A Study on an HIV Pathogenesis Model with Different Growth rates of Uninfected and Infected CD4+T cells
The objective of this paper is to discuss the dynamics of an HIV pathogenesis model with full logistic target cell growth of uninfected T cells and cure rate of infected T cells. Local and global dynamics of both infection-free and infected equilibrium points are rigorously established. It is found that if basic reproduction number R0â‰¤1, the infection is cleared from T cells and if R0>1, the HIV infection persists. Also, we have carried out numerical simulations to verify the results. The existence of non-trivial periodic solution is also studied by means of numerical simulation. Therefore, we find a parameter region where infected equilibrium point is globally stable to make the model biologically significant. From the overall study, it is found that proliferation of T cells cannot be ignored during the study of HIV dynamics for better results and we can focus on a treatment policy which can control the parameters of the model in such a way that the basic reproduction number remains less than or equal to one.
2010 AMS classification: 34A34, 34D23, 37C25
World Health Organisation HIV/AIDS Key facts. 2020 Jul https://www.who.int/newsroom/fact-sheets/detail/hiv
Douek DC, Brenchley JM, Betts MR, Ambrozak DR, Hill BJ, Okamoto Y, Casazza JP, Kuruppu J, Kunstman K, Wolinsky S, Grossman Z. HIV preferentially infects HIV-specific CD4+ T cells. Nature. 2002 May; 417(6884):95â€“8. PMid: 11986671. https://doi.org/10.1038/417095a
Perelson AS, Kirschner DE, Boer R De. Dynamics of HIV infection of CD4+ T cells. Math Biosci. 1993; 114:81â€“125. https://doi.org/10.1016/0025-5564(93)90043-A
Essunger P, Perelson AS. Modelling HIV infection CD4+ T - subpopulations. J Theoret Biol. 1994; 170:367â€“91. PMid: 7996863. https://doi.org/10.1006/jtbi.1994.1199
Wang L, Li MY. Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. Math Biosci. 2006; 200:44â€“57. PMid: 16466751. https://doi.org/10.1016/j.mbs.2005.12.026
Rong L, Gilchrist MA, Feng Z, Perelson AS. Modeling within host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility. J Theoret Biol. 2007; 247(4):804â€“18. PMid: 17532343 PMCid: PMC2265667. https://doi.org/10.1016/j.jtbi.2007.04.014
Perelson AS, Nelson Patrick W. Mathematical Analysis of HIV-1 dynamics in Vivo. SIAM Review. 1999; 41(1):3â€“44. https://doi.org/10.1137/S0036144598335107
Leenheer P. De, Smith HL. Virus dynamics: A global analysis. SIAM J Appl Math. 2003; 63:1313â€“27. https://doi.org/10.1137/S0036139902406905
Gao T, Wang W, Liu X. Mathematical analysis of an HIV model with impulsive antiretroviral drug doses. Math Comput Simul. 2011 Dec; 82(4):653â€“65. https://doi.org/10.1016/j.matcom.2011.10.007
Srivastava PK, Chandra P. Modeling the dynamics of HIV and CD4+ T cells during primary infection.
Nonlinear Anal Real World Appl. 2010 Apr; 11(2):612â€“8. https://doi.org/10.1016/j.nonrwa.2008.10.037
Chandra P. Mathematical modeling of HIV dynamics: In Vivo. Mathematics Student-India. 2009; 78(1):7.
Nowak MA, May RM. Virus dynamics. UK: Oxford University Press; 2000.
LaSalle JP. The stability of dynamical systems. SIAM; 1976. PMCid: PMC1411100. https://doi.org/10.21236/ADA031020
Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M. Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature. 1995; 373:123â€“6. PMid: 7816094. https://doi.org/10.1038/373123a0
Sachsenberg N, Perelson AS, Yerly S, Schockmel GA, Leduc D, Hirschel B, Perrin L. Turnover of CD4+ and CD8+ T lymphocytes in HIV-1 infection as measured by ki-67 antigen. J Exp Med. 1998; 187:1295â€“303. PMid: 9547340 PMCid: PMC2212238. https://doi.org/10.1084/jem.187.8.1295
Fan X, Brauner CM,Wittkop L. Mathematical analysis of a HIV model with quadratic logistic growth term. Discrete and Continuous Dynamical Systems Series-B. 2012; 17(7):2359â€“85. https://doi.org/10.3934/ dcdsb.2012.17.2359
Song X, Neumann AU. Global stability and periodic solution of the viral dynamics. J Math Anal Appl. 2007 May; 329(1):281â€“97. https://doi.org/10.1016/j.jmaa.2006.06.064
Wang X, Song X. Global stability and periodic solution of a model for HIV infection of CD4+ T cells.
Appl Math Comput. 2007 Jun; 189(2):1331â€“40. https://doi.org/10.1016/j.amc.2006.12.044
Culshaw RV, Ruan S. A delay-differential equation model of HIV infection of CD4+ T-cells. Math Biosci.
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Merdan M, Gokdogan A, Yildirim A. On the numerical solution of the model for HIV infection of CD4+ T cells. Comput Math. Appl. 2011 Jul; 62(1):118â€“23. https://doi.org/10.1016/j.camwa.2011.04.058
Zack JA, Arrigo SJ, Weitsman SR, Go AS, Haislip A, Chen IS. HIV-1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile, latent viral structure. Cell. 1990 Apr; 61(2):213â€“22. https://doi.org/10.1016/00928674(90)90802-L
Zack JA, Haislip AM, Krogstad P, Chen IS. Incompletely reverse-transcribed human immunodeficiency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral life cycle. J Virol. 1992 Mar; 66(3):1717â€“ 25. PMid: 1371173 PMCid: PMC240919. https://doi.org/10.1128/JVI.66.3.1717-1725.1992
Gradshteyn IS, Ryzhik IM. Routh-Hurwitz theorem, Tables of Integrals, Series and Products. San Diego: Academic Press; 2000.
Li MY, Muldowney JS. A geometric approach to global-stability problems. SIAM J Math Anal. 1996 Jul; 27(4):1070â€“83. https://doi.org/10.1137/ S0036141094266449
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Fiedler M. Additive compound matrices and inequalities for eigen values of stochastic matrices. Czech. Math J. 1974; 24(3): 392â€“402. https://doi.org/10.21136/CMJ.1974.101253
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Cube and Octahedron Sculpture
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This conceptual design for a possible geometrical sculpture contains a cube, an octahedron, and some fragments. First form an octahedron after expanding the six faces of a cube. Next, interconnect the cube with eight sides of the octahedron. Finally, construct a central cubic frame.
Contributed by: Sándor Kabai (August 2016)
Open content licensed under CC BY-NC-SA
This arrangement is useful for exploring the relationship of the octahedron and the cube, their possible decompositions, and the duality between them.


Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will count objects to name quantities from 11-20. Additionally, students can write these numerals if the teacher deems it appropriate (depending on students' prior experiences with these numbers).
Prior Knowledge: What prior knowledge should students have for this lesson?
This activity requires that students have an understanding of numbers 1-10 (identifying, counting, representing, etc.). To access and review this prior knowledge, show the students a group of counters of a quantity 0-10 and ask them to tell how many are shown. Ask for volunteers to tell how they know the amount. Additionally, you can have students use counters or drawings to model a given number, again 0-10. Though not required, it will also be helpful if students have some experience with rote counting to 20 (knowing number names and count sequence).
Guiding Questions: What are the guiding questions for this lesson?
- If we know that an amount of counters is ten and some more, how does that help us figure out how many there are in all?
- You say you have ___. How many would you have if I took away the group of ten? How do you know this?
- Do we always have to count all of the counters on your mat to find the total? Why or why not?
Teaching Phase: How will the teacher present the concept or skill to students?
- For this learning task, students will explore numbers 11-20 by counting 10 and then some more. Give each student pair a simple two-part mat (See attached file entitled Tens and Ones Mat, page 1) and a pile of counters (at least 20). Direct them to count out 10 counters and put them on the left side of their mat. Then ask them to count out four more and place them on the right side of their mat. Do this along with the students, displaying your counters with an overhead projector or a document camera.
- Direct the students to help you count to find the total number of counters (counting by ones). Then say, "Ten and four is fourteen." Have the class say it with you. Now turn the mat around and say, "Four and ten is fourteen." Again, have the class say it with you.
- Continue in this fashion to introduce or practice numbers 11-20 in random order, each time changing the ones number but leaving the group of ten on the mat. If using this to introduce numbers and number names for 11-20, this will require some direct instruction at first. If students have had prior experience with numbers 11-20 you can also model how to write numerals to represent these quantities and have students record the numerals on paper.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Once students have experience with numbers 11-20, they can work in pairs to roll dice, fill the mat, and count to find the total. Give each student pair a work mat (see attached file Tens and Ones Mat, page 1 simple mat or page 2 mat with ten frame), 20 counters, and 2 ten-sided dice (or 3 six-sided dice). Student A rolls one die and counts out that many counters. Student B rolls the other die and counts out that many counters. They count to find their total. Next they place their counters on the mat, first filling the left side with 10 and then placing the rest on the right side. In chorus they say, "Ten and ______ is ________," according to the amount shown. Again if students have had some prior experience with numbers 11-20 you can have them record the totals on paper. As students work, circulate to observe how students count and how they place counters on the mat. Ask probing questions such as:
- How are you counting?
- Is there another way to count? Show me and count aloud.
- What is your total amount? How do you know?
- How many would you have if I gave you 1 more? (or if I took away 1?)
- Add two more to the right side of the mat. How many do you have now? How do you know that is the correct amount?
- You say you have 18. How many would you have if I took away the group of ten?
- Do we always have to count all of the counters on your mat to find the total? Why or why not?
- How does knowing that this group is ten help us know how many we have in all?
- As student gain more experience with numbers 11-20, ask them to fill the mat before counting the numbers. Then see if they can tell you the total without counting all of the counters or by counting on from ten. (ex: ten and 2 more is 12, or ten, eleven, twelve)
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
- Tell students that they are going to help you create posters for a special event at your community center to raise money to help families in need. Explain that the community center will have games that children can play but each game requires one ticket. Tickets cost $1. However, if children bring canned and packaged food, they get free tickets. If a child brings a parent, they get 10 free tickets and they can get one additional free ticket for each food item they bring. Tell students that they will help you create posters to show how many free tickets children get depending on the amount of food items they bring.
- Give each pair of students several crayons and a poster of How Many Free Tickets Can You Get? (see attached file of same name). Note that there are two different posters, one with even numbers and one with odd numbers. You can decide to give all student pairs the same poster or give half of the pairs the odd numbers poster and give the other half the even numbers poster. Review the poster with the students, making sure they understand what information is represented. Ask them to figure out how many free tickets a child would get if they bring a parent and bring the number of food items represented in each row. Then ask them to write the corresponding total number of free tickets the child would get in the last column of each row. Guide the students through completing the first row to make sure they understand what to do.
- As students work in pairs, circulate and ask probing questions such as those listed in the Assessment sections or those listed in the Guided Practice section.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
- When all have finished, have students display their posters. Review answers. If students have different answers on their posters, ask for volunteers to explain and justify their answers. Facilitate corrective feedback and allow students to revise posters as needed.
- To summarize the lesson, ask students to discuss the following question with a partner, "If we know that an amount of counters is ten and some more, how does that help us figure out how many there are altogether?" Then ask for several volunteers to share their answers.
You can assess students' understanding in two ways. You can use the finished product of the posters to assess students' mastery of learning. Another alternative is to ask students to represent the last amount shown on their poster in a math journal with pictures and by writing the total amount under their picture as "Ten and ______ is ________." They can write the total using words, numerals or both. Provide a word bank to help with writing the words.
To access and review prior knowledge, show the students a group of 8 counters (or other number from 0-10) and ask them to tell how many are shown. Ask for volunteers to tell how they know the amount. Additionally, you can have students use counters or drawings to model a given number from 0-10. Though not required, it will also be helpful if students have experience with rote counting to 20 (knowing number names and count sequence). To assess their knowledge of the counting sequence, ask students to count out loud from 0-20. If any students struggle with any of these 3 tasks, they could need intervention before beginning the lesson or might need extra support during the lesson.
Feedback to Students
As students are working in partner pairs, observe how students model the various quantities, how they count, and how they describe the totals as ten and some more. For example, when counting the total some students might have to begin counting at one and others might be able to count on from ten. For students who always begin counting at one, ask questions such as, "How many are in this group? (pointing to the group of ten) How could you use that information to help you finish counting the rest to find the total? or Can you count on from ten?"
You might also ask questions such as:
- Describe what your model is showing.
- Does the number you wrote match your representation? How do you know?
- Do you notice a pattern in the numbers you are writing? Tell me about the patterns you see.
ACCOMMODATIONS & RECOMMENDATIONS
- Partner strong students with those who have more difficulty mastering new skills.
- If students are struggling with the counting sequence, have partners count aloud together from 0-20 before counting the counters. Then have then count out loud together as they determine the total.
- If students are struggling with counting larger quantities, give them more practice with smaller amounts (11-15).
- If students have a difficult time completing assignments within the lesson time frame, reduce the amount of items the students have to complete.
- If you have students record their totals using numerals 11-20 and some students struggle with this, allow them to tell you or their partner the quantity and record it for them. Then they can trace the numeral to get practice writing the amount. Note that some students might reverse numerals. This should not count against the student as handwriting is not the focus of the lesson.
Give students cards that have a different number written on each (11-20). Ask them to tell or show how many counters they would place on their work mats in terms of ten and some more. For example, given the number 17, they would tell or show that ten plus seven more would be placed on the mat.
Suggested Technology: Document Camera, Microsoft Office
Special Materials Needed:
- Tens and Ones Mat (attached file) - 1 for each pair of students
- 20 counters - 1 group for each pair of students
- 2 ten-sided dice -1 pair per student pair
- Paper & pencil or dry erase boards & markers for each student if you have them record numerals
- How Many Free Tickets Can You Get? (attached file) - 1 for each pair of students
- Crayons - a few for each pair of students
This lesson can be used to introduce numbers 11-20 or used for additional practice with these numbers. The attached Tens and Ones Mat file contains two work mats - a simple mat showing a column each for ten and some more or a work mat that uses a ten frame for the ten. Either mat can be used.
"Even though kindergarten, first-, and second-grade children daily experience numbers up to 20 and beyond it should not be assumed that they will automatically extend the set of relationships they have developed on smaller numbers to the numbers beyond 10. And yet these numbers play a big part in many simple counting activities, in basic facts, and in much of what we do with mental computation. Relationships with these numbers are just as important as relationships involving the numbers through 10." (John Van de Walle and Lou Ann Lovin, Teaching Student-Centered Mathematics Grades K-3, page 54)
In Kindergarten, it's not necessary for students to think about numbers 11-20 as one group of ten and an amount of ones, but it is important for them to understand that these numbers are ten and some more and to be able to count on from ten. This lays the foundation for future base ten understandings that will be developed in first grade.
SOURCE AND ACCESS INFORMATION
Name of Author/Source: Elizabeth Gehron
District/Organization of Contributor(s): Seminole
Is this Resource freely Available? Yes
Access Privileges: Public
* Please note that examples of resources are not intended as complete curriculum.


