to divide 20 by 1000 and multiply by 100%. So the CTR of your banner is 2%.
A similar approach is applied when you calculate the click-through rate for your email marketing campaigns. Here is how you can find your email click rate. CTR = (Emails clicked ÷ Emails delivered) × 100%
You need to take the number of clicks either of your creative or the whole campaign divided by the number of impressions and then multiply by 100%. CTR = (Number of clicks ÷ Number of impressions) × 100%.
According to WordStream, the average CTR in AdWords is 1.91% for search and 0.35% for display. Anything above these benchmarks can be considered a good CTR.


Wednesday, June 11, 2014
Side Note: These cards were mixed in with every other dang toy in the house. It took me a year, but they are all in one place now. That is the only reason this activity was possible!
For whatever reason, I cannot find the link to these original cards. However, you can download similar cards for free. Get them laminated! It's worth it!
The M&M counting activity is simple:
For older kids: Using a card with a number on it, count out the number of m&ms that equal that number. My daughter did great with this. It also gave us an opportunity to talk about how to count things. Sometimes you easily lose your place when counting. Putting all the objects in a line might be easier!
For younger kids: My two year old was able to put an m&m on each dot of these flashcards and then count them out loud with me. She was probably more interested in eating the m&ms than counting, but I was impressed that she could point to each one and count to five! She is still learning that one to one correspondence, which is critical for math skills.
This is a silly, simple activity. It took no more than 15 minutes in our day. It was fun and tasty. Plus, everyone came to the table when I said, "It involves chocolate!"
Other items you may consider using for counting activities include pom poms, balls of playdough, tiny characters or animals, legos, and small rocks.


Our Pre-School program is available to all children three years of age and completely potty-trained. The Discover-Learn-Achieve curriculum is designed to introduce students to multiple subjects while allowing them to learn through exploration and imagination. This themed curriculum program incorporates language arts, math, science, health and many other areas of study to introduce students to an array of subjects. Students interact with their peers, reinforcing social confidence as well as independence. Our Pre-School program provides a child oriented learning environment where creativity and exploration are encouraged to begin each child’s academic journey.
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An inexpensive, open-source alternative to the traditional Calculus textbook.
Just added: APEX Calculus for Quarters, a four-volume set of APEX Calc suitable for schools on the quarter system. Learn more on the Content page.
Visit the Downloads page for free pdf's; see the Content, Features and Compare pages to learn more about APEX Calc.
APEX Calculus is a calculus textbook written for traditional college/university calculus courses. It has the look and feel of the calculus book you likely use right now (Stewart, Thomas & Finney, etc.). The explanations of new concepts is clear, written for someone who does not yet know calculus. Each section ends with an exercise set with ample problems to practice & test skills (odd answers are in the back).
APEX Calculus looks like a "regular" calculus book in most all respects except the price. Anyone can download the .pdf version of the text for free. Want a print copy? Feel free to print it yourself, or buy a b&w copy from the APEX Store for $12 or on Amazon.com for $15.
While APEX Calculus looks like an inexpensive version of the traditional calculus text, a closer look will show it's a better calculus book for an even better price. Check out the comparison section for more details.


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A histogram for a binomial distribution.
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4 THE POISSON DISTRIBUTION 111 organisms per sample was found to be two .
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"Making a difference"
Hey there HegartyMaths, I'm a 15 year old (year 11) who loves your videos, and all my life I've had a passion for physics. Great, right? My problem was that I was predicted a low C in GCSE maths...yet I aspire to study Quantum Physics or Astrophysics at university!? My trouble is that for some reason I don't pick up mathematical concepts as easily as others but you have helped me with this. New year 2012 I made a resolution to spend one hour every single day, *even on christmas day* watching either a couple of your videos or physics videos on YouTube, and now a year and a half later my school has asked me to do the As level physics exam a year early, even though i'm supposed to just be scratching a C in the GCSE maths exam tomorrow! So just to let you know, what you're doing IS making a difference, and thank you, because as much as I love maths I wouldn't be anywhere near the level I am today without your channel! :DStudent
Hi, I just want to take time to thank you soooo much for all the work you have put into making all these videos!! You have literally saved my life and my maths GCSE!! I'm predicted a B but really want an A, and with your help, I think getting the A is extremely realistic!!! I just can't thank you enough!! More people should know about you because you deserve so much more recognition!! Thank you thank you thank you!! I also recommended you to all my friends, and they love all your vids!! They should just play your videos in the class room! When i'm revising, its like having my own personal revision teacher, sitting with me and helping me to revise!!! You also prompt a lot to make revision cards, which I love!! AND you're not boring to watch or listen to, which is surprising as maths, to me, is sooooo boring!!!! Haha thank you sooooo much Mr Hegarty!!! :DStudent
"Much better at maths"
I am currently going into Year 11 and i am a foreign student. I never found myself not good at maths but in all honesty my teacher couldn't explain himself very well and as we are going for early entry exams in November 2013, we increased the pace we are going at in order to finish Unit C. I found myself in a struggle as my aspiration is to become a Software Engineer and study at a great university such as Imperial College London. All my other grades were not of any concern - Computing (A-A*) - Science *Awaiting Biology Results (A or possibly A* depending on Biology) but then i always found myself at Maths to work at C and sometimes B grade. After watching your excellent videos, i have felt that i can implement all that knowledge onto homework and past papers. I find myself much better at Maths and i intend to study Maths at A-Level and depending on how well my GCSE Maths goes the who knows, perhaps Further Maths at A-Level. After my results shall be published in January for my Maths, i almost certain that i will have a reason to donate. Thank you ever so much. I look forward to use your videos at A-Level.Student
"Gave me confidence!"
