Inference and regression
Materials for class on Thursday, November 29, 2018
No slides for today, since there’s no formal class.
Inference and regression
Review of hypothesis testing and inference
ModernDive chapter 11 isn’t finished yet, but the principles it covers are relatively straightforward. When testing hypotheses with simulation, you follow this pattern (which you should probably have memorized by now):
- Step 1: Calculate \(\delta\). This is the thing you care about: the difference in means, the average, the median, the proportion, the difference in proportions, etc. You’re testing to see if this number is significantly difference from zero (or from some other number).
- Step 2: Invent world where \(\delta\) is null. Simulate what the world would look like if there was no difference between two groups, or if there was no difference in proportions, or where the average value is a specific number.
- Step 3: Look at \(\delta\) in the null world. Put the sample statistic in the null world and see if it fits well.
- Step 4: Calculate probability that \(\delta\) could exist in null world. This is your p-value, or the probability that you’d see a \(\delta\) at least that high in a world where there’s no effect.
- Step 5: Decide if \(\delta\) is statistically significant. Choose some evidentiary standard or threshold for deciding if there’s sufficient proof for rejecting the null world. Standard thresholds (from least to most rigorous) are 0.1, 0.05, and 0.01.
We care about statistical significance because of sampling error. If you measure something about a sample (like the proportion of blue M&Ms in a bag), you’re going to be off a little (or a lot if you have a small sample). If you see some number (i.e. if you find a small bag of M&Ms with 0 blue M&Ms in it), you want to know if that’s an anomaly, or if it’s something that could just happen by chance. If you had a fun-sized bag of 20 candies with 0 blue M&Ms in it, you could chalk that up to chance. If you had a giant Costco-sized bag of 2000 candies with 0 blue M&Ms in it, you’d be concerned. Such a finding would be statistically significant.
The same principle applies to calculating differences between groups. This is especially important in randomized controlled trials. Imagine a treatment group of 500 people is given some sort of new cold medicine and a control group of 500 people is given a placebo. All 1000 people somehow got the cold at the same time, and all 1000 people took their medicine (either the actual medicine or the placebo). Consider these two scenarios:
- Those in the control group were sick for an average of 7 days, while those in the treatment group were sick for an average of 6.875 days. The difference between the two groups is 0.125 days (or 3 hours). The 95% confidence interval for this difference ranges from -0.5 days to 1.25 days.
- Those in the control group were sick for an average of 7 days, while those in the treatment group were sick for an average of 4.5 days. The difference between the two groups is 2.5 days. The 95% confidence interval for this difference ranges from 1 day to 3.5 days.
There is a measurable difference between the average sickness duration of the two groups in both of these situations, but only the second group has a substantial and significant difference.
In the first situation, the confidence interval includes zero, which means the true unmeasurable difference between the two groups could potentially be zero. It could be negative. It could also be positive. We’re not sure. The probability of seeing a difference of 0.125 days in a world where there’s actually no difference is probably really high—that’s a typical number.
In the second situation, we’re very certain that the difference between the two groups is most definitely not zero. The probability that we’d see a difference of 2.5 days in a world where there’s actually no difference is really really low—this is like finding no blue M&Ms in a big bag.
In the first situation, we cannot say that the new cold medicine has a statistically significant effect on the duration of sickness—we don’t have enough evidence. In the second situation, we can.
Regression coefficients and inference
Just like averages, medians, proportions, differences in averages, and so on, regression coefficients are sample statistics. They are estimates of some unmeasurable population parameter that we can only calculate by taking samples from the population. Because of this, they also have standard errors and confidence intervals associated with them. They’re only estimates—they could be higher, or they could be lower.
With linear regression, what we generally care about is whether or not a coefficient is zero. If a coefficient could possibly be zero, it would mean that for every increase in X, Y might not actually change. If there’s very little chance that a coefficient could be zero, we can be pretty sure that for every increase in X, Y really truly responds and changes.
Here’s the process for testing hypotheses about coefficients:
- Step 1: Calculate \(\delta\). This is the coefficient you care about and want to test. You calculate it with the
- Step 4: Calculate probability that \(\delta\) could exist in null world. The
lm()function (in conjunction with
get_regression_table()) does this for you and provides p-values for each coefficient, or the probability that you’d see a coefficient at least that high in a world where the coefficient is actually zero.
- Step 5: Decide if \(\delta\) is statistically significant. Choose some evidentiary standard or threshold for deciding if there’s sufficient proof for rejecting the world where the coefficient is zero. Again, the standard thresholds are 0.1, 0.05, and 0.01.
Notice how steps 2 and 3 are missing here. That’s because the
infer package does not have a way to simulate a world where a coefficient is zero—there’s no
coefficient option in
specify() or anything like that. You can essentially do these steps in your head, though.
Here’s a full example using property tax data from Problem Set 3.
First we load the libraries and data we’ll need. We’ll divide the median home value variable by 100 so it’s a little easier to interpret (so we can say “for every $100 increase in median home value…” instead of "for every $1 increase in median home value)
taxes <- read_csv("data/property_taxes_2016.csv") %>% mutate(median_home_value = median_home_value / 100)
Next we build a regression model that predicts/explains per-household taxes based on median home value in each county, the proportion of households with kids in each county, and the state each county is in:
tax_model <- lm(tax_per_housing_unit ~ median_home_value + prop_houses_with_kids + state, data = taxes)
tax_model %>% get_regression_table()
tax_model %>% get_regression_summaries()
Up to this point in the semester, you’ve only looked at the estimate column. Note that there are columns named
p_value (the probability that you’d see this coefficient in a world where it’s actually zero),
lower_ci (the lower bound of the 95% confidence interval), and
upper_ci (the upper bound of the 95% confidence interval).
Here’s how we interpret all these results, now with hypothesis testing built in:
Median home value: Taking state and the proportion of households with kids into account (or controlling for state and households with kids), for every $100 increase in the median home value in a county, there is an associated increase in the median per-house tax burden of $0.40, on average. This effect is statistically significant (p < 0.001). The 95% confidence interval ranges between $0.37 and $0.44, which does not include $0, and the probability of seeing a coefficient as large as $0.40 in a world where it’s actually zero is practically 0.
The shorter way of saying all of that in a report or writeup is this:
Controlling for state and the proportion of households with kids, for every $100 increase in the median home value in a county, there is a statistically significant associated increase in the median per-house tax burden of $0.40 (p < 0.001).
Proportion of houses with kids: Controlling for state and median home values, for every percent increase in the proportion of households with kids, there is an associated increase in the median per-household tax burden of $14, on average. This coefficient is also statistically significant and has little chance of really potentially being zero. The 95% confidence interval ranges between $8.46 and $19.73 (and does not include zero) and the probability of seeing a coefficient as large as $14 in a world where it’s actually $0 is practically zero.
State effects: In Problem Set 3, we concluded that there are sizable state effects as well. This is quoted from the answer key:
“Controlling for households with kids and median home values, Californians see an average per-household tax burden of $123 more than their counterparts in Arizona. Utahans pay $213 less in taxes than Arizonans, on average. Nevadans pay $102 more than Arizonans on average, while Idahoans pay roughly the same as Arizonans (again, controlling for households with kids and median home values). A more practical, less number-focused interpretation of these numbers is that compared to Arizonans, Califorinians and Nevadans pay a lot more in taxes per household, Idahoans pay around the same amount, and Utahans pay substantially less, on average.”
However, this is not actually 100% accurate because of hypothesis testing and significance! The coefficient for California shows that Califorinians pay an average of $123 more than Arizonans. But based on the confidence interval, the uncertainty in that difference means that Califorinians might pay anywhere between $50 less than Arizonans to nearly $300 more than Arizonans. Or maybe the difference between the states is actually $0. We can’t tell. Notice the p-value—the probability of seeing a coefficient of $123 in a world where the true difference between the states is $0 is 0.164. That’s a little atypical, but not shocking—it’s higher than our lowest evidentiary standard of 10% (0.1). We thus cannot conclude that there’s a statistically significant difference in median property tax rates between California and Arizona, after controlling for home values and households with kids.
The same applies to both Idaho and Nevada. We originally said that Idahoans pay $9.50 more than Arizonans on average, but the probability of seeing a coefficient of 9.5 in a world where it’s actually 0 is 0.908. That’s a completely normal and expected number in a world where there’s no difference. Also note that the confidence interval ranges from -150 to 170—that’s all over the map and definitely includes zero. We can thus say that after controlling for home values and households with kids, there’s no significant difference between per-household property tax rates in Arizona and Idaho.
The regression model shows that Nevadans pay $102.50 more in taxes than Arizonans, but this too is not statistically significant (p = 0.299). The confidence interval includes 0, and the’s a 29.9% chance of seeing a difference as big as 102.5 in a world where it’s actually zero. Taking home values and households with kids into account, there’s no significant difference between per-household property tax rates in Arizona and Nevada.
Finally, we can do something with Utah. After controlling for home values and kids, people in Utah pay $213.20 less in property taxes than their counterparts in Arizona, and the difference is statistically significant (p = 0.021). There’s a 2.1% chance of seeing a difference at least as large as 213 in a world where it’s actually 0, and the confidence interval doesn’t include 0. Because 2.1% is lower than our typical standard of 5% (0.05), we can declare significance.
That’s a lot of really verbose writing. In real life, you’d probably only really be concerned about one or two of the coefficients and only include the others to control for other effects (like, in general, you don’t care about the exact nature of state effects, but they’re useful for picking up some of the variation in your outcome variable).
Here’s how you could write up this entire model with all coefficients in a complete, concise paragraph:
We used ordinary least squares (OLS) regression to explain the variation in per-household property tax rates in five Western states, based on the median home value and the proportion of households with kids in each county. We also controlled for state effects. With these three explanatory variables, our model explains nearly 84% of the variation in tax rates in these states. Home values and proportions of households with kids both have a statistically significant association with tax rates. Controlling for all other variables in our model, on average, each $100 increase in median home values is associated with a $0.40 increase in property taxes (p < 0.001), while every 1% increase in the proportion of households with kids is associated with a $14 increase in taxes (p < 0.001). With the exception of Utah, where property taxes are $213 lower than those in Arizona, on average (0 = 0.021), none of the individual state coefficients are statistically significant.
Now you get to interpret the coefficients for two regression models. You’ve already learned how to do the bulk of the interpretation: for every one unit increase in X there’s an associated β change in Y. Now you just need to incorporate hypothesis testing and determine if those effects are statistically significant.
Here are your basic templates:
- If statistically significant: After controlling for blah, a one unit change|increase in X is associated with a β change|increase|decrease in Y, and the coefficient|effect|association is statistically significant (p =
- If not statistically significant: After controlling for blah, X is not significantly associated with Y (p =
Open a new R Markdown file or R script and run the following code on your computer. Note that
read_csv() points to a URL on the internet. I did this so you don’t need to make a new RStudio project, download data, and put the data in that folder. This simplifies things.In real life, though, you’d want to put these files locally on your computer—imagine if you’re loading important data from some website that changes its URLs a year from now (or shuts down and stops existing!). It would be bad if your code tried to load data from dead URLs.
Interpret each of the coefficients in these two models. Go through each and determine if each is statistically significant or not. Try to write up a concise paragraph like the one above for each of the models.
The answers are at this page here. Don’t look at them until you’ve tried as a team.
We’ve worked with this world happiness data before. As a reminder, here’s what these different variables measure:
happiness_score: A composite index ranging from 1 to 8, with higher numbers indicating greater national happiness
life_expectancy: The average life expectancy, measured in years
access_to_electricity: The proportion of the population with regular access to electricity, measured on a 0-100 scale
region: Region the country is in. The base case is East Asia & Pacific, since it comes first alphabetically.
income: How rich a country is. There are four possible categories: High income, Upper middle income, Lower middle income, and Low income. High income is the base case because it’s first alphabetically.
library(tidyverse) library(moderndive) happiness <- read_csv("https://statsf18.classes.andrewheiss.com/data/world_happiness.csv") # The base case for region is "East Asia & Pacific" # The base case for income is "High income" model_happiness <- lm(happiness_score ~ life_expectancy + access_to_electricity + region + income, data = happiness) model_happiness %>% get_regression_table() model_happiness %>% get_regression_summaries()
We looked at this data during class 8. Here’s what these different variables measure:
leave_share: The proportion of the constituency that voted to leave the EU, measured on a 0-100 scale
con_2015: The proportion of the constituency that voted for the Conservative party in 2015, measured on a 0-100 scale
lab_2015: The proportion of the constituency that voted for the Labour party in 2015, measured on a 0-100 scale
ukip_2015: The proportion of the constituency that voted for UKIP in 2015, measured on a 0-100 scale
degree: The proportion of the constituency with a university degree, measured on a 0-100 scale
age_18to24: The proportion of the constituency that is between the ages of 18-24, measured on a 0-100 scale
born_in_uk: The proportion of the constituency that was born in the UK, measured on a 0-100 scale
unemployed: The proportion of the constituency that is unemployed, measured on a 0-100 scale
male: The proportion of the constituency that is male, measured on a 0-100 scale
results_brexit <- read_csv("https://statsf18.classes.andrewheiss.com/data/brexit_results.csv") model_brexit <- lm(leave_share ~ con_2015 + lab_2015 + ukip_2015 + degree + age_18to24 + born_in_uk + unemployed + male, data = results_brexit) model_brexit %>% get_regression_table() model_brexit %>% get_regression_summaries()