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Schiehallion's isolated position and regular shape led it to be selected by Charles Mason for a ground-breaking experiment to estimate the mass of the Earth in 1774. The deflection of a pendulum by the mass of the mountain provided an estimate of the mean density of the Earth, from which its mass and a value for Newton's Gravitational constant G could be deduced. Mason turned down a commission to carry out the work and it was instead coordinated by Astronomer Royal, Nevil Maskelyne. He was assisted in the task by mathematician Charles Hutton, who devised a graphical system to represent large volumes of surveyed heights, later known as contour lines.
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With simple linear regression, standard errors and confidence intervals for fitted (and predicted) values are easily computed. In R, we can use the
se.fit argument in
predict.lm, which returns the standard error for the fitted values, and
interval = "confidence" to return confidence intervals. With linear mixed-effects models, however, it is not so easy. Neither
lme4) provide these methods, as confidence intervals on mixed-effects model predictions are harder to produce.
The solution is to use the parametric bootstrap, which is conveniently implemented in
bootMer to be applied to models fit with the
lme4 package (
glmer). Here I describe a simple wrapper around
bootMer, providing an alternative for
predict.merMod that calculates standard errors (and confidence intervals) for predictions.
A side-effect of this implementation is that confidence intervals now appear in
visreg plots of mixed-effects models.
The code shown below is implemented as a very simple package,
bootpredictlme4, which can be installed with,
Note that the package cannot be used in conjunction with the
lmerTest package, since both packages replace the
predict.merMod function from
lme4. For this reason I will probably not attempt to submit this package to CRAN.
Suppose we fit a linear mixed-effects model, using the built-in
sleepstudy dataset (in the
lme4 package), which fits the reaction time of subjects as a function of the number of days since a sleep deprivation (3 hours per night) experiment.
library(lme4) fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
The structure allows for a random intercept and slope of reaction vs. days by subject. A
visreg plot (Fig. 1) shows the fitted model (for the average subject). Note that I have not yet loaded the
library(visreg) visreg(fm1, "Days", ylab="Reaction time (ms)")
If we are interested in the fitted value at some specified time, let’s say 5 days, we can use
re.form=NA to yield predictions for the average subject (i.e. not include random effects in the predictions). Note that we actually invoke
fm1 is fitted with
predict(fm1, newdata = data.frame(Days=5), re.form=NA)
## 1 ## 303.7415
Now we would like to calculate a confidence interval on this prediction, using the bootstrap. We can generate 100 bootstrap resamples with the
bootMer function, as follows. In this case, we apply the
predict.merMod to each of the bootstrapped fits. I am just using 100 fits to speed up this code, but you probably want to use 1000 samples or more.
b <- bootMer(fm1, nsim=100, FUN=function(x)predict(x, newdata=data.frame(Days=5), re.form=NA))
The resulting object
b contains lots of information, but most useful is the element
b$t, which lists the output of the function we applied to each of the bootstrap resamples, in this case the fitted value at
The following histogram shows the distribution of the fitted values, that is, the sampling distribution. Note that if you use more samples (which you should), the distribution will look more like a normal distribution.
hist(b$t, breaks=seq(250,350,by=5), ylim=c(0,25), main="", xlab="Reaction time at 5 Days (ms)", col="cornflowerblue") box()
The standard error of our predicted value can be estimated simply as the standard deviation of the sampling distribution, that is,
##  9.184015
and the confidence interval as the quantiles of the distribution (in this case, a 95% confidence interval).
quantile(b$t, probs=c(0.025, 0.975))
## 2.5% 97.5% ## 286.5467 321.5505
The example above is implemented in the
bootpredictlme4 package (as mentioned, install with
devtools::install_github("remkoduursma/bootpredictlme4")). When loading the package, the
predict.merMod function is replaced with a function that takes an
library(bootpredictlme4) predict(fm1, newdata=data.frame(Days=5), re.form=NA, se.fit=TRUE, nsim=100)
## $fit ## 1 ## 303.7415 ## ## $se.fit ## 1 ## 10.50841 ## ## $ci.fit ## 1 ## 2.5% 283.6878 ## 97.5% 323.8666
Two standard errors are computed, the
se.boot is the one described above (the standard deviation of the sampling distribution), and
se.fit is an effective standard error that reconstructs the bootstrapped confidence interval if we were to assume asymptotic normality. In the case of normality, the half-width of the confidence interval (for 95% coverage) is of course 1.96 times the standard error. Thus
se.fit is calculated as the half-width of the bootstrapped confidence interval, divided by 1.96. It is this standard error that is used by
visreg (see below), making sure that the resulting confidence interval is actually equal to the bootstrapped confidence interval.
In most cases
se.boot will be very similar, unless you used very few bootstrap samples, or the sampling distribution is very non-normal (which is certainly possible).
A major advantage is that
visreg automatically recognizes that we can compute standard errors, and now a confidence interval appears on the standard effects plot. Note that
bootpredictlme4 warns about using few bootstrap resamples. The default has a low number to allow rapid testing.
visreg(fm1, "Days", ylab="Reaction time (ms)")
## Number of bootstrap replicates very low. ## Set to higher value with e.g. options(bootnsim = 500)
The new predict method also allows more complicated
visreg plots, like the overlay plot (my personal favorite).
# Add a fake grouping variable to sleepstudy high <- with(sleepstudy, levels(reorder(Subject,Reaction,mean)))[1:9] sleepstudy$Group <- factor(ifelse(sleepstudy$Subject %in% high, "A", "B")) fm2 <- lmer(Reaction ~ Days*Group + (Days | Subject), sleepstudy) visreg(fm2, "Days", by="Group", overlay=TRUE, ylab="Reaction time (ms)")
## Number of bootstrap replicates very low. ## Set to higher value with e.g. options(bootnsim = 500) ## Number of bootstrap replicates very low. ## Set to higher value with e.g. options(bootnsim = 500)


Features aspects and solutions of problems of linear vibrating systems with a finite number of degrees of freedom. Starts with development of necessary tools in matrix theory, followed by numerical procedures for relevant matrix formulations and relevant theory of differential equations. Minimum of mathematical abstraction; assumes a familiarity with matrix theory, elementary calculus. 1966 edition.
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Metric Prefixes are “labels” that can be placed in front of base units. Whenever using the metric prefix chart, remember that “1” is associated with the metric prefix.
Conversion factors needed:
1 km (kilometer) = 103 m (meters)
A bedroom has a volume of 116 m3 .
What is its volume in cubic kilometers (km3)?
Frequently Asked Questions
What scientific concept do you need to know in order to solve this problem?
Our tutors have indicated that to solve this problem you will need to apply the Metric Prefixes concept. You can view video lessons to learn Metric Prefixes. Or if you need more Metric Prefixes practice, you can also practice Metric Prefixes practice problems.