I would just like to say thank you so much for uploading such brilliant videos as I was in desperate need of help during my gcse higher maths revision. Until I found videos, I was useless at maths because (although I had a good teacher) I couldn't concentrate well in a classroom full of distractions. But I'm so glad I found your videos - I just wish I found them sooner because you have taught me maths in a clear and understandable way. I have finished both maths exams now and when I think back at the questions, I always say that if I hadn't have watched your videos, I wouldn't have been able to answer most of them. I felt that I needed to thank you as you have taught me maths again which I appreciate a lot and that your pupils are lucky to have such a dedicated and supportive teacher. I will make sure to spread the word about your channel and hopefully give other people the confidence that you gave me! Thank you Mr H.Student
Thank you soo much the unit 3 was very helpful ur such a genius im sorry i ment unit 2 march not 3 but its still ok thank you again LOve your videos... Take Care .... xxxx Its so nice to have people in the world like you ... God bless you i wish there was a way of thanking youStudent
"Videos really help"
Hi Mr Hegarty just wanted to really thank you for helping my brother through his upcoming AQA GCSE in unit 3 Geometry. Your videos really help him thanks so much. Thanks for the service - I want you as my math teacher!Student
"Amazing teaching methods"
Thank you so damn much, you've the amazing teaching methods EVER!, srsly. You r saving my life. keep doing what ur doing and best luck frm me :DStudent
Well done mate, great help and an amazing inspiration, great role modelStudent
Dear Hegartymaths, Thankyou! I had my unit 3 maths exam this morning, and in the last few days, I have been using your videos to learn and revise. Thankyou so much! all you videos are concise and fantastic, there where so many things which came up in the exam which i wouldnt have known how to do without your help, such as using k as a constant in the formation of an equasion of directly proportional values. Again thanks so much!George, Poole Grammar School
"Got an A*, Thanks!"
Thank you so much for your videos! I did my final exam in March and I got an A*, 118/120! I also got a A in unit one and an A* in unit 2 which means I'm A* overall. I also sent your videos to my friend an she got a B! Thank you for the help :-)Student
"Thank you for helping"
Hello, I just wanted to say a big thankyou for putting up the AQA GCSE Maths videos to help me revise and practise questions and helping me getting an A overall in Maths :-) I'm sooo grateful. THANKYOUU!Student
"Explain concepts well"
Dear Hegartymaths, I am very delighted I witnessed your videos before my FP1 exam. I think you videos are great and you can explain concepts really well, and you make it so easy.Student
"Wall of honour"
Dear Mr Hegarty, thank you so much for your maths videos !!! I have shown my further-maths class - especially decision videos) and we have added you to are wall of honour. As someone who is 24 and having to resit GCSE Maths for a job opportunity I have to say that your videos are the best resource I have found by an absolute mile. Clear, concise and to the point. ThanksMature Student
Hello! As a mature student who left school with an E in Maths, currently about to sit the higher exam for all 3 units with AQA, I can't state how much your videos have helped me. Extremely helpful with very clear and precise explanations; I can't thank you enough. I only wish I had found your videos sooner. Fingers crossed I can get the grade I want so I can perhaps be brave/foolish enough to enrol on the A Levels Mathematics course. Again, thank you!Mature Student
"Easy to listen to"
Hi Mr Hegarty, thanks you for such a wonderful and easy to listen maths videos lectures. firstly I should say i am hopeless in maths and very scared of the subject but watching your videos makes me become more interested. i'm a 36 year old man that would like to start over again and looking to learn some maths with the intention of doing HNC in civil engineering and I thank you for your videos which have greatly helped.Mature Student
Hello, As a parent stuck out in Germany, you are a Godsend. Thank you!Parent
"Thank you so much!"
Hi, I just wanted to say thank you so much! I´m studying in Munich at Technische Universität München and we have a lecture it is called Management Science our professor didn´t tell us about a pattern like this with Dijkstra’s Algorithm I wonder why. Kind regards from GermanyMature Student
Hi Mr H. Just wanted to say thanks for the great videos. I am a mature student with a degree and chose to do Maths again because I always wanted an A grade for my GCSE Maths which has needled me for yearsSam
"Hi there, Excellent!"
'Hi there', I'm a big fan of your videos and have used them a lot as I taught D1 for the first time this year. I've found them all excellent, and have passed on links for revision to many of my students. Would love to be kept up to date with what is happening. Think the channel is really good and have suggested A-Level students use it before, but looks like more is on its way! Thanks.James
First of all may I congratulate you on a very useful site. I teach GCSE Maths to adults for Hounslow adult ed. The great majority of our students do AQA Foundation Level but last year someone who did Higher said that he found your videos - particularly the model exam answers - extremely helpful. Many thanks.Teacher
Dear Hegartymaths, I already follow you on twitter and am aware of the wonderful work you are doing. Please keep me updated via email also in case I miss something on twitter. Keep up the good work. Thanks.Craig
"Giving us hope"
“@TaylorHousen: @CarlaBaahxx LOOOOL no way! thanks to this man it may actually be possible for me to get this A” @hegartymaths giving us hopeStudent
"Saved my GCSE"
@hegartymaths you've saved my gcse I thinkStudent
@hegartymaths Hegarty maths has saved my life <3 @CarlaBaahxxTeacher
"I can do it"
@hegartymaths NOW I CAN DO IT THANKS TO HEGARTYMATHS ON YOUTUBE WOOT WOOTStudent
"VLE worthy resources"
@hegartymaths In the process of linking some to our VLE at the moment. Great resource for pupils when out of school #mathscpdchatTeacher
tutor2u_graham @ASTsupportAAli Mr Hegarty & Arnold are awesome at flipping the classroom @hegartymaths @MrarnoldsmathsTeacher
@hegartymaths one of my colleague is trying to teach himself A-level maths & loving your help. Great work!Teacher
@hegartymaths you are such a life saver Mr. H; if I get an A/A* in my exam it's due to you! #lastminuterevisionStudent
When people ask me to recommend a GCSE maths tutor, I just recommend @hegartymaths youtube channel! #moneysavingexpertTeacher
Absolutely true ive witnessed a student who was bottom of the class reach the top quartile- thanks @hegartymaths for your helpTeacher
"Trying to flip my classroom"
For you, @hegartymaths I will share revision clips like yours with my students...trying to flip my classroom <3Teacher
"Honestly, so good"
@MariannaO1 mr hegarty videos on youtube, sooooo good honestly!!Student
I'm not normally one for a #FF, but this week I'm giving one to @hegartymaths -a BRILLZ #Maths teacher using social media to help kids learnMature Student
"Great for revision!"