Senior Technical Team
Mason Macklem, PhD, Senior Scientist
Mason Macklem has over 15 years of research experience in high-performance computing, distributed computing networks, and visual data compression. He joined the Research and Development division of Packeteer Networks Inc. in 2000, focusing on perceptual data compression and human visual system modeling. From 2002 to 2004, he worked at the Centre for Experimental and Constructive Mathematics (CECM) and the CoLab Visualization Lab at Simon Fraser University, and from 2004 to 2008 at the Dalhousie Distributed Research Institute and Virtual Environment (D-DRIVE), focusing on numerical optimization and high-performance computing. Dr. Macklem completed his BA in Mathematics at University of British Columbia in 2000, his MSc in Mathematics at Simon Fraser University in 2002, his PhD in Computer Science in 2008 from Dalhousie University, and a Postdoctoral Fellowship at UBC-Okanagan from 2009-2010. Dr. Macklem is a two-time NSERC scholar and a three-time Killam Fellow. Dr. Macklem joined the Kelowna technology team in June 2011 as a Senior System Architect, designing server-side systems for a variety of data-centric projects, including highly-scalable automated systems for data ingestion and automated signal analysis, algorithm and system design for variable-priced computing resources, and perceptual video encoding.
Herre Wiersma, Senior Technologist
Herre Wiersma completed his BSc in Mathematics and Computer Science at Acadia University in Nova Scotia in 1999, and his MSc in Mathematics in 2002 at Simon Fraser University. Mr. Wiersma’s research concerned nonlinear functional analysis and convex optimization, with a focus on monotone operator theory. At SFU, Mr. Wiersma also worked at the Centre for Experimental and Constructive Mathematics (CECM) on stock option pricing models using entropy maximization, and distributed cluster-based software for parallel numerical optimization. In 2002, he moved to the Mathematics department at Dalhousie University, working with the Dalhousie Distributed Research Institute and Virtual Environment (D-DRIVE) research lab. At D-DRIVE, Mr. Wiersma engaged in a variety of projects, including: contributing to the Digital Library of Mathematical Functions (DLMF) project, a National Institute of Standards and Technology reference tool widely used by scientists and engineers; procuring, building and benchmarking the D-DRIVE compute cluster for use in distributed mathematical research; and further work in monotone operator theory, nonlinear functional analysis, and optimization. Mr. Wiersma joined the Kelowna technology and software development team in August of 2011, developing infrastructure for our content delivery network, distributed video trancoding systems, peer-to-peer data transfer protocols, and low-latency video communication.


Having spent a Sunday morning having myself filmed (see previous post) and my stroke deconstructed, I have since been trying to put into practice some of the advice I was given during and after that lesson.
It has not been easy trying to remember them all at once, but I am getting there slowly.
I thought it would be worthwhile giving myself a progress check the other day, so I decided to reproduce the 10 km session I did before work a few weeks ago, a little while before I had the lesson.
Basically, this means turning up at the pool at 6 AM, and swimming 10 km, split over 5 sets of 1 mile (64 lengths), and one set of 2 km (80 lengths), adding up to 10 km in total. No more than a minute between sets to have a nice cool drink. Similar to what you might try and do over the first three hours of a channel swim.
Let’s look at total swim time first:
Before lesson: 2h 50 m 49 s
After lesson: 2 h 52 m 50 s
Oh. That’s a bit disappointing. But hey ho, 2 minutes over 6 and a bit miles isn’t a complete disaster is it? This corresponds to an average 100 m time over the whole thing of 1:39 increasing to 1:40.
Now let’s look at strokes. My Poolmate Pro counts strokes as 1 per revolution of my left arm, so roughly double it and you get the actual strokes per length:
Before lesson: 9,10,9,9,9,9
After lesson: 8,8,8,8,8,8
Wow. So what I have done is lose slightly on the speed (roughly a quarter second per length), but knocked a couple of (real world) strokes off every length. So I guess I must be stroking slower then, for the maths to make sense?
Before lesson stroke rate: 22,24,22,22,22,22.
After lesson stroke rate: 20,19,19,19,18,19
Yep – all makes sense. I am sure I am not really going at 38 spm. In the open water this would be higher, without turning at the wall and pushing and gliding. Probably in the low 50s.
Finally, the software on Poolmate Pro calculates something called efficiency = strokes taken to swim the length + time taken to swim the length:
Before lesson efficiency: 33,35,34,34,34,34 – Average = 34
After lesson efficiency: 32,33,33,33,34,33 – Average = 33
So I am slightly more efficient over the 10 km. Well that’s nice.
Hopefully the improvements I am trying to make to my stroke will bed in a bit more over time. In particular I want to improve my relaxation on the recovery part of my stroke, and get every pull under the water giving me the maximal propulsion. I am conscious that on both occasions I felt quite fatigued after 10 km. I could have kept going for a while longer, maybe another hour, maybe another couple. But that’s only 16 km, less than halfway over the channel, and in ideal pool conditions with no waves, warm (maybe too warm) water. I have along way to go, but am starting to feel much more confident about Windermere next July.