An annuity is a financial product that pays out in equal intervals over time, such as but not limited to a retirement fund. This fund would pay the same value at a regular frequency until the annuity has been exhausted. This calculator assists in working out the future value of an annuity to gauge what the financial value will be after a set period of time, this can assist with decisions relating to the time value of money. The calculations for this future value of annuity calculator rely on knowing the current or present value of the annuity, the interest rate that will be given and the number of periods in which the interest will be accumulated - which in this case is years. We assume a static interest rate in this calculation as predicting the fluctuations of interest rates offered by financial institutions over time into the future is almost impossible.
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Nifty Applications in Discrete
Bill Marion, Doug Baldwin, Susanna
Epp, Peter Henderson
June 7-11, 2004
5/10/04 REGISTRATION IS NOW CLOSED
Have you ever struggled to come up with applications in a discrete
mathematics course which demonstrate to your students the relevance of
what seems like a rather disparate collection of topics to their study
of material in computer science (and, yes, even excite them too!)?
Then, this workshop is for you.
The workshop will provide participants with an in-depth look at a
number of problems and applications that arise in a typical discrete
math course (sequence) which is designed to meet the needs of computer
science majors. The problems and their solutions will be treated
from both a mathematical and a computational perspective.
Representative problems will be selected from the following domains:
encryption algorithms, finite-state automata and Turing machines,
analysis of algorithms, algorithm correctness, Boolean algebra and
logic circuits, graph algorithms, data base systems, functional and
logic programming, and computer graphics.
As a pre-workshop activity, participants will be given a collection of
possible problems to be discussed during the workshop, assigned some
readings on each of the problems and asked to choose which ones they
would like to see in more detail. At the workshop, for each of
the applications chosen, the presenters will provide background
information and the participants will work in teams to develop
solutions to a carefully chosen set of exercises. Participants will
return to their institutions prepared to implement the materials
developed. As follow-up, participants will be asked to contribute
their own examples and applications to an on-going, on-line repository
developed by the workshop leaders.
Return to the PREP front page


[Numpy-discussion] [Python-3000] PEP 31XX: A Type Hierarchy for Numbers (and other algebraic entities)
Fri Apr 27 13:59:07 CDT 2007
Jan Grant wrote:
> . . . the space of doubles include a small number of things
> that aren't representable by a rational (+/- inf, for instance).
Anton Sherwood, http://www.ogre.nu/
More information about the Numpy-discussion


This plugin enables mathjax (http://www.mathjax.org) functionality for WordPress (http://www.wordpress.org).
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8 months ago
Antidifferentiation of Linear Over Quadratic
9 months ago
Claremont McKenna College Department of Mathematical Sciences
Here are all of the videos that Mark Huber has uploaded to Vimeo. Appearances are videos that Mark Huber has been credited in by others.
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Professor Landon Gauthier came to Carthage from the University of Kentucky, where he earned his Ph.D. in mathematics with his advisor Dr. Russell Brown. His dissertation was on improving the regularity of coefficients on inverse problems involving the polyharmonic operator. On top of this, he is also interested in analysis, mathematical modeling, the history of mathematics, and curious problems in mathematics.
Prof. Gauthier went to UW-Platteville for his undergraduate degree. There, he became involved in the Mathematics Department where he helped in creating a math modeling club and participated in competitions across the Midwest. He is interested in helping the students at Carthage explore the opportunities that Carthage offers.
In his free time, Prof. Gauthier enjoys spending time with his wife and his two cats: Georg Cantor and Henrietta Lebesgue. He is an avid fan of watching all things sports and enjoys golfing.
- Ph.D. — Mathematics, University of Kentucky
- M.A. — Mathematics, University of Kentucky
- B.S. — Mathematics, the University of Wisconsin-Platteville
- MTH 1050
- MTH 1120
- NSG 2330
Real Analysis, Partial Differential Equations, Inverse Problems
Russell Brown, Landon Gauthier. Inverse boundary value problems for polyharmonic operators with non-smooth coefficients. Inverse Problems and Imaging, 2022, 16 (4) : 943-966. doi: 10.3934/ipi.2022006 Christopher Frayer. Landon Gauthier. “A tale of two circles: geometry of a class of quartic polynomials.” Involve 11 (3) 489 - 500, 2018. https://doi.org/10.2140/involve.2018.11.489


HOMEROOM STUDENTS: See Tab Below for Student Virtual Classroom & 7th Grade Electives Registration--- I miss you! Mrs. Flake
Canvas directions for Math Extended Learning-
Video directions to login to Canvas can be found HERE:
Once logged in you should be able to see the course. There's a link to the modules on the home page, but if that link does not work for some reason, please click the MODULES button on the left hand side. NOTE: If you only click ASSIGNMENTS you will miss the instruction and some activities. SO stick with going through the assignments as part of the MODULE!
Mrs. Cindy Flake, MAT
Interim 6th Grade Math Teacher
Mt. Pleasant Middle School
My Google Voice phone/text is: 980-222-4303


Math Series is designed keeping a preschoolers / kindergarten kids in mind. It is very ideal for kids between the ages 3 to 8. It is an excellent application for kids to have fun with numbers and math. Math Series is very ideal for parents since it engages kids while driving, in the airport or while waiting in the restaurants.
The default setup is to start the kids with simple number series (1's) and after 25 problems (first award), it will advance to next series and so on. If you choose random the series will be randomly selected. Additional options to setup the maximum start number and ascending or descending options are also provided.
Once the correct answer has been achieved, just press next button and a new series will appear. Five successful correct answer results in a colored star or a flower as a reward. The rewards icon keep changing. Math Series lets you adjust the levels to customize them for your child.