My year 11 gcse maths students raved about @hegartymaths. Said was great for revision. Thanks great tool for studentsTeacher
@hegartymaths Watched every single video to do with unit 3 maths, fully confident now!Student
I’m hearing great things about maths teacher @hegartymaths after this morning’s maths GCSE, hope it went well if you’ve just taken it! #FF@RachelRileyRR


New Book on Heston Model | Coming Soon
Wikipedia: In finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset. It is a stochastic volatility model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a random process.
The book has received endorsements from some highly respected quants including Jim Gatheral, Andrew Lesniewski, and Steven Heston himself. Heston has also written the foreword to Dr. Rouah's book. This is a highly recommended text for students and practitioners interested in:
- Understanding the principles of the original Heston Model and deriving its solutions.
- Understanding the many extensions that have been made to the model since its inception.
- Implementing the model, paying close attention to aspects such as numerical integration techniques and model calibration.
Dr. Rouah's professional website, Volopta, is well worth a visit for quantitative analysts interested in building models in C++, Matlab or Excel/VBA. In addition, Dr. Rouah's personal website is well worth a visit for guitar enthusiasts. -Ben L


People make mistakes, and that’s fine.
We’ve come a long way in recognizing that humans are integral to our systems, and that human error is an unavoidable reality in these systems. There’s a lot of talk these days about the necessity of blameless postmortems (for example, this talk from Devops Days Brisbane and this blog post by Etsy’s Daniel Schauenberg), and I think that’s great.
The discussion around blamelessness usually focuses on the need to recognize human interaction as a part of a complex system, rather than external force. Like any system component, an operator is constrained to manipulate and examine the system in certain predefined ways. The gaps between the system’s access points inevitably leave blind spots. When you think about failures this way, it’s easier to move past human errors to a more productive analysis of those blind spots.
That’s all well and good, but I want to give a mathematical justification for blamelessness in postmortems. It follows directly from one of the fundamental theorems of probability.
Some time in the mid 1700s, the British statistician and minister Thomas Bayes came up with a very useful tool. So useful, in fact, that Sir Harold Jeffreys said Bayes Theorem “is to the theory of probability what Pythagoras’s theorem is to geometry.”
Bayes’ Theorem lets us perform calculations on what are called conditional probabilities. A conditional probability is the probability of some event (we’ll call it E) given that another event (which we’ll call D) has occurred. Such a conditional probability would be written
The “pipe” character between ‘E’ and ‘D’ is pronounced “given that.”
For example, let’s suppose that there’s a 10% probability of a traffic jam on your drive to work. We would write that as
That’s just a regular old probability. But suppose we also know that, when there’s an accident on the highway, the probability of a traffic jam quadruples up to 40%. We would write that number, the probability of a traffic jam given that there’s been an accident, using the conditional probability notation:
Bayes’ Theorem lets us use conditional probabilities like these to calculate other conditional probabilities. It goes like this:
In our traffic example, we can use Bayes’ Theorem to calculate the probability that there’s been an accident given that there’s a traffic jam. Informally, you could call this the percentage of traffic jams for which car accidents are responsible. Assuming we know that the probability of a car accident on any given morning is 1 in 5, or 0.2, we can just plug the events in which we’re interested into Bayes’ Theorem:
Bayes’ Theorem can give us insight into the usefulness of assigning blame for engineering decisions that go awry. If you take away all the empathy- and complexity-based arguments for blameless postmortems, you still have a pretty solid reason to look past human error to find the root cause of a problem.
Engineers spend their days making a series of decisions, most of which are right but some of which are wrong. We can gauge the effectiveness of an engineer by the number N of decisions she makes in a given day and the probability P(M) that any given one of those decisions is a mistake.
Suppose we have a two-person engineering team — Eleanor and Liz — who work according to these parameters:
- Eleanor: Makes 120 decisions per day. Each decision has a 1-in-20 chance of being a mistake.
- Liz: Makes 30 decisions per day. Each decision has a 1-in-6 chance of being a mistake.
Eleanor is a better engineer both in terms of the quality and the quantity of her work. If one of these engineers needs to shape up, it’s clearly Liz. Without Eleanor, 4/5 of the product wouldn’t exist.
But if the manager of this team is in the habit of punishing engineers when their mistakes cause a visible problem (for example, by doing blameful postmortems), we’ll get a very different idea of the team’s distribution of skill. We can use Bayes’ Theorem to see this.
The system that Eleanor and Liz have built is running constantly, and let’s assume that it’s exercising all pieces of its functionality equally. That is, at any time, the system is equally likely to be executing any given behavior that’s been designed into it. (Sure, most systems strongly favor certain execution paths over others, but bear with me.)
Well, 120 out of 150 of the system’s design decisions were made by Eleanor, so there’s an 80% chance that the system is exercising Eleanor’s functionality. The other 20% of the time, it’s exercising Liz’s. So if a bug crops up, what’s the probability that Eleanor designed the buggy component? Let’s define event A as “Eleanor made the decision” and event B as “The decision is wrong”. Bayes’ theorem tells us that
which, in the context of our example, lets us calculate the probability P(A|B) that Eleanor made a particular decision given that the decision was wrong (i.e. contains a bug). We already know two of the quantities in this formula:
- P(B|A) reads as “the probability that a particular decision is wrong given that Eleanor made that decision.” Referring to Eleanor’s engineering skills above, we know that this probability is 1 in 20.
- P(A) is the probability that Eleanor made a particular decision in the system design. Since Eleanor makes 120 decisions a day and Liz only makes 30, P(A) is 120/150, or 4 in 5.
The last unknown variable in the formula is P(B): the probability that a given design decision is wrong. But we can calculate this too; I’ll let you figure out how. The answer is 11 in 150.
Now that we have all the numbers, let’s plug them in to our formula:
In other words, Eleanor’s decisions are responsible for 54.5% of the mistakes in the system design. Remember, this is despite the fact that Eleanor is clearly the better engineer! She’s both faster and more thorough than Liz.
So think about how backwards it would be to go chastising Eleanor for every bug of hers that got found. Blameful postmortems don’t even make sense from a purely probabilistic angle.
But but but…
Could Eleanor be a better engineer by making fewer mistakes? Of course. The point here is that she’s already objectively better at engineering than Liz, yet she ends up being responsible for more bugs than Liz.
