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August 2019
Variable selection in multiple regression
If you’ve been following this series, you now know that multiple regression can be very useful but that its usefulness depends on overcoming several challenges. One of those challenges is that if we
use all of the covariates available to us and some of them are highly correlated with one another, our assessment of which covariates have an association with the response variable may be misleading
and any prediction we make about new observations may be very unreliable. That leads us to the problem of variable selection. Rather than using all of the covariates we have available, maybe we’d be
better off if we used only a few.
In this R notebook, I explore a couple of approaches to variable selection:
1. Restricting the covariates to those we know have an association with the response variable.^1
2. Identifying clusters of covariates that are highly associated with one another, (relatively) unassociated with those in other clusters, and picking one covariate from each cluster for the
As you’ll see for the sample data set we’ve been exploring in which there are two clusters of covariates having strong associations within clusters and weak to non-existent associations between
clusters, neither of these approaches serves us particularly well. The next installment will explore another commonly used approach – principal components regression.
1. There’s at least one obvious problem with this approach that I don’t discuss in the notebook. In the work I’ve been involved with, we rarely know ahead of time which covariates, if any, have
“real” relationships with the response variable. Most often we’ve measured covariates because we anticipate that they have some relationship to what we’re interested in and we’re trying to figure
out which one(s) are most important. ↩
2. This approach has some practical problems that I don’t discuss in the notebook. How strong do associations have to be to be “highly associated”? How weak do they have to be to be “(relatively)
unassociated”? What do we do if there isn’t a clear cutoff between “highly associated” and “(relatively) associated”? ↩
The Marist Mindset List for the Class of 2023
Yes, you read that right. It’s the Marist Mindset List, not the Beloit Mindset List. It’s the same Mindset list as before, but it now has a new home. If you’ve never heard of the Mindset List before,
here’s the full press release. The short version is
The Marist Mindset List is created by Ron Nief, Director Emeritus of Public Affairs at Beloit College, along with educators McBride and Westerberg, Shaffer, and Zurhellen. Additional items on the
list, as well as commentaries and guides, can be found at www.marist.edu/mindset-list and www.themindsetlist.com.
As always, I enjoy looking over the list, even though it makes me feel really old. Here are a few of the items I found particularly striking this year.
• Like Pearl Harbor for their grandparents, and the Kennedy assassination for their parents, 9/11 is an historical event.
• The primary use of a phone has always been to take pictures.
• The nation’s mantra has always been: “If you see something, say something.”
• They are as non-judgmental about sexual orientation as their parents were about smoking pot.
• Apple iPods have always been nostalgic.
You can find the full list at www.marist.edu/mindset-list. Enjoy!
Challenges of multiple regression (or why we might want to select variables)
Variable selection in multiple regression
We saw in the first installment in this series that multiple regression may allow us to distinguish “real” from “spurious” associations among variables. Since it worked so effectively in the example
we studied, you might wonder why you would ever want to reduce the number of covariates in a multiple regression.
Why not simply throw in everything you’ve measured and let the multiple regression sort things out for you? There are at least a couple of reasons:
1. When you have covariates that are highly correlated, the associations that are strongly supported may not be the ones that are “real”. In other words, if you’re using multiple regression in an
attempt to identify the “important” covariates, you may identify the wrong ones.
2. When you have covariates that are highly correlated, any attempt to extrapolate predictions beyond the range of covariates that you’ve measured may be misleading. This is especially true if you
fit a linear regression and the true relationship is curvilinear.^1
This R notebook explores both of these points using the same set of deterministic relationships we’ve used before to generate the data, but increasing the residual variance.^2
1. The R notebook linked here doesn’t explore the problem of extrapolation when the true relationship is curvilinear, but if you’ve been following along and you have a reasonable amount of facility
with R, you shouldn’t find it hard to explore that on your own. ↩
2. The R-squared in our initial example was greater than 0.99. That’s why multiple regression worked so well. The example you’ll see here has an R-squared of “only” 0.42 (adjusted 0.36). The “only”
is in quotes because in many analyses in ecology an evolution, an R-squared that large would seem pretty good. ↩
What is multiple regression doing?
Not long after making my initial post in this series on variable selection in multiple regression, I received the following question on Twitter:
The short answer is that lm() isn’t doing anything special with the covariates. It’s simply minimizing the squared deviation between predictions and observations. The longer version is that it’s able
to “recognize” the “real” relationships in the example because it’s doing something analogous to a controlled experiment. It is (statistically) holding other covariates constant and asking what the
effect of varying just one of them is. The trick is that it’s doing this for all of the covariates simultaneously.
I illustrate this in a new R notebook by imagining a regression analysis in which we look for an association between, say, x9 and the residuals left after regressing y on x1.
Collecting my thoughts about variable selection in multiple regression
I was talking with one of my graduate students a few days ago about variable selection in multiple regression. She was looking for a published “cheat sheet.” I told her I didn’t know of any. “Why
don’t you write one?” “The world’s too complicated for that. There will always be judgment involved. There will never be a simple recipe to follow.” That was the end of it, for then.
From the title you can tell that I decided I needed to get my own thoughts in order about variable selection. If you know me, you also know that I find one of the best ways to get my thoughts
straight is to write them down. So that’s what I’m starting now.