Definition: A Greek astronomer whose heliocentric theory predated Copernicus by almost 2000 years. Background: Aristarchus of Samos was a Greek astronomer and mathematician belonging to the Pythagorean school. They attempted to understand the universe in terms of geometry and arithmetic. Aristarchus proposed that the earth rotates on its axis daily and that heavenly bodies, including the earth, revolve around the sun. This heliocentric view predated Nicolaus Copernicus (1473-1543) by almost 2000 years! He is also known as the first astronomer to calculate astronomical distances using mathematics, which up to that time had been a matter of guesswork and speculation. A prominent lunar crater (the brightest feature on the moon) is named after him. Further Reading:
Dictionary of Scientists (1999). Aristarchus of Samos [On-line] Available: http://www.xrefer.com/entry/493941
Heath, T. (1913). Aristarchus of Samos: The ancient Copernicus. Oxford: Clarendon Press.
O'Connor, J. J., & Robertson , E. F. (1999). Aristarchus of Samos [On-line] Available: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aristarchus.html
Related Terms: Copernicus, Nicolaus (1473 - 1543)
Pythagoras (cs. 485 - 415 B.C.)
Self-Instructional Resources: Take a 1-item self-test over this concept.
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HomeLiteratureCOMPUTATIONAL PARTIAL DIFFERENTIAL EQUATIONS LANGTANGEN PDF
COMPUTATIONAL PARTIAL DIFFERENTIAL EQUATIONS LANGTANGEN PDF
September 26, 2020
Computational Partial Differential Equations: Numerical Methods and Diffpack . However, Langtangen is quick to point out that if you take the time to learn C++. Computational Partial Differential. Equations. Numerical Methods and Diffpack Programming. 2nd Edition. Hans Petter Langtangen. Computational Partial Differential Equations: Numerical Methods and Diffpack Hans Petter Langtangen, Otto Munthe, Solving systems of partial differential.
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Software tools for multigrid methods. Communications in Applied Numerical Methods, 4: Texts in Computational Science and Engineering, vol 1. Numerical methods for incompressible viscous flow. A flexible software design for non-standard finite element methods.
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A comparison of methods for the statistics of slow-drift oscillations. Advances in Water Re- sources, A software framework for mixed finite element programming. Kluwer Academic Publishers, Lecture Notes in Computational Science and Engineering. Research interests – methods for creating flexible scientific – software object-orientation, scripting- the generic programming environment Diffpack, numerical methods for flow problems e.
Error estimates for the linear Navier-Stokes equations. A Python framework for verifying numerical solutions of partial differential equations.
This textbook introduces several major numerical methods for solving pagtial partial differential equations PDEs in science and engineering, including elliptic, parabolic, and …. A finite element method for fully nonlinear water waves. Increasing the efficiency and reliability of numerical software development.
Computational Partial Differential Equations. An investigation of domain decomposition methods for one-dimensional dispersive long wave equations.
A unified finite element model for the injection molding process. Texts in Com- putational Science and Engineering, vol 3. Journal of Computational Physics, Performance modeling of PDE solvers. Bruasetlangtqngen prize, Norsk Eksportrad og Norsk Dataforenings kreativitetspris Award for research in information technology with A.
A software framework for easy parallelization of PDE solvers. The basic equations dirferential eulerian continuum mechanics. This book will have strong appeal to interdisciplinary audiences, particularly in regard to its treatments of fluid mechanics, heat equations, and continuum mechanics.
Finite element modeling of elastic structures. International Center for Scientific Research. A method for smoothing derivatives of multilinear finite el- ement fields. Implicit finite element methods for two-phase flow in oil reservoirs.
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Papers in Books and Proceedings with Review H. A math- ematical model for the melt spinning of polymer fibers. Springer, 2nd edition,
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The Science of Santa
By John Michael Keller
Based on an original letter in the US Skeptic 2(3), 6-7, 1994, here updated and converted into metric equivalents.
Thanks to time zones, and assuming travel is east to west (which seems logical), Santa has a maximum of 31 hours in which to visit every good child. So he needs to average 130 million / 31 hours = 4.2 million home visits per hour or 1160 per second. This means that for every home with a good child, Santa has less than 1/1000th of a second in which to park, jump down the chimney, distribute presents, eat the snacks left for him, get back up the chimney, and move on to the next home. Your camera will need a fast shutter speed.
Again, to keep it simple, assume that these 130 million homes have the same average density as homes in the USA (10 per sq km), which means that on average the homes are (1/10)0.5 = 0.32 km apart. However, only one-third of the Earth is land, which will increase the average distance apart to 3 x 0.32 = 1.0 km. So Santa has to travel a total of 130 million x 1.0 = 130 million km in 31 hours, not counting stops for reindeer games etc.
In other words Santa's sleigh has to average 130 million / 31 = 4.2 million km per hour or 1160 km per second, which is roughly 1160 / 0.344 = 3370 times the speed of sound and 1160 / 300,000 = 1/250th the speed of light. For comparison, the fastest space probe moved at a poky 44 km per second. A conventional non-Santa reindeer can reach 24 km per hour, at which speed it would take 130 million / 24 = 5.4 million hours or 615 years to reach every good child. Yes, flying at Mach 3370 is quicker. However, the acceleration needed to reach 1160 km per second in less than 1/1000th of a second is more than 1160 x 1000 / 0.0098 = 118 million g, so Santa has to hold on tight.
What payload is the sleigh carrying? Assuming that each child gets no more than a medium-sized set of Lego weighing 2 kg, the sleigh at lift-off is carrying 2 x 130 million kg or 260,000 tonnes, not counting Santa, who is traditionally described as overweight (perhaps inevitably after 130 million snacks). On land a conventional reindeer can pull about 150 kg or 0.15 tonne. Assuming that a flying reindeer can lift ten times this or 1.5 tonnes, to lift 260,000 tonnes Santa will need 260,000 / 1.5 = 170,000 flying reindeer.
Assuming each flying reindeer weighs 0.5 tonne, 170,000 will increase the total takeoff weight to nearly half a million tonnes, which is roughly five times the weight of the world's largest passenger cruise ship. Now half a million tonnes travelling at 1160 km per second will encounter the same enormous air resistance as a spacecraft re-entering the earth's atmosphere. The lead pair of reindeer will burst into flame almost instantaneously, exposing the next reindeer pair, and so on, creating a deafening sonic boom at the same time.
In no time at all the entire reindeer team, Santa, and 130 million presents will be vaporised in a huge fire ball. If for some reason the good children in your household received no presents this Christmas, that's why.


How to calculate load factor on a per tenant basis for a commercial office space?
Asked 3 months ago
Hi, I manage an office building and I want to understand if the space is being used efficiently and if the rental income is optimized for each tenant. I’d like clarification on how to calculate the load factor on a per-tenant basis, as our tenants occupy different spaces with different layouts and configurations. Can anyone provide guidance on this? Many thanks.
Tuesday, March 21, 2023
To find the load factor, you’ll need two figures: The tenant's usable square footage and the entire building's rentable square footage. The basic method to calculate the load factor involves simply dividing the two figures.
For example, if your building has 10,000 square feet of space, of which one tenant uses 1,000 square feet, the load factor for that tenant would be 10.
By calculating the load factor for each tenant, you can identify how much space they occupy versus the space available. POC Systems allows you to visualize load factors graphically.
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Writing & Language
The creation of today's animated films, frequently hailed for their lifelike artistry, relies heavily on mathematics. Aspiring animators would benefit from a strong background in math if they wish to create, for instance, the rounded shapes making up an old man's knobby knuckles or a young girl's bouncy hair. One of the algorithms now considered an industry standard repeatedly subdivides a polygon by splitting and averaging the midpoint of a line. Developed by Pixar Animation Studio's Tony DeRose, this algorithm eliminated the problem of discernible edges of polygons by replacing them with the seamlessly rounded curves of the parabolas that emerge as a result of the polygon subdivision process. Innovations such as DeRose's depend on an understanding of applied mathematics in areas such as geometry, linear algebra, trigonometry, and calculus.
Which choice most effectively sets up the information that follows in the paragraph?
B.Animators use algorithms, or sequences of mathematical steps,
C.The animation industry has grown increasingly dependent on computer-generated imagery
D.In fact, animators draw on a variety of skills, many of them in fields outside of a traditional understanding of art,


Linear Algebra (subject)
Linear algebra is a branch of abstract algebra where we study the structure on, or structure resulting from vector spaces connected (via the arrows or homomorphism of) linear maps, these are simply maps that preserve the structure of vector spaces.
After that comes bilinear maps (a special case of multilinear maps) and onwards via the tensor product. This has links to analysis via differentiation as a vector space equipped with a norm are the minimum requirements to be able to differentiate
The linear algebra project contains:


YOU DO NOT HAVE TO TURN IN MANILA PATTERNS.
11:00 Eunice and Eliva
11:15 Kathy and Zin
11:30 Angel and Linet
11:45 Alexandria and Mable
12:00 Bridget and Brittany
12:15 Amy and Stef
Hi Guys, I’m out sick today. Please see Yoli or Veronica on the third floor for a key. Dr. Davis might also be around.
See you Tuesday for presentations.
In preparation for Project #5, this is a great opportunity to make your visit to the IFF!
Annular Hyperbolic Surfaces
Saturday, January 26 from 3pm – 6pm
Mathematicians long believed it was impossible to make models of hyperbolic space. Then the great Cornell geometer Bill Thurston discovered he could make them from circular rings of paper. Thurston’s model in turn inspired Dr. Daina Tamina to invent hyperbolic crochet. In this workshop we offer a rare opportunity to construct Thurston’s model of hyperbolic space. Led by mathematician Darryl Yong and using beautiful die-cut rings stamped specially for this event, we’ll construct hyperbolic surfaces. This is a one-off workshop. Seating is limited.
Dr. Darryl Yong is a professor of mathematics at Harvey Mudd College with a special interest in math education and outreach.