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Unformatted text preview: ogica
Laws on working hours, wages, etc.
Communication Exponential Change
Exponential rate of change --- change that happens at a faster and faster
Linear rate of change --- change that remains fixed
Example of cassettes on page 16 Exponential change
What is exponential change?
If you buy 2 books every month, how many books you would
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have after 6 months? 2×6=12 If you buy books the first month and you double the amount
If you buy 2 books in the first month, and you double the amount
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Amount you buy: 2 – 4 – 8 – 16 – 32 - 64
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The following events are upcoming and recurring events that support the advancement and participation of minorities in the mathematical sciences. More information may be found on the calendar.
If you are interested in having your conference or workshop represented here, first check that your event meets the guidelines outlined here, then contact us at email@example.com with a description of your event.
Summer 2021 Center Sponsored Workshops
Creating a Better Summer Experience: A DEI Workshop for REU Directors and Faculty Mentors
This workshop, designed for REU directors and faculty mentors, will provide participants with the tools to assist with mentoring underrepresented and minoritized students in mathematics. Sessions will address building supportive communities of scholars, the science of mentoring, and a summary of lessons learned from NSF CORE grant on supporting students of color in the mathematical sciences.
Conference Date: May 22nd, 2021
Registration Deadline: May 9th, 2021
Future/Recurring Conferences and Workshops
Conference for African-American Researchers in Mathematical Sciences (CAARMS)
CAARMS is an annual conference that spotlights the accomplishments of mathematicians from underrepresented minority groups. The event includes invited speakers, tutorials, and graduate student poster sessions and provides a forum where minority researchers in the mathematical sciences can meet, learn about work across different mathematical fields, and mentor minority graduate students.
The EDGE Summer Session
The EDGE Summer Session is for women entering PhD programs and especially encourages the participation of women from underrepresented minority groups. The program includes mini-courses and guest lectures given by faculty researchers and mentoring sessions with faculty and graduate student mentors.
Application date: Available December 2020
Blackwell-Tapia Conference and Award Ceremony
The Blackwell-Tapia Conference is a biannual conference in honor of David Blackwell and Richard Tapia. The conference includes scientific talks, poster sessions, panel discussions, networking opportunities, and the awarding of the Blackwell-Tapia Prize. The conference aims to recognize and showcase mathematical excellence by minority researchers and to recognize and disseminate successful efforts to address under-representation. Participants are invited from all career stages and represent institutions of all sizes across the country, including Puerto Rico.
Richard Tapia Celebration of Diversity in Computing Conference
The Tapia Conference brings together undergraduate and graduate students, faculty, researchers, and professionals in computing from all backgrounds and ethnicities. The conference aims to celebrate diversity in computing, connect people from common backgrounds, ethnicities, disabilities, and gender, obtain advice from leaders in academia and industry, and be inspired by presentations and conversations.
Conference date: September 16-19, 2020
Regular Registration date: August 31, 2020
Infinite Possibilities Conference (IPC)
The IPC is a national conference designed to promote, educate, encourage, and support minority women underrepresented in mathematics and statistics. The conference includes research talks, poster sessions, mentoring and career breakout sessions, professional development, and community. Participants represent women in high school, college, graduate school, postdocs, early faculty, and senior researchers in academia, industry, and government.
Latinx in the Mathematical Sciences Conference (Latmath)
Latmath showcases the achievements of Latinx in the mathematical sciences. The conference aims to encourage Latinx to pursue careers in the mathematical sciences, to promote the advancement of Latinx, to showcase research conducted by Latinx, and to build a community around shared academic interests. Activities include research talks, panel discussions, student poster sessions, and pre-conference activities for undergraduate and graduate students.
Conference date: March 4-6, 2021
Spring Opportunities Workshop
The Spring Opportunities Workshop aims to prepare women and underrepresented minorities for work in academia, industry, and government laboratories. The workshop provides an opportunity for women in the mathematical sciences to improve their non-technical skills such as public speaking and professional networking, working in teams, ethical reasoning. Participation is encouraged by graduate students, postdoctoral fellows, junior researchers, and professionals in the mathematical sciences. The conference includes research presentations, professional development sessions, motivational keynote speakers, Graduate School & Career Expo Hall, multicultural celebrations and traditions, and an inclusive and welcoming community.”
SACNAS National Diversity in STEM Conference
The SACNAS conference is the largest multidisciplinary and multicultural STEM diversity even in the country and serves to equip, empower, and energize participants for their academic and processional paths in STEM. The conference include research presentations, processional development sessions, motivational keynote speakers, Graduate School & Career Expo Hall, multicultural celebrations and traditions, and an inclusive and welcoming community.
Conference date: October 22-24, 2020
Regular Registration date: September 26, 2020
Location: Long Beach, CA
Modern Mathematics Workshop
The Modern Mathematics Workshop is a two-day workshop held in conjunction with the national meeting of SACNAS. The workshop showcases the contemporary research happening at the NSF Mathematical Institutes. Activities include research expositions aimed at graduate students and researchers, mini-courses aimed at undergraduates, a keynote lecture by a distinguished scientist, and a reception for participants to learn about the Mathematical Institutes.
Workshop Celebrating Diversity at SIAM Annual Meeting
The Workshop Celebrating Diversity is held in conjunction with the SIAM Annual Meeting. The workshop provides a chance for students to listen to technical talks presented by minority graduate students. The workshop aims to send a clear, explicit message of support from SIAM to members of underrepresented groups, to bring together a mixture of people of different ages and professional experiences, to provide an opportunity for minority graduate students to present their research and learn about careers in applied and computational mathematics.
Modern African Diaspora Joint Mathematics Workshop (ADJOINT) Mathematics Workshop
The African Diaspora Joint Mathematics Workshop (ADJOINT) is a two-week summer activity designed for researchers with a Ph.D. degree in the mathematical and statistical sciences who are interested in conducting research in a collegial environment. The main objective of ADJOINT is to provide opportunities for in-person research collaboration to U.S. mathematical and statistical scientists, especially those from the African Diaspora, who will work in small groups with research leaders on various research projects.
Application: Tuesday, December 15, 2020
Dates: June 21 – July 2, 2020
Location: Berkeley, California


Parameter Estimation and Inverse Problems3rd Edition
Richard C. AsterBrian BorchersClifford H. ThurberPrice: USD: 99.95ISBN: 978-0-12-804651-7Copyright date: Oct 23, 2018
Elsevier Science & Technology
Parameter Estimation and Inverse Problems, Third Edition, is structured around a course at New Mexico Tech and is designed to be accessible to typical graduate students in the physical sciences who do not have an extensive mathematical background. The book is complemented by a companion website that includes MATLAB codes that correspond to examples that are illustrated with simple, easy to follow problems that illuminate the details of particular numerical methods. Updates to the new edition include more discussions of Laplacian smoothing, an expansion of basis function exercises, the addition of stochastic descent, an improved presentation of Fourier methods and exercises, and more.
Graduate and advanced undergraduate students taking courses in geophysical inverse problems. It is also used as a reference for geoscientists and researchers in academe and industry
1. Introduction2. Linear Regression3. Rank Deficiency and Ill-Conditioning4. Tikhonov Regularization5. Discretizing by Basis Functions6. Iterative Methods of Solving Linear Problems7. Additional Regularization Techniques8. Fourier Techniques9. Nonlinear Regression10. Nonlinear Inverse Problems11. Bayesian Methods12 Adjoint Methods


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In this quick check math assessment on number sense, students demonstrate knowledge of one more/one less and ten more/ten less than a given number. Included are directions, number cards, checklist, and activity sheet.