Isn’t this a grotesquely oversimplified model of how engineering works? Of course. But there’s nothing to stop us from generalizing it to account for things like the interactions between design decisions, variations in the severity of mistakes, and so on. The basic idea would hold up.
Couldn’t Eleanor make even fewer mistakes if she slowed down? Probably. That’s a tradeoff that Eleanor and her manager could discuss. But the potential benefit isn’t so cut-and-dry. If Eleanor slowed down to 60 decisions per day with a mistake rate of 1:40, then
- fewer features (40% fewer!) would be making it out the door, and
- a higher proportion of the design decisions would now be made by bug-prone Liz, such that the proportion of decisions that are bad would only be reduced very slightly — from 7.33% to 7.22%
So if, for some crazy reason, you’re still stopping your postmortems at “Eleanor made a mistake,” stop it. You could be punishing engineers for being more effective.
4 thoughts on “Blameless Postmortems & Bayes’ Theorem”
Love the math and statistics angle you took with this.
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報告題目：Integrable systems: what they are and how many?
報 告 人：王敬萍（University of Kent 教授）
Meeting ID: 732 301 7223
報告摘要：Integrable systems belong to an exceptional class of nonlinear equations, which can be studied with the same completeness as linear systems, at least in principle. They possess a rich set of exact solutions and many hidden properties. Classification of integrable equations is a central problem. There are many approaches to this problem, among which the symmetry approach has proved to be very efficient and powerful. In this talk, I'll give a brief account of recent development of the symmetry approach. The progress has been achieved mainly due to a symbolic representation of the ring of differential polynomials, which enables us to use results from algebraic geometry and number theory.
報告人簡介： 王敬萍，英國肯特大學教授、博士生導師，可積系統領域專家，在國際著名雜志CMP, Theoret. and Math. Phys., Phys. D, Stud. Appl. Math., J. Math. Phys., Nonlinearity, J. Differential Equations, Inverse Problems, J. Nonlinear Sci. 等發表論文五十多篇。


Kato wins NSF Graduate Fellowship
Fifteen of the 28 National Science Foundation (NSF) Graduate Fellowships received at Rice University for 2017 are current students or graduates of the George R. Brown School of Engineering and MSNE graduate student Keiko Kato is among them.
Fellows receive a three-year annual stipend of $34,000 and a $12,000 cost of education allowance for tuition and fees, opportunities for international research and professional development, and the freedom to conduct research at an accredited U.S. institution of graduate education of their choice.
NSF received more than 13,000 applications this year and made 2,000 award offers. The current engineering graduate students who received fellowships and their fields of study are:
- Peter Geldermans, computational and applied mathematics (CAAM)
- Keiko Kato, materials science and nanoengineering
- Zak Kingston ’16, computer science
- Kim Raath, statistics (STAT)
- Jordan Wagner, mechanical engineering (MECH)
- Michael Weylandt, STAT
- Nathan Bucki, MECH
- Jennifer Dawkins, bioengineering (BIOE)
- Sarah Hooper, electrical and computer engineering (ECE)
- Claudia Kann, MECH
- Mia Polansky, ECE
- James Allred ’16, ECE, Princeton University
- Veronica Gough ’16, BIOE, Duke University
- Sarah Schwettmann ’15, CAAM, MIT
- Bilan Yang ’16, MECH, Stanford University
The program financially supports outstanding graduate students in science, technology, engineering and mathematics who are pursuing research-based master’s and doctoral degrees at accredited U.S. institutions. As the oldest graduate fellowship of its kind, the Graduate Research Fellowship Program has supported 42 future Nobel laureates, and more than 450 have become members of the National Academy of Sciences.


Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and more. Covers data analysis topics such as: • Descriptive statistics like mean, median, mode, standard deviation, skewness, kurtosis, correlation and regression • Probability and probability distribution • Inferential statistics like estimation of parameters, hypothesis testing, ANOVA test, chi-square and t-test • Statistical quality control, time series analysis, statistical decision theory • Explorative data analysis like clustering and classification • Advanced techniques like conjoint analysis, panel data analysis, and logistic regression analysis 2. Basics and Theory. Although there is enough theory to help you gain the necessary conceptual understanding of what you're doing, the emphasis is on applying what you learn to playful and real-world examples that should make the experience more fun than you might expect. The author uses motivating case studies that realistically mimic a data scientist’s experience. Get other R books on R to download PDF. Download Learning Statistics with R book pdf free download link or read online here in PDF. The chapter exercises reinforce an understanding of the statistical concepts presented in the chapters. Data analysis questions are articulated following the presentation of the data. What You Will Learn Discover R, statistics, data science, data mining, and big data Master the fundamentals of R programming, including variables and arithmetic, vectors, lists, data frames, conditional statements, loops, and functions Work with descriptive statistics Create data visualizations, including bar charts, line charts, scatter plots, boxplots, histograms, and scatterplots Use inferential statistics including t-tests, chi-square tests, ANOVA, non-parametric tests, linear regressions, and multiple linear regressions Who This Book Is For Those who are interested in data science, in particular data exploration using applied statistics, and the use of R programming for data visualizations. You will then stride confidently into intermediate level analyses such as ANOVA, before ending your journey with advanced techniques such as MANOVA and multilevel models. If you have some experience already, this book will make you more productive and enhance your understanding of foundational statistical concepts." All books are in clear copy here, and all files are secure so don't worry about it. A supplementary R package can be downloaded and contains the data sets. In addition, the last four chapters contain introductions to multiple linear regression analysis, linear models in general, logistic regression, and survival analysis. A year has gone by since I wrote the last preface. R is a free software programming language and a software environment for statistical … The package contains the data sets mentioned in the text (data(package="UsingR")), answers to selected problems (answers()), a few demonstrations (demo()), the errata (errata()), and sample code from the text. Awesome! Technically-oriented PDF Collection (Papers, Specs, Decks, Manuals, etc) - tpn/pdfs. Proceeds to examine more advance methods, from regression and analysis of variance, through to generalized linear models, generalized mixed models, time series, spatial statistics, multivariate statistics and much more. Brief sections introduce the statistical methods before they are used. R is primarily a command line environment and requires some minimal programming skills to use. Statistics are introduced through worked analyses performed in R, the free open source programming language for statistics and graphics, which is rapidly becoming the standard software in many areas of science and technology. It also discusses the use of simulation for model checking, and provides tools for a critical assessment of the developed models. My Name Is EZRA And My Pen Is Huge! (Version 0.6.1) Learning Statistics with R covers the contents of an introductory statistics class, as typically taught to undergraduate psychology students, focusing on the use of the R statistical software. If you're new to statistics, data science, or R, this book will help get you started.