2 edition of new mathematics dictionary and handbook. found in the catalog.
new mathematics dictionary and handbook.
Robert W. Marks
|Series||Bantam science and mathematics|
Books about all areas of mathematics including the history of mathematics, recreational mathematics, and any other book exploring the wonders of math. Score A book’s total score is based on multiple factors, including the number of people who have voted for it and how highly those voters ranked the book. Book definition: A book is a number of pieces of paper, usually with words printed on them, which are | Meaning, pronunciation, translations and examples.
This book is about the topic of mathematical analysis, particularly in the field of engineering. This will build on topics covered in Probability, Algebra, Linear Algebra, Calculus, Ordinary Differential Equations, and others. ( views) Handbook of Engineering Mathematics by Walter E. Wynne, William Spraragen - Van Nostrand, book: [noun] a set of written sheets of skin or paper or tablets of wood or ivory. a set of written, printed, or blank sheets bound together between a front and back cover. a long written or printed literary composition. a major division of a treatise or literary work. a record of a business's financial transactions or financial condition.
Handbook of Mathematics, Sixth Edition by I. N Description. This guide book to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. as well as Information Processing. Besides many enhancements and new. 2. Gallery of official portraits of the Presidents of the United states -- Social studies -- Science and mathematics -- English and literature -- Education and careers -- Webster's new world dictionary, second concise edition -- v. 3.
Drilling for scientific purposes: report of the International Upper Mantle Symposium, Ottawa, 2-3, September, 1965. edited by D.C. Finlay and C.H. Smith
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The new mathematics dictionary and handbook Hardcover – January 1, by Robert W Marks (Author) See all 5 formats and editions Hide other formats and editions. Price New from Used from Hardcover "Please retry" $ — $ Author: Robert W Marks.
Buy The New Mathematics Dictionary and Handbook on FREE SHIPPING on qualified orders. The new mathematics dictionary and handbook by Marks, Robert W.
Publication date Topics Mathematics, Mathematics Publisher New York, Bantam Books Borrow this book to access EPUB and PDF files. IN COLLECTIONS. Books to Borrow. Books for People with Print Disabilities.
Internet Archive : New mathematics dictionary and handbook. book new mathematics dictionary and handbook. [Robert W Marks] New mathematics dictionary and handbook.
New York, Bantam Books  (OCoLC) Document Type: Book: All Authors / Contributors: Robert W Marks. Find more information about: OCLC Number: This book helps the student complete the transition from purely manipulative to rigorous mathematics, with topics that cover basic set FREE Artinian Modules over Group Rings.
Mathematics Dictionary and Handbook by Eugene D. Nichols, Sharon Schwartz and a great selection of related books, art and collectibles available now at ISBN: OCLC Number: Description: vi, pages: illustrations (some color), color map ; 22 cm: Responsibility: Eugene D.
Nichols. ‘The depth of information provided is admirable’ New ScientistAuthoritative and reliable, this A-Z provides jargon-free definitions for even the most technical mathematical terms. With 3, entries ranging from Achilles paradox to zero matrix, it covers all commonly encountered terms and concepts from pure and applied mathematics and statistics, for example, linear algebra, optimisation.
New Listing Webster's New World College Dictionary, Fourth Edition [Book with CD-ROM] [ The out of 5 stars 7 product ratings 7 product ratings - Webster's New World College Dictionary, Fourth Edition [Book with CD-ROM] [ The. Handbook of the Mathematics of the Arts and Sciences Sriraman, B.
(Ed) () The goal of this Handbook is to become an authoritative source with chapters that show the origins, unification, and points of similarity between different disciplines and. New mathematics dictionary and handbook. Here, in handy dictionary form, are the answers to your questions about mathematics, presented in simple, clear nontechnical language.
Show health and safety information. Please be aware that resources have been published on the website in the form that they were originally supplied. This means that. Here is an unordered list of online mathematics books, textbooks, monographs, lecture notes, and other mathematics related documents freely available on the web.
I tried to select only the works in book formats, "real" books that are mainly in PDF format, so many well-known html-based mathematics web pages and online tutorials are left out. Results 1 - 50 of 65 for Reference Textbooks 1. Quick Review Math Handbook hot words hot topics by Glencoe McGraw-Hill Staff ISBN: List Price: $ Mathematics Dictionary book.
Read 2 reviews from the world's largest community for readers. Significant updates and revisions in this edition accommodate /5. Handbook of Mathematics is an important, useful and compact reference book suitable for everyday study, problem solving or exam revision for class XI – XII.
This book is a multi-purpose quick revision resource that contains almost all key notes, terms, definitions and formulae that all students & professionals in mathematics will want to have. U.S. Department of Transportation Federal Aviation Administration Independence Avenue, SW Washington, DC () tell-FAA (() ).
This book represents a comprehensive handbook on mathematics covering analysis, algebra, geometry, foundations of mathematics, calculus of variations and optimization, theory of probability and mathematical statistics, numerical mathematics and.
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Grade 12 Mathematics Handbook - Free download Ebook, Handbook, Textbook, User Guide PDF files on the internet quickly and easily. This book is with me since I needed to have a Reference book for Mathematics. The Preface says: This handbook is a continuation of the Handbook of Elementary Mathematics by the same author and includes the material usually studied in a mathematics course in Written:.
Click to read more about The new mathematics dictionary and handbook by Robert W. Marks. LibraryThing is a cataloging and social networking site for bookloversAuthor: Robert W. Marks.The IB Diploma Programme is aimed at students aged 16 to 19, for the final two years of high school.
We have developed a series of textbooks to align with the new Diploma Programme Mathematics courses for first teaching in August Our current editions for the IB Middle Years Programme (MYP) have been written to accompany the framework.This book is intended to be used by children ages 5 to 6.
Other age groups will also benefit from the book. Anyone can use this book globally, although the curriculum may differ slightly from one region to the other. This is so because the core content of Mathematics is the same around the world.


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Derivatives of inverse trigonometric functions Calculator online with solution and steps. Detailed step by step solutions to your Derivatives of inverse trigonometric functions problems online with our math solver and calculator. How to Use the Additive Inverse Calculator? Follow these simple steps to use the additive inverse calculator: Step 1: Enter the number whose additive inverse you want, in the input box. Step 2: Click on "Calculate" to find the additive inverse of the number.
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Find more Engineering widgets in Wolfram|Alpha. 2021-4-10 Inverse Cosine Calculator You can enter input as either a decimal or as the adjacent over the hypotenuse There are 2 different ways that you can enter input into our arc cos calculator. Method 1: Decimal This inverse matrix calculator help you to find the inverse matrix. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the inverse matrix using Gaussian elimination.
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Definite Integral Calculator; Derivative Calculator; Double Integral Calculator
Inverse of Matrix Calculator The calculator will find the inverse of the square matrix using the Gaussian elimination method, with steps shown. Get the free "Inverse Function Calculator - Math101" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.
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Dɪsᴄᴏᴠᴇʀ Mᴏʀᴇ Aᴛ Tʜᴇ Cᴀʟᴄᴜʟᴀᴛᴏʀ Gᴜɪᴅᴇ You're probably going to have to explain what you mean by "inverse function". Both the AWD calculator and wavescan calculators have 1/x buttons - which to me Matrices, when multiplied by its inverse will give a resultant identity matrix.
Enter the value to be used in the Inverse Sine (arcsin) trigonometric function Calculate
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iMatrix - Matrix Calculator av Mobixee Ltd - iPhone/iPad
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This is a free online Inverse Sine (arcsin) calculator. You can calculate the value of Inverse Sine (arcsin) trigonometric function instantly using this tool.