Benoît Caya October 29, 2020 Reading Worksheets
On the first day of each month the canteen cooks hamburgers. The Cook needs 1/10 of a kg of mince, 1/8 of a lettuce and 1/5 of a lettuce for each burger. How much of each item does the Cook need to have on hand if every student (except 5) order a hamburger? It takes Principal Jones 20 minutes to walk from his house to the school. If he walks to the school in the morning and home in the afternoon. How long will he walk in total in 1 school week? How long will he walk in total in a 9 week term? The school day starts at 9:00 a.m. First break is from 11:00 a.m to 11:20 a.m. The second break is from 1:00 p.m to 1:40 p.m. The school day finishes at 3:00 p.m. How much time do the students spend in class in one day? How much time do the students spend in class in one week? How much time do the students spend on break in one day? How much time do the students spend on break in one week?
1st grade math worksheets and my Mom has math teaching style. Math will not be as terrible as it seems if parents take interest in preparing their little ones for math before school age. I grew up not understanding how it is that people talk about math as difficult as they do, it was my best subject at school. It was easy because of my upbringing that ensured that math and I got acquainted long before school. My mother who was a primary grade teacher told me how she began teaching me math in different guises at home before I got to school age. I remember that with my Mom everything was somehow connected to math. She made me count the buttons in my shirt as she dressed me up, asked questions that demanded answers that are related to sums, like how many pair of shoes do you have? How many buttons are there on your Daddy has shirt? Count all the furniture in the living room and several math games.
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Remember to select worksheets that are the right level difficulty for your child. Get something too hard, and your child will become discouraged. Make it too easy, and they will not learn much. Home-school worksheets are far more than busy work. They are an important part of making sure that the concepts you teach in your home school stick with your child. Just make sure you have a good source for providing the worksheets, and that they are the right ones for your kid or kids.
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In our area, in the middle of Wayne National Forest, hunting is such a part of the culture that the first day of gun season, the Monday after Thanksgiving, is known in the schools as "Deer Day." That day is written into the school calendar as a day when schools are closed so students can go hunting. Naturally, my son wanted to join in, so he took the required "Hunters Safety Course" offered by the Ohio Department of Natural Resources. He earned a 98% on the test. His internal motivation was obviously stronger than anything we could do or offer.
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By Jörg Bewersdorff
Read Online or Download Algebra für Einsteiger: Von der Gleichungsauflösung zur Galois-Theorie (German Edition) PDF
Similar algebra books
The speculation of generalized inverses of genuine or complicated matrices has been expertly built and documented. however the generalized inverses of matrices over earrings have obtained complete remedy just recently. during this ebook, the writer, who contributed to the study and improvement of the idea, explains his effects.
This 2002 ebook offers the reader with mathematical instruments taken from matrix calculus and zero-one matrices and demonstrates how those instruments vastly facilitate the applying of classical statistical methods to econometric versions. The matrix calculus effects are derived from a number of simple principles which are generalizations of the foundations of normal calculus.
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Additional info for Algebra für Einsteiger: Von der Gleichungsauflösung zur Galois-Theorie (German Edition)
Algebra für Einsteiger: Von der Gleichungsauflösung zur Galois-Theorie (German Edition) by Jörg Bewersdorff


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BuzzMath focuses on helping middle schoolers apply their math expertise. It accommodates high-quality issues, provides instant and detailed feedback, and lets students progress at their own pace. Randomly generated values let students retry problems to acquire mastery. Teachers additionally receive detailed outcomes that assist them guide and monitor pupil progress. Learn the talents that will set you up for achievement in decimal place worth; operations with decimals and fractions; powers of 10; volume; and properties of shapes. Learn Algebra 2 aligned to the Eureka Math/EngageNY curriculum —polynomials, rational features, trigonometry, and more.
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Students can earn ninja belts of different colors for their ability degree. In-depth classes with videos, guided practice, interactive self-tests, and extra. Check out the lots of of games on this web site and on Coolmath4kids. Personalized math curriculum that includes digital classes and small-group instruction. Also has on-line modules, workbooks and reply keys, and professional growth. Here’s an online learning house that is participating, supportive, and designed to get youngsters excited about math.
This interactive online device helps students grasp fundamental facts. Set up as a whole-class or small-group competitors, students earn ninja playing cards and observe outcomes on a ninja scoring poster. Two board video games that train operations are also obtainable for purchase. Through ongoing research, MIND Research Institute continues to analyze key questions about studying, arithmetic, and the way the mind works. ST Math is their pre-K–8 visual tutorial program, helping teachers engage children extra deeply in math learning.
The many topics embody Algebra, Analysis, Game Theory, Graph Theory, Optimization, and Statistics. Statistics is a useful level of math, because it involves gathering and analyzing numbers and data. Statistics is mentioned last in this math information because seniors normally take it in college as a last math course. You can be taught much more with MathHelp, a website that gives resources and tips to improve your test-taking expertise.
Make math about more than numbers with engaging items, real-world eventualities, and limitless questions. Teachers choose the strand after which arrange college students to work independently. Istation makes customized learning straightforward with computer-adaptive instruction, assessments, personalized data profiles, and instructor sources. Includes digital classes and face-to-face teaching methods.
Learn the abilities that will set you up for fulfillment in numbers and operations; fixing equations and systems of equations; linear equations and functions; and geometry. Aplus Math supplies Interactive math sources for academics, mother and father, and college students that includes free math worksheets, math games, math flashcards, and more. Tons of enjoyable and educational on-line math games, from basic operations to algebra and geometry. Teachers know that top-of-the-line methods to verify studying sticks is through track.
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Presentation exponential growth and decay section 3.5. objectives. Carbon 14 dating math central. The half life of carbon 14 is 5730 years? yahoo answers.
Before reading more about carbon-14 dating, do the following activity to see how In mathematics, a smooth, declining curve of the type just drawn is called, not central to understanding how carbon dating is used to establish the age of an
Radiocarbon Dating and American Evangelical Christians
properties play a central role in their solution. Of course, a .. functions in calculus and higher mathematics. . Archaeology—Carbon 14 Dating. As long as a guide to dating in college dating divas favorite things done


The United States currently imports all of it's coffee. The annual demand for coffee by US consumers is given by the demand curve Q = 250 - 10P, where Q is quantity (in millions of pounds) and P is the market price per pound of coffee. World producers can harvest and ship coffee to us distributors at a constant marginal (= average) cost of $8 per pound. US distributors can in turn distribute coffee for a constant $2 per pound. The US coffee market is competitive. Congress is considering a tariff on coffee imports of $2 per pound.
a) If there is no tariff, how much do consumer pay for a pound of coffee? What is the quantity demanded?
b) If the tariff is imposed, how much will consumers pay for a pound of coffee? What is the quantity demanded?
c) Calculate the lost consumer surplus
d) Calculate the tax revenue collected by the government
e) Does the tariff result in a net gain or a net loss to society as a whole?
I'm not certain if I am correct on the following or not.
a) Consumers will pay $10 dollars per pound and the quantity demanded would be 23.936 million pounds.
b) I am not sure how to get started on this part.© BrainMass Inc. brainmass.com October 9, 2019, 7:04 pm ad1c9bdddf
Calculate the lost consumer surplus.


We?ve written before about howusing yourhands can help your memory. Are your kids?or you?havingtrouble with math? Another reason to use your hands!
Psychologists have discovered that kids who are told togesture while doing math are more likely to solve mathproblems. Researchers conducted two studies with a total of176 children in late third and early fourth grade. SaraBroaders and her colleagues randomly assigned the studentsto do different manipulations. Some were told to gesture,others were told not to gesture, and some were not toldanything either way (they were the controls). These were allkids had made mistakes in solving math problems in the past.
Students had to solve six math problems on a chalkboard andexplain to an experimenter how they solved each problem. Theresearchers coded the children?s videotaped efforts,analyzing gestures and utterances that conveyedproblem-solving strategies.
Children who were told to move their hands when explaininghow they?d solve a problem were four times as likely tomanually express correct new ways to solve problems. Andeven though they didn?t give the right answer, theirgestures revealed an implicit knowledge of mathematical ideas.
“Mankind is trapped. I want to help you spring the trap.””The veil between the worlds can fall. The undiscoveredcountry can become your backyard.””Your destiny, each of you, is to become all of God.”Find out whosaid these provocative words.
NOTE: This news story, previously published on our old site, will have any links removed.
Subscribers, to watch the subscriber version of the video, first log in then click on Dreamland Subscriber-Only Video Podcast link.