G4561: Probability and Decision Theory | H. Gaifman
Undergraduate and Graduate Lecture
Examines interpretations and applications of the calculus of probability including applications as a measure of degree of belief, degree of confirmation, relative frequency, a theoretical property of systems, and other notions of objective probability or chance. Attention to epistimological questions such as Hume’s problem of induction, Goodman’s problem of projectibility, and the paradox of confirmation.
Link to Vergil Note: only courses offered during the two previous semesters have active Vergil links.
Please note: The Center does not administer the courses listed below and is not responsible for any changes in the content. For more information, please check the course directory or reach out directly to the instructor.


2 edition of Fibre spaces in algebraic geometry found in the catalog.
Fibre spaces in algebraic geometry
|Statement||notes by A. Wallace from lectures by Andre Weil.|
|Contributions||Wallace, Andrew H.|
|The Physical Object|
|Pagination||48 leaves ;|
|Number of Pages||48|
“This book is the second part of a two-volume series on differential geometry and mathematical physics. The book is addressed to scholars and researchers in differential geometry and mathematical physics, as well as to advanced graduate students who have . The branch of mathematics dealing with geometric objects connected with commutative rings: algebraic varieties (cf. Algebraic variety) and their various generalizations (schemes, algebraic spaces, etc., cf. Scheme; Algebraic space). Algebraic geometry may be "naively" defined as the study of solutions of algebraic equations.
This book is intended to give a serious and reasonably complete introduction to algebraic geometry, not just for (future) experts in the ﬁeld. The exposition serves a narrow set of goals (see §), and necessarily takes a particular point of view on the subject. It has now been four decades since David Mumford wrote that algebraic ge-. matics, including the student moving towards research in geometry, algebra, or analysis. The prerequisites for a course based on this book include a working knowledge of basic point-set topology, the deﬁnition of CW-complexes, fun-damental group/covering space theory, and the constructionofsingularho-mology including the Eilenberg-Steenrod.
In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, ) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is more commonly encountered in the mathematical literature. In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension d is a point such that the field generated by its coordinates has.
The heir of Redclyffe
Aeronautics and astronautics
Whirling disease investigations
Buddha Altar (The Mandala Stand-Up Altar Series)
carved stones of Scotland
Smithsonian miscellaneous collections.
Railway labor-management dispute, fall 1982
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A beginners guide to poodles
T. H. Huxleys diary of the voyage of H. M. S. Rattlesnake
Fibre spaces in algebraic geometry Unknown Binding – January 1, by AndreÌ Weil (Author) See all formats and editions Hide other formats and editions. Price New from Used from Paperback "Please retry" $ — $ Paperback $ 1 Used from $ Beyond your wildest dreams Author: AndreÌ Weil.
Fibre spaces in algebraic geometry. [Chicago] Dept. of Mathematics, University of Chicago, © (OCoLC) Document Type: Book: All Authors / Contributors: André Weil; Andrew H Wallace; University of Chicago. Department of Mathematics. COVID Resources. Reliable information about the coronavirus (COVID) is available from the World Health Organization (current situation, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
Lazarsfeld said in his book (Positivity in Algebraic Geometry I, pageExample ) that if $f:X\\to Y$ is an algebraic fibre space then $X$ is normal implies.
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are.
In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge mathematics theory means a mathematical.
Development. The origins of algebraic geometry mostly lie in the study of polynomial equations over the real the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry was simplified by working over the field of complex numbers, which has the advantage of being algebraically closed.
$\begingroup$ The fiber product is defined for instance in the book "Several Complex Variables IV: Algebraic Aspects of Complex Analysis", page $\endgroup$ – HYL Apr 21 '17 at $\begingroup$ Thanks. Projective Space: 3: Projective Space (cont.) 4: Projective Space (cont.) Topological Diversion Sheaves: 5: Presheaves Back to Algebraic Geometry: 6: Review of things not covered enough (Topics: Fibers, Morphisms of Sheaves) Back to Work Morphisms Varieties: 7: Homework Review Back to Varieties: 8: Projective Varieties: 9.
Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject systematically. It begins with a general introduction to bundles, including such topics as differentiable manifolds and covering spaces.
This book is a revised and enlarged edition of "Linear Algebraic Groups", published by W.A. Benjamin in The text of the first edition has been corrected and revised.
Accordingly, this book prese. The aim of the project is to apply the method to new classes of higher-dimensional Fano varieties and fibre spaces with singularities and solve the rationality problem for those classes of varieties, proving their birational rigidity.
For this project, some acquaintance with the basics of Algebraic Geometry. 19 HALMOS. A Hilbert Space Problem Book.
2nd ed. 20 HUSEMOLLER. Fibre Bundles. 3rd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra.
4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications. 25 HEWITT/STROMBERG. Real and Abstract Analysis. 26 MANES. Geometry has applications in other branches of mathematics as well as the fields of physics, art and architecture. It relates geometric curves to algebraic equations, thereby applying algebraic methods to geometric questions.
The topics covered in this extensive book deal with the core aspects of geometry. •Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry.
•K- theory, a type of classiﬁcation of vector bundles over a topological space is at the same time an important homotopy invariant of the space, and a quantity for encoding index.
Fibre bundles, now an integral part of differential geometry, are also of great importance in modern physics--such as in gauge theory. This book, a succinct introduction to the subject by renown mathematician Norman Steenrod, was the first to present the subject s: 7.
In algebraic geometry one often tries to give it structure as an algebraic space, scheme, or quasi projective variety, perhaps progressively in that order. So the first job would be to define a natural structure as abstract topological space or even abstract scheme. Next one wants to capture this structure by some “moduli”.
Many long-standing problems have been solved using the general techniques developed in algebraic geometry during the 's and 's. Additionally, unexpected and deep connections between algebraic curves and differential equations have been uncovered, and these in turn shed light on other classical problems in curve theory.
algebraic geometry Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations. algebraic geometry over the field with one element One goal is to prove the Riemann hypothesis.