“Agewmetrhtoz mhdeiz eisitw”
[If one were to convert the font above to Greek letters, it would read:]
“Only he who is familiar with geometry shall be admitted here.”
(Apparently a knowledge of the Greek language was also important.)
This website is not quite so stringent, and will happily admit any she who is interested in entering. (The guys will, of course, still need to know some geometry.) Alternatively, anyone who can carry a tune may also enter — in a bucket or by any other means. Music is, after all, based on geometry and mathematics.
For anyone who doesn’t think of themselves as being “familiar with geometry,” the words attributed to Plato’s school might be considering insulting, and thereby construed to have little or no validity. Any grapes from the discourteous must inevitably be sour.
This circumstance is indeed unfortunate in that anyone who is not familiar with geometry, anyone who does not appreciate the premise that Philosophy is about understanding the meaning of a mathematical ratio (the Golden Mean), and anyone who adheres exclusively to a literal interpretation of spiritual teachings… Such an individual might find it difficult to fully comprehend and appreciate the exquisite beauty of the universe. One might even miss the underlying delights in sacred geometry jewelry.
One appealing aspect of geometry is that it constitutes A Graphic Description of the universe, one which is easily visualized, and in some cases can even be aesthetically pleasing. Mathematics in general can often get bogged down in jargon, i.e. abstract symbols which tend to hide the meaning of an equation. But geometrical figures have body, depth, width and the advantage that a picture is worth a thousand words.
At its most fundamental level, Philosophy may be said to involve the use of reason and interactive communications in seeking truth and knowledge. This laudable goal is undertaken in an attempt to understand reality: i.e. the causes and nature of things, the principles governing existence, the material universe, the perception of physical phenomena, and the analysis of human behavior. Which, when you think about it, is quite inclusive. Philosophy is also notably distinct from the mundane — the dull, routine, of-this-world boredoms, all of which include: work, sports, politics, money, lack of money, and whatever else that distracts us from studying philosophy.
The word, philosophy, derives from the Latin philosophia, and can be written as phi-lo-sophia in order to investigate the meaning of the word based on its parts (using the philosophical tools of “reason and interactive communications”). The first portion, “phi” is simply the Greek letter F or f (equivalent to F or f in the English alphabet). Meanwhile, “lo” (according to the Oxford dictionary) is an archaic form of “calling attention to an amazing sight” (as in “lo and behold”). “Sophia”, on the other hand — besides being the Goddess of Wisdom (and to which the biblical Book of Proverbs is essentially dedicated) — is considered to be a combining form meaning “knowledge”, “thought”, or “wisdom”.
The end result is that philosophia — or Phi lo Sophia — can be viewed as the wisdom, knowledge, or calling attention to the amazing sight of Phi. Which is precisely what sacred geometry is all about.
A dedicated study of a simple Greek letter might appear to the unenlightened as something short of an illuminating quest. This allegedly simple Greek letter, however, is a mathematical symbol for what is known as the Golden Mean — a mathematical ratio which is enormously influential in understanding everything from the universe and nature, to ancient cultural monuments and esoteric traditions, to the rise and fall of hemlines or stock prices. The Golden Mean (or phi) is by its nature the essential ingredient in Sacred Geometry, Numerology, and as it turns out, Connective Physics and science in general.
The inherent connection between philosophy, sacred geometry, and numbers is manifested in numerous ways. There is the Geometry of Alphabets, i.e. the Hebrew, Greek, and Arabic alphabets being sacred and the first five books of the Bible being representations of sacred geometry. Much of this understanding was contained in the Ha Qabala, an aspect of Jewish and Christian mysticism which is worthy of prolonged and concentrated study.
It has also been said that “Sacred Geometry exists beyond religion, belief or nationality. It unites us all in the belief of the one. By understanding the concept of the one each of us can take control over our own lives and break free of the circle of chaos in the material world which Judaism calls the world of lies.” [quoted from ka-gold’s Age of Aquarius article] This is significant stuff!
For just an inkling of what Sacred Geometry is all about, consider just one facet of it, the Golden Mean or Golden Number. The latter website provides a wealth of information in a very nicely designed web-package.
The idea that philosophy is inherently mathematical might not be construed as good news for mathematicallyuninclined philosophers. But as pointed out in A Non-Mathematical Digression, it might be a good idea to momentarily address this potential obstacle.
Given that disclaimer, the amazing phenomenon of Fibonacci Numbers (complete with “bunnies”), the Golden Mean (a very rewarding subject), Golden Mean Mathematics (for those so unwise or so inclined), The Great Pyramids (a really big subject), Nines (e.g. cloud addresses), Transcendental Numbers (for those who really want to get above the crowd), Cycles, Music, and the Harmony of the Spheres (harmonies of all kinds being a good thing)… All of these glories await your click of the mouse.
For the more sophisticated mathematicians, an excellent beginning commentary can be found at b>http://mathworld.wolfram.com/GoldenRatio.html>. As an initial example, the referenced website describes theGolden Mean (Ratio) as “A number often encountered when taking the ratios of distances in simple geometric figures such as the pentagram, decagon and dodecagon. It is denoted phi, or sometimes tau (which is an abbreviation of the Greek “tome,” meaning “to cut”). Phi is also known as the divine proportion, golden mean, and golden section and is a Pisot-Vijayaraghavan constant. It has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers.” Wow.
2003© Copyright Dan Sewell Ward, All Rights Reserved
Highly recommend following the links for further information, this is a great site 🙂


A floating-point unit (FPU, colloquially a math co-processor) is a part of a computer system specially designed to carry out operations on floating-point numbers. Typical operations are addition, subtraction, multiplication, division, and square root. Some FPUs can also perform various transcendental functions such as exponential or trigonometric calculations, but the accuracy can be very low, so that some systems prefer to compute these functions in software.
In general-purpose computer architectures, one or more FPUs may be integrated as execution units within the central processing unit; however, many embedded processors do not have hardware support for floating-point operations (while they increasingly have them as standard, at least 32-bit ones).
When a CPU is executing a program that calls for a floating-point operation, there are three ways to carry it out:
- A floating-point unit emulator (or floating-point library), such as SANE, SoftwareFPU, or PowerFPU.
- An add-on FPU, a dedicated hardware unit such as the Motorola 68881 or 68882.
- An integrated FPU, included in processors such as the Motorola 68040, and all PowerPC and Intel processors used by Apple, which made separate add-on FPUs obsolete.
- "The IBM System/360 Model 91: Floating-Point Execution Unit", IBM Journal of Research and Development, January 1967, pp. 34–53.
- Bruce Dawson (2014-10-09). Intel Underestimates Error Bounds by 1.3 quintillion. randomascii.wordpress.com. Retrieved on 2020-01-16.
- FSIN Documentation Improvements in the “Intel® 64 and IA-32 Architectures Software Developer’s Manual”. intel.com (2014-10-09). Retrieved on 2020-01-16.


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MyMaths is an interactive online teaching and homework subscription website for schools that builds pupil engagement and consolidates maths knowledge. It is used in over 70 countries by approximately four million students each year!
Through homework help online services, students can get help with homework in any of the subjects that they are studying. They will also have access to information that is useful through online dictionaries, encyclopedias, atlases, thesauruses, and almanacs.
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- This event has passed.
Mathematical Discourse: Using and Connecting Mathematical Representation
January 30 @ 4:00 pm - 5:30 pm
Prepare yourself to unleash the full potential of mathematical communication! Join us as we delve into the fascinating world of mathematical representations and explore how they can deepen our understanding of concepts. Engage in a mathematical task using a variety of representations to enrich your understanding and spark meaningful discourse. Reflect on the incredible impact different representations can have on mathematical discourse and discover practical strategies for fostering connections and enhancing mathematical communication skills. This is a unique opportunity to discover the exhilarating possibilities of mathematical discourse!


How well do time series (intrinsic) and cross-sectional (relative) momentum work for different types of currency exchange rates? In their April 2017 paper entitled “Momentum in Traditional and Cryptocurrencies Made Simple”, Janick Rohrbach, Silvan Suremann and Joerg Osterrieder compare the effectiveness of time series and cross-sectional momentum as applied to three groups of currency exchange rates: G10 currencies; non-G10 conventional currencies; and, cryptocurrencies.
To measure momentum they employ three pairs (one fast and one slow) of exponential moving averages (EMA) spanning short, intermediate and long horizons.
AHL Explains - Momentum
When the fast EMA of a pair is above (below) the slow EMA, the trend is positive (negative). They extract a momentum signal for each exchange rate from these three EMA pairs by:
- For each EMA pair, taking the difference between the fast and slow EMA.
- For each EMA pair, dividing the output of step 1 by the standard deviation of the exchange rate over the last three months to scale currency fluctuations to the same magnitude.
- For each EMA pair, dividing the output of step 2 by its own standard deviation over the last year to suppress series volatility.
- For each EMA pair, mapping all outputs of step 3 to signals between -1 and 1.
- Averaging the signals across the three EMA pairs to produce an overall momentum signal.
The time series portfolio holds all currencies weighted each day according to their respective prior-day overall momentum signals.
The cross-sectional portfolio is each day long (short) the three currencies with the highest (lowest) overall momentum signals.
Key performance metrics are annualized average gross return, annualized standard deviation of returns, annualized gross Sharpe ratio (assuming risk-free rate 0%) and maximum drawdown. Using daily foreign currency exchange rates for 23 conventional currencies and seven cryptocurrencies versus the U.S.
dollar as available through late March 2017, they find that:
- For G10 currencies (data start July 1974):
- The time series (cross-sectional) portfolio generates annualized average gross return 22% (7%), with annualized standard deviation 41% (24%) and annualized gross Sharpe ratio 0.53 (0.27).
- Maximum drawdown of the time series (cross-sectional) portfolio is about -17% (-35%).
This drawdown for the cross-sectional portfolio commences in 2001 and has not yet recovered its high water mark.
- For 13 non-G10 conventional currencies (data start January 1995):
- The time series (cross-sectional) portfolio generates annualized average gross return 31% (21%), with annualized standard deviation 38% (28%) and annualized gross Sharpe ratio 0.82 (0.78).
- Maximum drawdown of the time series (cross-sectional) portfolio is about -25% (-35%).
- For seven cryptocurrencies (data start March 2015):
- The time series (cross-sectional) portfolio generates annualized average gross return 299% (359%), with annualized standard deviation 185% (242%) and annualized gross Sharpe ratio 1.62 (1.48).
- Maximum drawdown of the time series (cross-sectional) portfolio is about -40% (-70%).
- Across the samples:
- Time series portfolios offer higher risk-adjusted performances returns than corresponding cross-sectional portfolios.
- Strategies work well during calm markets but suffer crashes during unusual events.
In summary, evidence indicates that time series momentum generally works better than cross-sectional momentum across different groups of currency exchange rates on a gross risk-adjusted basis.
Cautions regarding findings include:
- As noted in the paper, return calculations are gross, not net.
Accounting for trading frictions (likely substantial due to daily portfolio reformation) would reduce all returns. Since turnovers may vary by currency and by strategy (time series or cross-sectional), net findings may differ from gross findings.
- There may be an issue of performing all required calculations and executing portfolio reformation in a timely manner on a daily basis.
- Assuming a risk-free rate of 0% in calculating Sharpe ratio seems unreasonable, especially for the sample of G10 currencies that commences in the mid-1970s.
- Signal generation rules are elaborate, suggesting potential for snooping bias in rule construction and/or parameter settings and therefore overstatement of expected returns.
- As noted in the paper, the sample period for cryptocurrencies is extremely short in terms of variety of currency market and economic conditions.
See also “When Carry, Momentum and Value Work”.