Combino is an hybride game composed by three games an edition, a 2D game cards and a 3D construction blocs.
Dyscalculia is difficulty in learning or comprehending arithmetic, such as difficulty in understanding numbers, learning how to manipulate numbers, and learning facts in mathematics. It is generally seen as a specific developmental disorder. Dyscalculia is a persistent disorder.
For a dyscalculia child to perform an operation as simple (in appearance) as 1 + 1, or to associate two fundamental forms, is a complicated task, which can handicap them heavily.
Combino is a dyscalculia game for children from 2 to 7 years. You can practice Combino with a speech therapist and continue the rehabilitation at home with your children using the poster and the explanation book called "Plans de Jeu". Is compose by three independent games : an edition print in risography, a 2D cards game printed on PVC with a UV proccess and a construction blocs made of polyethylene injected (bi-colors. You can Combine them or play separately which makes Combino an hybrid game.
The full Combino game
One part of the seconde game, children must find the geometric model hidden in the edition with 2D cards
Superposition of colors : we speak of inclusion of colors
© Photo Credit Miko/Miko Studio
© 2018 par Eisenlohr Adrien


RAND Function – Generate Random ‘Decimal’ Numbers In Excel
In my last article I discussed about how to generate RANDOM numbers in Microsoft Excel which is very easy using the function RANDBETWEEN. You type the function in excel and provide the Bottom and Top number (as lower and upper limit) and you get the Random numbers.
But if you notice carefully all these randomly generated numbers are integer type numbers, therefore, in this article I will discuss with you how to generate Decimal Random Numbers.
One solution is to divide the output of RANDBETWEEN numbers by 10, 100 or 1000, i.e., in multiple of 10.
But the problem is the integer part of these numbers will reduce and the number will not be within bottom and top range.
For example, suppose the output of RANDBETWEEN is 4387 and we divide it by 10, it will turn into 438.7, divide by 100 it will be 43.87, further divide by 1000, the output will be 4.387.
Which is not desirable.
The solution is to use RAND function.
Lets first discuss about the RAND Function.
The syntax for the RAND function in Microsoft Excel is very simple.
Parameters or Arguments
No parameters or arguments require for the RAND function.
Note : The Microsoft Excel RAND function returns a random number which is greater than or equal to 0 and less than 1. The RAND function returns a new random number each time your spreadsheet recalculates, similar to RANDBETWEEN function.
To use RAND function type =RAND() and press enter and you are done.
But did you noticed integer part is missing, why ?
Because RAND function output is from 0 to 1.
So, you can modify your formula by multiplying 10 and you may or may not get the decimal number having integer part.
This is because if the output of RAND function is starting with 0 then the number will be without having any integer part.
Also notice that we have no control over these Random numbers i.e., no upper or lower boundary or limit is implemented.
The Solution :
The solution is to modify the formula as below :
Where B represents Upper limit or Top number and A presents the Lower limit or bottom number.
What is the function of (B-A)+A in RAND formula ?
The Ordering of Mathematical Operation teaches us that multiplication is first to be calculated in this formula and then addition and then Subtraction will be applied.
The steps are
1. RAND generates a random number between 0 to 1
2. This number gets multiplied by B and A
3. New numbers are subtracted
4. then summed up by A
Hope you have enjoyed this article …
Keep reading, sharing & Stay blessed …


Calculus Optimization: Cone of Maximal Volume Inscribed in a Sphere
Lots of classical calculus problems involve finding the optimal dimensions of a shape inscribed in a sphere so that the shape has the largest possible volume. One of the easiest examples of such a problem is finding the dimensions of a cylinder inscribed within a sphere. Here we tackle a very similar problem, finding the maximum volume cone that can fit inside a sphere. Though a cone has two independent measurements, base radius and height, the constraint that it must fit inside a unit sphere (sphere of radius 1) lets us transform a two-variable calculus problem into a simpler single-variable calculus problem.
Step 1: Transforming a 2-Variable Equation Into a 1-Variable Equation
The formula for the volume of a cone is V(r, h) = (π/3)hr^2. Here, the function V is a function of r and h. But when a cone must fit exactly inside a unit sphere, we can discover a relation between r and h that allows us to rewrite r as a function of h, or h as a function of r. The figure below shows how we can relate r and h for a cone inscribed in a unit sphere.
When a cone is inscribed in a unit sphere, the relation between h and r is
(h-1)^2 + r^2 = 1
which can be simplified to
h^2 - 2h + 1 + r^2 = 1
h^2 - 2h + r^2 = 0
r^2 = 2h - h^2
r = sqrt(2h - h^2)
Now we can rewrite the cone volume formula solely in terms of h:
V(r, h) = (π/3)hr^2
V(h) = (π/3)h*[sqrt(2h - h^2)]^2
= (π/3)h(2h - h^2)
= (π/3)(2h^2 - h^3)
Step 2: Taking the Derivative, Finding h and r
The last step is to take the derivative of the volume function V(h), set the derivative expression equal to zero, solve for h, and then use our value of h to back solve for r. Doing so gives us
V'(h) = (π/3)(4h - 3h^2)
(π/3)(4h - 3h^2) = 0
4h - 3h^2 = 0
4h = 3h^2
4 = 3h
h = 4/3
So the cone with a maximal volume that can be inscribed within a unit circle has a height of 4/3. To figure out the value of r, we use the expression r = sqrt(2h - h^2) that we found in Step 1. Plugging h = 4/3 into this expression gives us
r = sqrt(8/3 - 16/9)
Finally, we can write the solution as h = 4/3 and r = sqrt(8/9). The volume of this cone works out to be
V = (π/3)(4/3)(sqrt(8/9))^2
The volume of a unit sphere is 4π/3, which means the cone of maximal volume takes up 8/27 or about 29.63% of the unit sphere's volume.
Comparison to the Cylinder Problem
In the closely related problem of a finding a maximal volume cylinder inscribed within a unit sphere, the optimal dimensions turned out to be h = 2/sqrt(3) and r = sqrt(2/3), with a total cylindrical volume of V = 4π/[3*sqrt(3)] ≈ 2.4184.
Solving as a 2-Variable Calculus Problem
Instead of transforming this into a single-variable problem, you can solve it as a 2-variable problem using the technique of Lagrange multipliers. This is a technique covered in multivariable calculus or Calc III. With Lagrange multipliers, we call the constraint equation g(r, h) = r^2 - 2h + h^2 = 0 and solve the system of equations
∂V/∂r = λ*∂g/∂r
∂V/∂h = λ*∂g/∂h
g = 0
which is the system
(2π/3)hr = 2λr
(π/3)r^2 = 2λ(1-h)
r^2 - 2h + h^2 = 0
Since it is a system of non-linear equations, you cannot use matrices to solve it, but instead substitution.