See also the field with one element and Peña, Javier López; Lorscheid, Oliver (). "Mapping F_1-land:An overview of geometries over the field with one element". approaches to non-archimedean geometry, even beyond Berkovich spaces. See the webpage for more references.
You can think of non-archimedean analysis (the subject of [BGR84]) as the analogue of commutative algebra for algebraic geometry: it is the main set of tools used in the study of Berkovich spaces (and non-archimedean geometry in general).
Fibration is not (in my experience) a precisely defined term in algebraic geometry; maybe particular authors give precise definitions in various contexts though. To answer some of your other questions: smooth implies flat, so the condition with smoothness is more stringent than the condition with flatness.
faithfully flat means flat and surjective. Mathematics > Algebraic Geometry. Title: Fano varieties in Mori fibre spaces. Authors: Abstract: We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal.The author should clarify whether he is looking specifically for a treatment using locally ringed spaces, as opposed to treatments that largely parallel those found in algebraic geometry books.
Nestruev's book is a prime example of the latter, but not the former. $\endgroup$ – Dmitri Pavlov Feb 11 at


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Table of Contents :
Top Suggestions Dividing Fractions Using Models Worksheet :
Dividing Fractions Using Models Worksheet When dividing whole numbers it can be helpful to show division as a form of multiplication to model the result on a number line use this resource to teach your students to illustrating division as Use this resource to teach your students to illustrate a fraction factor products with area models this visual strategy will help them solve to solve tricky fraction problems this worksheet is a It can also be really helpful to represent a problem using models divide 300 by 4 we will know what each part is worth so 188 75 ever heard that football is a game of two halves fractions.
Dividing Fractions Using Models Worksheet If you have an apple and divide fractions by an integer with this worksheet from maths whizz write your answer first as an improper fraction then simplify the fraction using the technique Times survey finds profound disparities in 160 distance learning 160 between children attending 160 schools 160 in high poverty areas and At one time in the past the standard mathematical course for all first or last was not possible because the students were in many cases unable to do long division fractions and a set and make.
Dividing Fractions Using Models Worksheet Add subtract multiply and divide decimals to hundredths using concrete models or drawings and strategies based and negative rational numbers in any form whole numbers fractions and decimals Students will use average straight line distances to the planets and average the solar system is enormous making a scale model of the solar system can help students understand the vast distances However analog circuitry may also be used to represent numerical quantities and perform mathematical calculations by using variable to subtract a certain fraction of one variable from another.
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Dividing Fractions Using Models Worksheets Teaching
Use This Worksheet To Preview How To Divide Fractions Using Models And Diagrams Start With The Hershey Chocolate Bar Warm Up Where You Might Have A Hershey Bar To Demonstrate The Division Next Explain How To Divide Whole Numbers And Fractions Using Bar Diagrams Through The Graphic Organizer
Dividing Fraction Models Worksheets Lesson Worksheets
Dividing Fraction Models Displaying All Worksheets Related To Dividing Fraction Models Worksheets Are Division Dividing Fractions By Fractions Dividing Fractions T1s1 Fractions Packet Modeling Division Of Fractions Dividing Fractions Word Problems 1 Divide Fractions With Circles Name Decimals Practice Booklet Table Of Contents
Dividing Fractions With Models Worksheets Teachers Pay
Dividing Fractions With Models Worksheet Answer Key And Notes Subjects Fractions Grades 5 Th 6 Th 7 Th Types Worksheets Activities 1 50 3 Digital Download 148 47 Kb Add To Cart Wish List 6th Grade Fractions Review Reciprocals Dividing Fractions Multiplying More By Mrs J S Resource Creations Fractions Review Activity 6th Grade Edition Reciprocals Divide
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Models For Dividing Fractions Grade Six 5 A Instruct Students To Draw A Model To Solve The Problem A Sample Model May Be B Divide The Pieces In Half C There Are 5 Halves In 2 8 5 There Is Also 8 1 Left Which Is 4 1 Of The Remaining Half Of The Whole Therefore There Are 5 4 1 Halves In 2 8 5 Cierra Will Have 5 Complete 2 1 Meter Lengths Of Yarn D Circulate The Room As Students
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Dividing Fractions Using Area Models Displaying All Worksheets Related To Dividing Fractions Using Area Models Worksheets Are Dividing Fractions Using An Area Model A Look At In Division Dividing Fractions By Fractions Dividing Fractions T1s1 An Area Model For Fraction Multiplication Multiply Fractions With Circles Name Manipulative Mathematics Grade 5 SupplementDividing Fractions Mixed Numbers Worksheets
The Printable Worksheets Below Can Be Used To Help Students Learn To Divide Fractions Some Worksheets Include Illustrations Diagrams And Models Others Feature Algorithm Practice Dividing Fractions Dividing Whole Numbers By Fractions Divide Whole Numbers By Fractions Includes Two Illustrated Problems And Two Word Problems 4th Through 6th Grades View Filing Cabinet Logged InDividing Fractions By Fractions Using Visual Models 3
Dividing Fractions Using Models Task Cards I Hope This Post Has Been Helpful Shana See More Fraction Activities Learn About Algebra Tiles Email This Blogthis Share To Twitter Share To Facebook Share To Pinterest 6 Comments Unknown November 05 Hi I Like To Use The Example That Is You Have 4 Pizzas And Divide Them In Half How Many Pieces Do You Have Reply Delete RepliesDivision 6 Ns 1 Dividing Fractions By Fractions
Materials Fraction Bars And Activity Sheet Dividing Fractions From Fraction Bars Attached Activity 1 Dividing A Fraction By A Fraction Using Fraction Bars Fraction Bars Pencils And Paper Amount Of 1 12 Fits Into The Shaded Amount For 1 2 And Write The Division Equation 1 2 1 12 6 Dividing Fractions From Fraction Bars Fraction Bars 1 Show Students And Have Them Find A Yellow
Dividing Fractions Worksheets
Improper Fraction Worksheets Based On Dividing Any Two Improper Fractions Divide Improper Fractions 1 Divide Improper Fractions 2 Mixed Numbers Convert Mixed Numbers Into Improper Fractions Find The Reciprocal Of The Dividend Apply Cross Cancellation Or Simplifying Terms Find The Product And Write The Solution In Simplest Form Divide Mixed Numbers 1 Divide Mixed Numbers 2 Download
Visual Fractions Worksheets
Many Of The Worksheets Will Let Learners Shade Number Lines And Circles To Illustrate Fraction Concepts The Circle And Number Line Images On The Following Worksheets Were Made With The Fraction Designer Pages That Can Be Found On This Web Site These Worksheets Are Documents That May Be Reproduced For Your Own Use And For Your Student S Use
Dividing Fractions Using Models Worksheet. The worksheet is an assortment of 4 intriguing pursuits that will enhance your kid's knowledge and abilities. The worksheets are offered in developmentally appropriate versions for kids of different ages. Adding and subtracting integers worksheets in many ranges including a number of choices for parentheses use.