Welcome to TOMAC's official website! TOMAC (Test Of Mathematical Competence) is a mathematical test based on score. Unlike other mathematical tests, TOMAC doesn't have the notion of pass or fail. Instead, the examinee's mathematical ability will be measured by score. Since the score will be the proof of capability in mathematics, some companies and schools uses TOMAC for the entrance examination. Unfortunately, this test is given exclusively in Japan. Currently, there is no web based registration. For more information please contact us by e-mail to info.
The Six Characteristics of TOMAC
1. Presents mathematical profiency by score
As mentioned above, TOMAC doesn't have the notion of passing or failing the test. The test result will be estimated as score/grade. For more details, please refer to the "Details of Grades and Scores" section below. The score is reliable since it's converted on absolute scale and based on the criteria of Item Response Theory. From this score, the examinees will know their level of mathematical profiency, and thus be the solution for enriching mathematical skills in future.
2. Discover examinee's arithmetical potentiality
In order to acquire mathematical abilities, it's important for each person to discover the potential abilities of mathematics. TOMAC emphasize this and therefore, each problem has the factor of deriving ones such abilities.
3. Unique listening problems
Unique mathematical listening problems are given during the test.
4. License card styled certificate
Examinees could carry their certificate easily, since its size fits in the wallet perfectly like the other license cards. We also issue the "Score Certificate" to prove the participants' ability for school/employment admissions.
5. Issue of the detailed report
The detailed report of the test will be sent with the certificate described above. Not just indicating the scores, this report will also present the examinee's mathematical ability from five aspects: calculation, making relational formula of quantity, shapes, analyzation/reasoning and processing information to mathematics. These analyzation will help examinee to strengthen their mathematical abilities.
6. Participation in volunteer activities
Part of the test fee will be donated to support volunteer activities and mathematical research institutes.
Details of Grades and Scores
The grades and scores table
Note: All of the criteria of each grade is based on Japanese mathematical curriculum decreed by the Japanese Ministry of Education, Culture, Sports, Science and Technology.
Details of the score system
Prof. Satoshi Kawanobe
Prof. Toshio Sawada
Prof. Yasuo Iwai
Who We Are
TOMAC is administrated by The Society for Mathematics and Science Certification (Surikentei Kyokai) in Japan. Since the establishment in 1995, the mission was to cultivate mathematical ability for Japanese people. Before the birth of TOMAC, there was no mathematical test that measures such ability by scores; the notion of scoring mathematical test was such a revolution in Japan. Hoping to extend people's potential abilities of mathematics. Besides TOMAC, we also prepare a custom-made mathematical test for schools and companies in Japan.
Introduction of the board member


This book offers effective, research-based strategies that can be mixed and matched to differentiate mathematics instruction for high school students through four different learning styles.
Learn From the Experts!
Sign up for a Math Professional Development Institute in your area―visit www.ThoughtfulClassroom.com/events
Explains how the strategies address NCTM process standards and students” learning styles.
Includes step-by-step directions, examples and planning considerations for each strategy.
Provides reproducible forms for implementing the strategies.
Offers variations and ways to adapt each strategy to meet a variety of instructional demands.With assessment
Sub-Title – 21 Techniques for Differentiating Instruction and Assessment.


Short codes I have developed as part of my research. Please let me know if you decide to use one or have any questions.
Function to bin data by ω500 velocities, as done by Bony et al. (2004) and many others.
This script includes functions to compute the power spectrum of a signal and to perform a coherence analysis of two time-series using the multi-taper method. These are basically just Python versions of the Matlab functions available from Peter Huybers' website. Requires the spectrum Python module.
Function for calculating eddy flux co-spectra, following method of Hayashi (1971) (see also Randel and Held (1991)). Includes functions to calculate space-time cross-spectra and phase-speed cross-spectrum.
Stand-alone Python script for a "1-dimensional" 2-layer quasi-geostrophic model described in Panetta and Held (1988). The model is truncated in wavenumber space to retain only the zonal-mean flow and one non-zero zonal wavenumber (the default set-up retains the most unstable wave) and can be quickly run on a laptop. Non-linear terms are de-aliased using the Orszag (1971) "two-thirds" rule. The domain is doubly-periodic and the model is forced by an unstable meridional temperature gradient and by lower-layer Ekman friction. Note that all variables are non-dimensionalized.


FM’s arithmetic correct, her mathematical conclusion wrong: Chidambaram
Chidambaram added: "Regrettably, the FM thinks that the average Indian has below-average Intelligence in understanding numbers relating to finances."
New Delhi: Senior Congress leader P. Chidambaram on Tuesday criticised Finance Minister Nirmala Sitharaman for getting ‘wrong’ mathematical conclusion in the developmental expenditure.
He said, “The FM’s arithmetic is correct. Her mathematical conclusion is wrong!”
Chidambaram added: “Regrettably, the FM thinks that the average Indian has below-average Intelligence in understanding numbers relating to finances.”
He alleged that the Finance Minister has tweeted that her government spent Rs 90.9 lakh crore on developmental expenditure in 8 years compared to Rs 49.2 lakh crore spent by the UPA government in 10 years
But Chidambaram flags that the size of the Budget is bigger every year and the total expenditure is bigger every year. Developmental expenditure or social services expenditure will also be bigger every year, he said.
He said the correct mathematical analysis will require the Finance Minister to compare the ‘proportion’ of any head of expenditure to total expenditure. Such a comparison will reveal that, as a proportion, social services expenditure has plummeted after 2014-15.
“The FM’s numbers are in the RBI’s Handbook of Statistics. It will show that the proportion has fallen from an average of 9% of total expenditure under UPA to an average of 5% of total expenditure under NDA,” Chidambaram tweeted.
The Finance Minister had tweeted a day before, “The developmental expenditure incurred by the Modi Govt of Rs 90.9 lakh cr so far is over and above the Rs 93,685.68 crore already spent between 2014-22 on repaying the UPA-era oil bonds. Further, an additional Rs 1.48 lakh crore will be paid by 2026.”
She further said, “The expenditure incurred by the Modi Govt includes Rs 24.85 lakh crore spent so far on food, fuel and fertiliser subsidies and Rs 26.3 lakh crore on capital creation. Over the 10 years of UPA, only Rs 13.9 lakh crore was spent on subsidies.”
She had also slammed the UPA for spending less, “RBI data shows total developmental expenditure incurred by Modi Govt in 2014-22 was Rs 90.9 lakh cr, far higher than is being alleged by some sections of the Opposition. In contrast, only Rs 49.2 lakh crore was spent on this during 2004-14.”


Instituto de Matemáticas, UNAM, México
His research interests include the theory and applications of Operations Research and Combinatorics; developing models and methods -both exact and heuristic- to solve various optimization problems in Physics, Biochemistry, Finance, and Political Sciences, as well as in Industrial, Chemical and Electrical Engineering.
Dr. Romero graduated as Civil Engineer from the National Autonomous University of Mexico (UNAM). He studied Computer Sciences at the University of Liege, Belgium, and earned his PhD in Applied Mathematics (Operations Research) at the University of Grenoble, France. Later, he spent some exchange time at the Polytechnic of Central London, England. He currently serves as a full-time researcher at the Institute of Mathematics, UNAM, in Cuernavaca, Mexico.
Dr. Romero has published over 50 research papers, receiving more than 400 citations, which include those by Kenneth Arrow, 1972 Nobel laureate in Economics. He has delivered about 100 lectures in conferences, has taught 90 courses, mostly at graduate level, and has participated in the organization of 50 academic meetings. His career includes positions as a researcher at the Institute of Electrical Research, Cuernavaca, Mexico, and the University of Ottawa, Canada.
He was General Director in INEGI, Mexico, and Head of the Cuernavaca branch of the Institute of Mathematics, UNAM. Dr. Romero has participated in projects that require the development of mathematical models and methods, to provide scientific basis to the decision making process in various public and private institutions.
Dr. Romero is member of the Mexican System of Investigators (SNI Level III).