At Place Farm Primary Academy, we teach Maths through an engaging and accessible style, using the Mathematics Mastery programme. Mathematics Mastery was developed by the ARK schools and inspired by Maths teaching from Shanghai and Singapore. At Place Farm Primary Academy, Mathematics Mastery is taught from Years One to Six.
The Mathematics Mastery curriculum follows the National Curriculum objectives and has been developed to ensure that every child can achieve excellence in Maths. It provides children with a deep understanding of the subject, through the use of a concrete, pictorial and abstract approach. This ensures that children fully understand what they are learning, and are able to explain their methods.
Mathematics Mastery provides fewer topics to be taught in greater depth, and aims for children to calculate equations with confidence. The lessons are taught in six parts, enabling children to explore and discuss concepts with their peers, before working independently. This is invaluable, because it ensures that the children have a secure understanding of each new concept.
To reinforce our teaching of Mathematics Mastery, there are many ways that you can support your child at home;
- Practising multiplication tables
- Exploring telling the time
- Weighing and measuring
- Playing money-focused games
- Playing board games
- Looking for shapes
- Discussing the days of the week and the months of the year
There is also a wealth of information available on the Mathematics Mastery website, which can be found at: www.mathematicsmastery.org
Miss Lister is the Maths Leader at Place Farm Primary Academy.


Bass numbers of semigroup-graded local cohomology
Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology HIi(M) at any graded ideal I in terms of Ext modules. We use this method to obtain flniteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Qsat is simplicial, and a finitely generated module M over k[Q] that is graded by Qgp, the Bass numbers of HIi(M) are finite for any Q-graded ideal I of k[Q]. Conversely, if Qsat is not simplicial, we find a graded ideal I and graded k[Q]-module M such that the local cohomology module HIi(M) has infinite-dimensional socle. We introduce and exploit the combinatorially defined essential set of a semigroup.
Volume / Issue
Start / End Page
International Standard Serial Number (ISSN)
Digital Object Identifier (DOI)


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course 3 chapter 3 equations in two variables worksheet answers lesson 4 homework practice mathcounts4evercourse 3 • chapter 3 equations in two variables name date period lesson 5 homework practice graph a line using intercepts state the x and y intercepts of each function 1 6x 8y = 24 x intercept —4 y intercept 3 2 x gy = 18 x intercept 24 y intercept 3 4 10a loy = 20 x intercept 48 y intercept 36 x lntercept 2 y intercept 2 3 x y = 12 5 x y = 1 x intercept 1 y.
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- July 25, 2019 at 11:28 pm
I know this seems like an elementary question, but I’m not having any luck accomplishing my task. So if I have a circle and I want to add a line from the center of the circle and have it intersect with the circle, how do I do it? I’ve tried everything I can think of and it won’t make a complete path. I’m wanting it to combine so when I send it to the CNC it will start at the point in the middle of the circle and then travel along the straight line and complete the circle. I can not for the life of me get it to work. Please help!!!
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Read the problem looking for keywords and values:
- What type of variable is the focus of the problem? Is it quantitative, e.g. a mean, or categorical, e.g. a proportion or percent?
- How many variables are of concern?
- Is(are) the population(s) standard deviation, sigma, given?
- Are sample variances equal or are you instructed to assume they are equal?
- What is(are) the sample size(s)?
- Is a claim or proposition mentioned?
- Are you given summary statistics or raw data?
At many golf clubs, a teaching professional provides a free 10-minute lesson to new customers. A golf magazine reports that golf facilities that provide these free lessons gain, on average, $2100 in green fees, lessons, or equipment expenditures. A teaching professional believes that the average gain exceeds 2,100, and collects data (shown below in the solution examples) on the gain from 15 clubs in his area. At the 0.05 level of significance, does the data support the claim?
What should you glean from reading the problem statement?
- This is a single-sample test about a mean.
- The teaching professional believes that average gain exceeds $2100. Thus the claim is that the mean gain exceeds $2100. The phrase “exceeds $2100” indicates the claim contains the “>” math operator which is a form of inequality. Thus, the claim must be the alternative hypothesis, Ha: μ > $2100, because the null must be an equality.
- The null hypothesis is always the complement of Ha and is Ho: μ <= $2100.
- Because Ha contains the > operator, this is a right-tailed test. Remember the mnemonic that the math operator in the alternative points toward the tail of the test.
- We are given raw data, N = 15, and we do not know the population standard deviation, sigma. Because we do not know sigma and N is less than 30, we must use a t-test for a small sample.
- The degrees of freedom for the t-test are N-1 or 14.
- If the t-test statistic for the data is greater than 1.76 (or p<0.05), we reject the null and accept the alternative.
- Using technology, see below, the test statistic is 1.85 which is greater than 1.76 and falls in the rejection region. Accordingly, the p-value is 0.0425 which is less than alpha = 0.05. Both methods indicate the null hypothesis should be rejected in favor of the alternative.
- Because the claim is the alternative, which has been supported, the conclusion is “There is sufficient evidence to support the claim that the average gain from giving free lessons to new customers is greater than $2100.”
StatCrunch: The data is in column “gain.” 1. Select Stat>T Stats>One Sample>With Data. 2. In the One Sample T dialog box (2), click on “gain” in the Select Column(s) section, select the Perform Hypothesis test for µ radio button. Enter the null hypothesis, Ho: μ= 2100. Select > in the Ha: μ > 2100 and click Compute. 3. The results of the hypothesis test are in the box (with Options button) that opens. The t-test statistic is 1.85 and the p-value is 0.0425.
The data is in column A. Select PHSTAT>One-Sample Tests>t-test for Mean, sigma unknown. In the dialog box that opens, enter the Null Hypothesis of 2100, the Level of Significance of 0.05, click on Sample Statistics Unknown, enter the range of the data values, check that the First cell contains a label, select an Upper-tail Test and click OK.
A new tab will be inserted with the name Hypothesis. The t-test statistic is 1.85 and the p-value is 0.0425.


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Hypothesis testing relates to the basic concepts of Statistics. Some of the basic statistics are Parametric, Non Parametric tests, Confidence intervals, alternate hypothesis, Significance of tests and many other tests. Thus, you may get that the related topics are simple but sometimes these can provide a complex problems that need appropriate solutions. Thus, the requirement of Hypothesis Testing Homework Help is increasing day by day.
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Independent simple random samples of 950 men and 2500 women are tested. Among the men in the sample, 81 have red/green color blindness. Among the women in the sample, 9 have red/green color blindness.
The value of the test statistic is z= (round to at least three decimal places).
A sample of 56 students who live on-campus has a mean GPA of 83.7 with a sample standard deviation of 7.44. A independent sample of 42 students who live off-campus has a mean GPA of 82.4 with a standard deviation of 5.45
The value of the test statistic is
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In practice structural equations are often estimated by least-squares, thus neglecting any simultaneity. This paper reveals why this may often be justifiable and when. Assuming data stationarity and existence of the first four moments of the disturbances we find the limiting distribution of the ordinary least-squares (OLS) estimator in a linear simultaneous equations model. In simple static and dynamic models we compare the asymptotic efficiency of this inconsistent estimator with that of consistent simple instrumental variable (IV) estimators and depict cases where -- due to relative weakness of the instruments or mildness of the simultaneity -- the inconsistent estimator is more precise. In addition, we examine by simulation to what extent these first-order asymptotic findings are reflected in finite sample, taking into account non-existence of moments of the IV estimator. By dynamic visualization techniques we enable to appreciate any differences in efficiency over a parameter space of a much higher dimension than just two, viz. in colored animated image sequences (which are not very effective in print, but much more so in live-on-screen projection).
invalid instruments simultaneity bias weak instruments 4D diagrams due to relative weakness of the instruments or mildness of the simultaneity