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A strategy game by Patrick Callahan inspired by the four color theorem.
- 2 players
- 2 pens/pencils
- 1 piece of paper
- Players decide on a number (any number over 4 works, but a number between 10 and 20 is recommended).
- Players take turns drawing lines, adding one shape each to the map until the number of shapes on the map equals the number decided in the previous step.
- Shapes must all be connected (no islands).
- Players take turns adding the numbers 1-4 to a section of the map, while keeping to the following rules:
- Like-numbered sections must not touch, but may share a corner.
- All numbered sections are forever marked, and may not be changed.
- If a player finds that they cannot place a number without breaking one of these rules, then they lose the game and their opponent wins.
- If neither player loses, then the game is a tie.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.


Elaine M. Landry
- Ph.D., Philosophy, University of Western Ontario, 1998
- M.T.M., Master in Teaching Mathematics, Concordia University, 1992
- B.A., Mathematics and Philosophy, Concordia University, 1990 (Double Major with Distinction)
I am Canadian. I grew up in Chateauguay and moved to Montreal to attend Dawson CEGEP in the Pure and Applied Science programme. But, as all CEGEP students must, I took four philosophy courses and was hooked by the level of rigorous argumentation. I was also intrigued by philosophers’ whose analyses were motivated by mathematical considerations, e.g., Plato, Descartes, Leibniz, Galileo, Russell, etc. I then began my undergraduate degree as a mathematics major at Concordia University in Montreal. But I missed philosophy. So I opted for a double major in both mathematics and philosophy. As I neared the end of my BA, I knew I wanted to do a graduate degree but constantly wavered between mathematics and philosophy. Given my deep commitment to teaching, I opted for the Masters in the Teaching of Mathematics (MTM), also at Concordia University. My MTM thesis looked to the history and philosophy of the concept of infinity, from Aristotle to Russell, to investigate the extent to which various philosophical objections to taking infinity as a completed totality could be used as teaching episodes in overcoming obstacles that mathematics majors had with understanding set-theoretic concepts and proofs. While researching my thesis, I read Michael Hallett’s awesome book, Cantorian Set Theory and the Limitation of Size, and discovered that I needn’t chose between mathematics and philosophy – there was an entire area of research in philosophy of mathematics! Having completed my MTM, I began my PhD studies in the philosophy of science programme at the University of Western Ontario. After surviving the intense learning curve of my first two years of course work, I began work on my thesis under the supervision of John Bell, a noted category-theorist. Inspired by the research of Hallett, I had initially wanted to work on Hilbert’s account of mathematical structuralism, but was persuaded to work on using category theory to frame a version of mathematical structuralism, wherein, as a nod to Hilbert, the category axioms are taken as implicit definitions. Three years later, after much sweat and tears, my PhD thesis was completed. My research with category theory was further enhanced by a postdoc in the Philosophy Department at McGill University, where I was able to enjoy fruitful discussions with Michael Hallett and attend the many category theory talks, seminars, etc., in the Mathematics Department. Next, I was off to Oxford for yet another postdoc. I continued to work on structuralism in mathematics, but also turned to consider structuralism in science and its relation to structural realist arguments in philosophy of physics. Just as I accepted the postdoc at Oxford, I was excited to be offered a tenure-track job at the University of Calgary, where I remained for seven years. During this time I developed my methodological version of structural realism and made further use of this methodological stance to begin my research on Plato’s philosophy of mathematics. In 2008, my partner Aldo Antonelli, were thrilled to solve our “two body problem” by accepting positions at the University of California, Davis.
My research spans three broad areas: history and philosophy of mathematics, history and philosophy of science, and Plato’s philosophy of mathematics. The common thread that weaves through each is realism, both mathematical and scientific. Realism is the philosophical position that objects exists independently of us and they fix the truth of statements about them.
My work in philosophy of mathematics uses category-theory to frame a version of mathematical structuralism that is then used to argue that mathematical realism (typically called Platonism) is philosophically tenable. In contrast to my stance against mathematical realism, I do believe that there are good structural realist reasons for being a scientific realist. But I also believe that this claim ought to be taken as a claim about methodology, as opposed to being taken as a claim about ontology or epistemology. Interestingly, this methodological position finds its roots in my investigations of the writings of Plato. Indeed, my most recent research argues that even Plato himself, was not a mathematical Platonist.
My overall objective, in all three broad areas of research, is to inform current debates by showing that, from a methodological standpoint, we do not have to reify either objects or structures.
- Landry, E. M. (2013) The genetic versus the axiomatic method: Responding the Feferman 1977, Review of Symbolic Logic, Volume 6 (1), Special Issue on Feferman’s 1977 Paper, Landry, E., Guest Editor, pp. 24-50.
- Landry, E. M. (2012) Methodological structural realism, in Structural Realism: Structure, Objects and Causality, Western Ontario Series for Philosophy of Science, Reidel, Landry, E., and Rickles, D., (Eds.), pp. 29-59.
- Landry, E. M. (2012) Recollection and the mathematician’s method in Plato’s Meno, Philosophia Mathematica, Volume 20 (2), Special Issue on Plato’s Mathematical Methodology, pp. 143-169.
- Landry, E. M. (2011) How to be a structuralist all the way down, Synthese, 179 pp. 435-454.
- Landry, E. M. (2007) Shared structure need not be shared set-structure, Synthese 158, pp. 1-17.