Math has never been easy for me. I spent most of my childhood and teen years and even some of college, in tears, because math was so dang hard for me. I have found myself frustrated at times trying to teach my son when it comes to math, because to this day, when I see math problems my brain goes into panic mode in a split second. It seems no matter how many times I sit and work on math I forget formulas every single time.
We are now starting a new curriculum through Oak Meadows distance learning. Lesson one begins with fractions, so right off the bat I am feeling discouraged, but thankfully this curriculum comes with a teacher's manual. However, after going through the manual and instructions, I seemed to be spacing out from the get go. I might as well be trying to read Chinese. It took me an hour, but I got through two problems and refreshed my memory on a couple simple fractions. The second one ended up being an improper fraction and I had to remember how to simplify that. After spending fifteen minutes trying to remember, and messing up over and over, I finally looked for a Youtube video, In the end we got through 12 problems, no tears, and had fun.
So if you need a refresher yourself, here ya go, or maybe just a laugh at my expense. ;)


We propose a method for 3D point cloud surface fitting. This method, called DeepFit, incorporates a neural network to learn point-wise weights for weighted least squares polynomial surface fitting.
About three months ago, I hit one of my first career goals – have a paper accepted to CVPR (achievement unlocked ! ). Last week I had the privilege to present a poster of our work on normal estimation for unstructured 3D point clouds using CNNs at CVPR 2019 in Long Beach California (16-20.6.19). In […]
This post will show you a good way to visualize normal vectors on 3D point clouds. At the beginning of my master’s degree, I was working on a project where I used normal vectors on 3D point clouds to perform 3D point cloud over-segmentation. One of the first things I did was to try and […]