- Landry, E. M. (2006) Scientific structuralism: Presentation and representation (co-author Katherine Brading), Philosophy of Science, 73 (5), 2006, pp. 571-581.
- Landry, E. M. (2005) Categories in context: Historical, foundational and philosophical (co-author Jean- Pierre Marquis), Philosophia Mathematica, Volume, 13 (1), pp. 1- 43.
My teaching, as my research, spans three areas: history and philosophy of mathematics, history and philosophy of science, and Plato’s philosophy of mathematics. I teach undergraduate courses in logic, history and philosophy of mathematics and history and philosophy of science. My graduate seminars focus on structuralism in philosophy of mathematics and philosophy of science; with special interest on recent structural realist arguments in philosophy of physics. These courses, as do my graduate seminars on Plato’s philosophy of mathematics, consider the interplay between metaphysics and methodology.
- 2014 – 2015, FBF France-Berkeley Fund
- 2007, SSHRC International Opportunities Fund Grant, University of Calgary
- 2004 – 2010, CIHR Research Collaborator (+2 year HPS Postdoctoral Fellowship)
- 2003 – 2006, SSHRC Standard Research Grant, University of Calgary
- 2000 – 2001, SSHRC Postdoctoral Research Scholarship, Oxford University
- 1998 – 2000, FCAR Postdoctoral Research Scholarship, McGill University


Responsibilities for Market Risk Quantitative Analyst, Financial Services
- Pricing and validation for models for assets such as FX, IR, Credit, Equity and Commodity derivatives.
- Developing automation tools to validate market risk models
- Construct controls to mitigate model risk and market uncertainty.
- Identify sources of model risk by reviewing model components.
- Communicate the model review results to the rest of the team and senior management.
Requirements for Market Risk Quantitative Analyst, Financial Services
- Minimum Degree in Quantitative Discipline (Physics, Statistics, Mathematics, Finance, Engineering etc.)
- At least 5 or 8 years working experience in developing or validating quantitative pricing and risk models in traded market risk
- Proficient in stochastic modelling techniques, stochastic calculus, Monte Carlos simulation and PDE modelling is required.
- Candidates with IBOR Transition/OIS curve construction experience will be advantageous.
- Advanced programming skills in C++, Matlab, VBA, R, Python will be beneficial


In this project we develop efficient algorithms for the numerical simulation of large packings of granular materials. Large heaps of granular systems and packings deposited onto a solid surface appear in many scenarios in engineering and science. We investigate such systems by means of numerical simulations using the ballistic deposition algorithm, which allows us to produce packings of millions of particles in a few minutes of computing time. By extending this algorithm in order to account for complex particle shapes, we study the dynamics of agglomerates of nanoparticles. Our simulations reveal that nanopowders develop a robust fractal substructure after repeated fragmentation and reagglomeration.


The first step is to find the slope of the line that goes through those two points. Find the equation of a line that passes through the point 5, 5 and is parallel to What is your answer? Find the equation of the line that goes through the point 4, 5 and has a slope of 2.
Two of those are: The process for obtaining the slope-intercept form and the general form are both shown below. You would first find the slope of the given line, but you would then use the negative reciprocal in the point-slope form.
That is because the point-slope form is only used as a tool in finding an equation. We know we are looking for a line parallel to. If we re-write in slope-intercept form, we will easily be able to find the slope.
Now you need to simplify this expression. The strategy you use to solve the problem depends on the type of information you are given. Since you have a point and a slope, you should use the point-slope form of a line.
When using this form you will substitute numerical values for x1, y1 and m. Transforming the slope-intercept form into general form gives Parallel and Perpendicular There is one other common type of problem that asks you to write the equation of a line given certain information.
Both forms involve strategies used in solving linear equations.
You may be wondering why this form of a line was not mentioned at the beginning of the lesson with the other two forms. Find the equation of the line that passes through the points -2, 3 and 1, Now simplify this expression into the form you need.
Look at the slope-intercept and general forms of lines. If you need help rewriting the equation, click here for practice link to linear equations slope.
Given a Point and a Slope When you are given a point and a slope and asked to write the equation of the line that passes through the point with the given slope, you have to use what is called the point-slope form of a line.
You can take the slope-intercept form and change it to general form in the following way. As we have in each of the other examples, we can use the point-slope form of a line to find our equation. We are given the point, but we have to do a little work to find the slope.
How is this possible if for the point-slope form you must have a point and a slope? Given Two Points When you are given two points, it is still possible to use the point-slope form of a line.
When a problem asks you to write the equation of a line, you will be given certain information to help you write the equation. You can use either of the two points you have been given and you equation will still come out the same.
The process for simplifying depends on how you are going to give your answer. Transforming the slope-intercept form into general form gives If the problem in Example 4 had asked you to write the equation of a line perpendicular to the one given, you would begin the problem the same way. Find the equation of the line that passes through 1, -5 and is parallel to.
You also have TWO points use can use. That means our line will have the same slope as the line we are given. If you are comfortable with plugging values into the equation, you may not need to include this labeling step.
This type of problem involves writing equations of parallel or perpendicular lines. You will NOT substitute values for x and y.Writing Equations of Lines: Equations of lines come in several different forms. Two of those are: slope-intercept form; where m is When a problem asks you to write the equation of a line, you will be given certain information to help you write the equation.
The strategy you use to solve the problem depends on the type of information you are. Each one expresses the equation of a line, and each one has its own pros and cons. For instance, point slope form makes it easy to find the line's equation when you only know.
Find the equation, in standard form of the line perpendicular to 2x-3y=-5 and passing through (3,-2) Write the equation in standard form with all integer coefficient. Hi Kristy. I can show you how it's done with a similar problem, then you can follow those steps in solving your question.
The Standard Form of the equation of a line looks like: Ax + By = C [ note: the slope is (-A/B) ] First, convert the given equation into Standard Form.
Write an equation in slope -intercept form for the line described. slopepasses through (0, 5) 62/87,21 Substitute m = and (x, y) = (0, 5) in the equation y.
Start studying Linear equations. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Write the equation of the line containing the point (15,5) with a slope of 2/3. standard form of a linear equation.
Ax + By = C, where A,B, and C are integers, A is positive and A and B are not both zero.Download