Epidemic is applied to a disease which, distributing widely, onslaughts many individual at the same clip. Epidemic is a widespread eruption of an infective disease. When an epidemic exists, it will impact many human populations. There were many factors to imitate the rise of epidemics such as hapless population wellness, in-migration and failure of public wellness plans.
Harmonizing to the universe wellness organisation appeared, disease should hold the undermentioned conditions: a new pathogens, disease can take to infection causes a serious complications, pathogen spread easy, particularly in interpersonal communicating. By and large talking, this disease is due to some powerful infective micro-organisms, and the infections caused by viruses, bacteriums.
Historically talking, awful epidemics have reoccurs over and over once more. Some epidemic diseases, such as the variola, pestilence, and grippe, have been persisted in the history. Smallpox was uprooted worldwide by 1980. In the eighteenth century, the universe ‘s major trade paths, several destructive appear cholera pandemic caused great infective diseases.
In the past worldwide, the chief cause of many people were killed is infective diseases and there are more deceases than all the war, such as the Black Death that struck Europe in 1347 had kill between one-third and one-half of the people in many metropoliss and towns severally, this ill-condition seized the advancement of civilisation for several coevalss.
To analyze the SIR theoretical account ( developed by Kermack and McKendrick ) in the signifier of the system of nonlinear ordinary differential equations.
To work out the SIR theoretical account numerically utilizing Mathematica and imitating the Agent Based Modeling utilizing NetLogo.
To look into the infection spread under the SIR theoretical account
To construe the consequences of epidemic job based on these two theoretical account.
1.3 Research Question
What is an epidemic?
Can an epidemic be avoided or command?
Which is better to work out the SIR theoretical account, Equation based Model or agent-based theoretical account?
What are the important parametric quantities that govern the two theoretical accounts?
1.4 Epidemic Problem mold in Mathematica Program
Mathematica plan is a general computing machine package system and linguistic communication in used for Mathematica plan and other applications. Mathematica plan non merely for usage in calculation, it besides use for mold, simulation, development and deployment, visualizationand certification. Mathematica calculations can be divided into 3 chief categories which are Numeric, Graphical and Symbolic.
Different occupations covering with different things, but the Mathematica plan is a comprehensive system to supply unprecedented work flow, dependability, sustainability and invention. In this undertaking Mathematica plan is used as a mold and information analysis the rate of epidemic. The inquiry can be answered by making the theoretical account of an epidemic with variables matching to the different reaction of a population and the features of a virus.
1.5 Epidemic Problem Simulating in NetLogo
NetLogo is multi-agent scheduling linguistic communications and incorporate mold environment and a platform specifically for agent-based mold. NetLogo is most suited for complex system patterning development. Model can steer 100s or 1000s of “ agent ” all operating independently.
NetLogo besides let pupils can imitate and “ drama ” , explore their behaviour in different conditions. NetLogo has extended files and tutorial. It besides comes with a theoretical account library, which is a big aggregation of pre-written simulation, it can be used and alteration.
If an epidemic occurs, the variables matching to a population reaction and features of disease will impact its continuance and badness. In attempts to command the spread of the disease, we must take an optimum solution for the maximal public wellness benefits. In NetLogo scheduling, system kineticss can utilize a alone scheduling. To find the influence of assorted factors have on the continuance and serious infective disease, we can alter the variable and expression at the form of the graph differs between tallies in NetLogo.
1.6 Prilimanary Research
The Ebola virus patterning in Mathematica 7
The Ebola virus imitating in NetLogo
This thesis is divided into five chapters. In the first chapter, we discuss the debut of epidemic. For chapter two, we introduce the General Epidemic theoretical account by Kermack and McKendrick ( 1927 ) . In this chapter, we show how to deduce the theoretical account. For chapter three, we discuss the Mathematical Modeling. In chapter four, we will discourse epidemic theoretical account patterning in Mathematica plan. In chapter five, we will discourse SIR theoretical accounts imitating in NetLogo plan. In chapter four and five, we will plot the solution for the theoretical account. Finally is chapter six. In this chapter, we will make an readings and decision about the consequence of epidemic theoretical account.
2.0 SIR Model
In 1927, W. O. Kermack and A. G. McKendrick created a theoretical account of epidemic. The independent variable for this theoretical account is clip ( T ) . Assume the population is a disjoint brotherhood, there are three dependent Variables:
1. S = S ( T ) , which is the figure of susceptible individuals
2. I = I ( T ) , which is the figure of septic individuals
3. R = R ( T ) , which is the figure of cured individuals
The entire population = S ( T ) + I ( T ) + R ( T ) .
SIR theoretical account was based on the theoretical account in the spread of disease of the population. SIR theoretical account is a simple but good theoretical account of infective diseases, such as rubeolas, chicken-pox and German measles, which one time the individual infected with, will non be infecting once more.
2.2 Premises of the SIR theoretical account
SIR theoretical account is based on some premises. Suppose the population measure is immense and changeless. Because we ignore births and in-migration, therefore cipher is added to the susceptible group. Since the lone manner to go forth the susceptible groups will be infected, we assume the time-rate of alteration forA the figure of susceptible depends to the figure of people who already susceptible, the figure of individuals that already infected and the sum of the susceptible individuals contact with septic individual.
In add-on, we are hypothesis each septic people have a fixed value I? contact per twenty-four hours, and there are adequate sufficient to distribute the disease. Not all these contacts are with susceptible people. If we assume that the population is homogenous commixture, the fraction of these contacts that are with susceptible isA S ( T ) . Therefore on an mean, each infected individual will bring forth I?A S ( T ) A of new septic individuals per twenty-four hours.
We besides assume that a fixed fractionA I? in the septic group will retrieve bit by bit in any given twenty-four hours. For illustration, if the mean continuance of infection is four yearss, so on norm, one-quarter of the population under infected will retrieve each twenty-four hours.
2.3 SIR Formulas
There are three basic dependent differential equations:
S ‘ ( T ) = – I? S ( T ) I ( T )
I ‘ ( T ) = I? S ( T ) I ( T ) – I? I ( T )
R ‘ ( T ) = I? I ( T )
The theoretical account starts with some basic notation. That are S ( T ) is the figure of susceptible individuals at clip T, I ( T ) is the figure of septic individuals at clip T, and R ( T ) is the figure of cured individuals at clip T
These equations describe the passages of individuals from S to I to R. By adding the three equations, the size of the population is changeless and equal to the initial population size, which we denote with the parametric quantity N. Therefore the entire population
N= S ( T ) + I ( T ) + R ( T ) .
We call the parametric quantity I? the infection rate and the parametric quantity I? the recovery rate with I? and I? must be or greater to zero. The term I? is a standard kinetic footings, based on the thought that the figure of unit clip to meet between the susceptible and infective will be relative to the Numberss value. The infection I? is determined by both the brush frequence and the efficiency of distributing the diseases per brush.
If we imagine the procedure in a disease that a really suited for the SIR model, we will acquire a flow of people from the susceptible group will travel to the infection group, so to the removed group. i‚?
The diagram of SIR theoretical account
I? S I R I
The individual perchance moves from the susceptible to the infected group when person comes in contact with an septic individual. Qualify as a contact in the population, depends on the disease. For HIV virus a contact may be sexual contact or a blood transfusion. For Ebola virus it contact with septic organic structure ‘s funeral, and contact with septic individuals without exercising proper cautious.
2.5 Derivations of the SIR theoretical account
The theoretical account is described by three ordinary differential equations:
For the susceptible differential equation,
When we plotting the graph of S ( T ) versus T with I? and I? is a invariables, that is a negative exponential relationship between S and t. Since S ( 0 ) a‰ 0 when T = 0, , therefore the graph will started with a?z .
The graph of S versus T
2. For the septic differential equation,
When we plotting the graph of I ( T ) versus T with I? and I? is a invariables, that is a exponential relationship between I and t. Since I ( 0 ) a‰ 0 when T = 0, , therefore the graph will non started with 0.
The graph of I versus t
3. For the cured differential equation,
When we plotting the graph of R ( T ) versus T with I? and I? is a invariables, really clearly, that is a additive relationship between R and t. Since R ( 0 ) = 0 when T = 0, the graph will started with 0.
The graph of R versus T
4 Vector Notation
If work outing with numerical values for the invariables a and B, utilizing vector notation can do the system easier to cover with.
2.6 A Graphic Solution to the SIR Model
To demo a solution to the SIR theoretical account, we try to plot the differential equations with value a = B = 1 and allow the initial value S ( 0 ) = 5, I ( 0 ) = 0 and R ( 0 ) = 0.
S ‘ ( T ) = – S ( T ) I ( T ) ( Moore, 2000 )
I ‘ ( T ) = S ( T ) I ( T ) – I ( T )
R ‘ ( T ) = I ( T )
The three populations versus clip give the end product. The septic is relative to the alteration in clip, the figure of septic and the figure of susceptible. The alteration in the septic population addition from the susceptible group and lessening into the cured group.
3.0 Mathematical Modeling
Mathematical mold is a replacing of an object studied by its image. The mathematical mold is the method of making a mathematical theoretical account of a job, and utilizing it to analyse and work out the job.
In a mathematical theoretical account, mathematical variables represented the explored system and its properties, maps are represented the activities and equations relationships.
Quasistatic theoretical accounts and Dynamic theoretical accounts represent the two major type of mathematical mold. Quasistatic theoretical accounts shows the relationships between the system attributes approximate to equilibrium. The national economic system theoretical accounts is one of quasistatic models.Dynamic theoretical accounts describe the fluctuation of maps change over the clip. The spread of a disease is one of the dynamic theoretical accounts.
Mathematical theoretical accounts are used peculiarly in the scientific disciplines and technology, such as natural philosophies, biological science, and electrinic technology but besides in the societal scientific disciplines, such as economic sciences, sociology and political scientific discipline ; physicists, applied scientists, computing machine scientists, and economic experts are the most widely used mathematical theoretical account.
3.2 Important of Mathematical Modeling
Mathematical mold is an interdisciplinary topic. Mathematicss and specializers in different Fieldss portion their cognition and experience to uninterrupted betterment on extant merchandises, make sooner develop, or predict the certain merchandise ‘s behavior.
The most of import of mold is to derive understanding. If a mathematical theoretical account is reflects the indispensable behaviour of a real-world system of involvement, we will easy to derive understanding about the system than utilizing an analysis of the theoretical account. In add-on, if we want to construct a theoretical account, we need to happen out which factors in the system are most of import, and how the different facet of the relevant system.
We need to foretell or imitate in the mathematical mold. We ever want to cognize what is the real- universe system will make in the hereafter, but it is expensive, impractical or unable to experiment straight with the system. Finally, we need to gauge the large values in the mathematical mold.
3.3 Methodology of Mathematical Modeling
Agent Based Modeling ( ABM ) and Equation Based Modeling ( EMB ) are the attacks of mathematica modeling.
ABM and EBM portion some common concerns, but in two different ways: the basic relation theoretical account between entities, and do them the degree at which they their focal point. These two attacks have recognized that the universe has two sorts of entities: observables and persons.
EBM start with a set of equations that express relationships among observables. The rating of these equations produces the development of the observables over clip. These equations may be algebraic, or they may capture variableness over clip or over clip and infinite. The modeller may acknowledge these relationships result from the meshing behaviours of the persons, but those behaviours have no obvious representation in EBM.
ABM do n’t get down with equations that relate observables to one another, but with behaviours via the interact between persons with one another. These behaviours may affect more personal straight or non straight through sharing environment. The modeller devising much attending to the observation as the theoretical account runs, and may value a inferior history of the dealingss among those observation, but the history is due to the mold and simulation of motion, non its get downing point. The modeller doing start representative of each single behaviour, so turns them over the interaction
In decision, EBM work outing the theoretical account from macroscopic degree to microscopic degree by utilizing the system of ordinary differential equations ( ODE ) and partial differential equations ( ODE ) . Besides that, ABM work outing the theoretical account from microscopic degree to macroscopic degree by utilizing the complex dynamical system ( CDS ) .
4.0 Modeling in Mathematica Programme
Mathematica package consists of wolfram research company. Mathematica 1.0 version released on June 23, 1988. After the release in scientific discipline, engineering, media, and other Fieldss caused a esthesis, considered a radical betterment. Several months subsequently, in all over the universe have 1000s of Mathematica users. Today, in all over the universe have Mathematica 1000000s of loyal clients.
Mathematica 7 usage letters, Numberss and other mathematical symbols or inequality, constitute the equation, images or with diagrams of mathematical logic to depict the features of the system. Mathematica is studied and the motion regulations of system is a powerful tool, it ‘s analysis, design, prediction and anticipation and command existent system.
When we use Mathematica input the epidemic job, it will be use as a numerical and symbolic reckoner and print out the reply.
4.2 Graphic Interface of Mathematica
In most computing machine systems, Mathematica supports a “ notebook ” interface in which we interact with Mathematica by making synergistic paperss.
If use computing machine via a strictly graphical interface, we normally double-click the Mathematica icon to get down with the Mathematica. If use computing machine via a textually based in the operating system, we can normally input the bid mathematica to get down Mathematica.
When Mathematica starts up, it normally gives a clean notebook. When we enter Mathematica input into the notebook, so type Shift-Enter ( keep down the Shift key, so imperativeness Enter. ) to do Mathematica procedure the input.
In add-on, we besides can fix the input by utilizing the criterion redacting maps of graphical interface, which may travel on for several lines. After send Mathematica input from the notebook, Mathematica will label the input with In [ n ] : = . It labels the corresponding end product Out [ n ] = .
When type 2 + 2, so stop the input with Shift-Enter. Mathematica will treat the input, and so adds the input label In [ 1 ] : = , subsequently gives the end product.
Throughout this book, “ duologues ” with Mathematica are shown in the undermentioned manner:
With a notebook interface, we merely type in 2 + 2 and so type Shift-Enter. Mathematica so adds the label In [ 1 ] : = , and print out the consequence.
In [ 1 ] : = 2 + 2
Out [ 1 ] =
5.0 Imitating in NetLogo Programme
NetLogo is a programmable mold environment for imitating complex scientific phenomena, both natural and societal. It is one of the most widely used multi-agent modeling tools today, with a community of 1000s of users worldwide. Its “ low-threshold, noceiling ” design doctrine is inherited from Logo. NetLogo is simple plenty that pupils and instructors can easy plan and run simulations, and advanced plenty to function as a powerful tool for research workers in many subjects. Novitiates will happen an easy-to-learn, intuitive, and well-documented scheduling linguistic communication with an elegant graphical interface.
Experts and research workers can utilize NetLogo ‘s advanced characteristics, such as automatic running experiments, 3-D support, and user expansibility. NetLogo besides includes HubNet, which prepare a web of scholars to collaboratively, explore and command a simulation. NetLogo connects NetLogo Lab by external physical devices utilizing the consecutive port, and a System Dynamics Modeler do assorted agent-based and polymerisation representations.
NetLogo has extended certification, including a library with more than 150 sample theoretical accounts in a series of sphere, tutorials, a simple vocabulary, and sample codification illustrations. This package is free and works on all major calculating platforms. Fabrication, make the system kineticss assorted agent-based
5.2 Graphic interface of NetLogo
This theoretical account simulated the transmittal and saving of all people are infected with the virus. Ecological life scientists suggested several influence factors within a population infected straight. This theoretical account is initialized with 150 people, including 10 are infected.
Peoples of the universe indiscriminately move in one of the three provinces below:
healthy but susceptible to infection ( green ) ,
sick and infective ( ruddy ) ,
healthy and immune ( Grey ) . Peoples may decease of infection or a natural decease.
The factors in this theoretical account are summarized below with an account
Controls ( BLUE ) – allow to run and command the flow of executing
1. SETUP button
resets the artworks and secret plans
distributes with 140 green susceptible people and 10 ruddy septic people
2. Travel button
get down the simulation.
Settings ( GREEN ) – allow to modify parametric quantities
3. Peoples slider
Density of the population
Population denseness frequently affect infection, immune and susceptible personal contact each other.
Some familiar virus easy spread.
Some viruses spread from every smallest contact
Others ( illustration: the HIV virus ) require important contact before the virus transmitted.
Classify the people that had into group of susceptible, septic and immune.
Determined the opportunities of people dices of the virus or a natural decease.
All of the new born people replace those who decease.
Duration of infectiousness
Time of the virus infected wellness people.
Duration of a people infected before they recover or decease.
Number of hebdomad in the clip graduated table.
Positions ( BEIGE ) – allow to expose information
3 end product show show the per centum of population is infected and immune, and the figure of old ages have already passed.
I ) proctors – expose the current value of variables
two ) plots – show the history episode of a variable ‘s value
three ) artworks window – the chief thought of the NetLogo universe
The secret plan shows ( in their several colourss ) the figure of people which is susceptible, infected, and immune. It besides shows the entire figure of people in the population.
5.3 The epidemic simulated by NetLogo Programme
1. The HIV virus simulated by NetLogo Programme
Let the theoretical account is initialized with 150 people, of which 10 are infected and 1 old ages that have passed.
The HIV virus has a really long continuance, a really low recovery rate, but a really low infectiousness value.
2.The Ebola virus simulated by NetLogo Programme
Let the theoretical account is initialized with 150 people, of which 10 are infected and 1 old ages that have passed.
The celebrated Ebola virus in cardinal Africa has a really short continuance, a really high infectiousness value, and a really low recovery rate.
The celebrated Ebola hemorrhagic fever in cardinal Africa a really short clip, really high infectiousness value, an highly low recovery rate.