Expect to see a new entry every week or so. I’ll be posting the details in R notebooks so that you can download the code, run it yourself, and play around with it if you’re so inclined.^1 As I
develop notebooks, I’ll develop a static page with links to them. Unlike the page on causal inference in ecology, which links to blog posts, these will link directly to HTML versions of R notebooks
that will show discuss the aspect of the issue I’m working through that week along with the R code that facilitated my thinking. All of the source code will be available in a Github repository, but
you’ll also be able to download the .Rmd file when you have the HTML version open simply by clicking on the “Code” button at the top right of the page and selecting “Download Rmd” from the dropdown.
If you’re still interested after all of that. Here’s a link to the first installment:
Why multiple regression is needed
1. You’ll get the most out of R notebooks if you work with them through RStudio. Fortunately, the open source version is likely to serve your needs, so all it will cost you is a little bit of disk
space. ↩ | {"url":"https://darwin.eeb.uconn.edu/uncommon-ground/blog/2019/08/","timestamp":"2024-11-08T12:00:32Z","content_type":"text/html","content_length":"51587","record_id":"<urn:uuid:8ceca16e-b802-4ef3-ad76-ce4162b67107>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00351.warc.gz"} |
1 Digit By 4 Digit Division Worksheets - Divisonworksheets.com
1 Digit By 4 Digit Division Worksheets
1 Digit By 4 Digit Division Worksheets – Utilize division worksheets to help your child learn and refresh division concepts. Worksheets come in a wide range of styles, and you can even design your
own. They’re great because you can download them and modify them to your preferences. They’re perfect for second-graders, kindergarteners, and first-graders.
huge numbers by two
The worksheets are able to assist children in dividing huge numbers. A lot of worksheets are limited to three, two, or even four different divisors. This method means that the child doesn’t have to
be concerned about missing a division or making mistakes with their times tables. To help your child learn this mathematical ability you can either download worksheets or find them on the internet.
Kids can work on and build their comprehension of the subject by using worksheets on multi-digit division. It is a fundamental maths skill that is essential for many calculations in daily life, as
well as complex topics. These worksheets provide an interactive set of questions and activities that help students understand the concept.
Students struggle to divide large numbers. These worksheets utilize a standard algorithm as well as step-by–step instructions. It is possible for students to lose the intellectual understanding
required. Long division can be taught by using the base ten blocks. After students have learned the steps involved, long division will become a natural to them.
Large numbers that are divided could be practiced by pupils using a variety of worksheets and exercises. The worksheets also provide fractional results in decimals. There are worksheets that allow
you to calculate hundreds ofths. This is particularly helpful when you need to divide large sums of money.
Sort the numbers into smaller groups.
It can be difficult to arrange a number into small groups. It can appear appealing on paper, but many participants of small groups dislike this process. This is the way the human body evolves. The
process can aid in the Kingdom’s unending expansion. It inspires others and motivates people to reach out to those who are forgotten.
It can be a useful tool for brainstorming. It is possible to form groups of people who have similar traits and experiences. This will let you think of new ideas. Reintroduce yourself to each person
once you’ve created your groups. It’s a good activity that stimulates creativity and innovation.
It can be used to split massive numbers into smaller units. If you are looking to make an equal amount of things for different groups, it can be helpful. You could break up the class into five
groups. Combine these groups and you’ll get the initial 30 students.
Remember you are able to divide numbers by using two different types of numbers: the divisor, as well as the quotient. The result of dividing one by another is “ten/five,” while dividing two by two
yields the same result.
Powers of ten can be used for huge numbers.
To help us compare the large number of numbers, we can divide them into powers of 10. Decimals are a very regular element of shopping. These can be seen on receipts and food labels, price tags and
even receipts. The petrol pumps also use them to show the cost per gallon, as well as the amount of gasoline that flows through the funnel.
There are two ways to divide large numbers into powers of ten. The first is by moving the decimal left, and then multiplying it by 10-1. The other method utilizes the power of ten’s associative
feature. Once you’ve learned the associative property of powers of 10, you will be able to divide an enormous number into smaller powers that are equal to 10.
The first method is based on mental computation. Divide 2.5 by the power of 10 to find the pattern. The decimal position shifts to one side for each tenth power. This concept can be applied to solve
any problem.
The other is to mentally divide very large numbers into powers of 10. You can then quickly express large numbers using scientific notation. When using scientific notation, huge numbers should be
written in positive exponents. It is possible to convert 450,000 numbers into 4.5 by moving the decimal point 5 spaces to the left. To break large numbers down into smaller powers, you can apply the
exponent 5.
Gallery of 1 Digit By 4 Digit Division Worksheets
Fourth Grade Math Worksheets Free Printable K5 Learning 4 Digit By 1
4 Digit By 1 Digit Long Division With Grid Assistance And NO Remainders
Free Division Worksheets
Leave a Comment | {"url":"https://www.divisonworksheets.com/1-digit-by-4-digit-division-worksheets/","timestamp":"2024-11-07T05:58:31Z","content_type":"text/html","content_length":"64766","record_id":"<urn:uuid:6e530310-35e7-4190-9108-757f763e059f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00805.warc.gz"} |
Identifying Right Angles in a Given Figure
Question Video: Identifying Right Angles in a Given Figure Mathematics
How many right angles are there in the figure?
Video Transcript
How many right angles are there in the figure?
We’re gonna walk through the picture and label the right angles. These two window pieces are rectangles, which means they have right angles. The same thing is true for the right window. Each of the
windows had four right angles. Our door is a rectangle; there were four right angles inside the door. We also can look at the space between the door and the bottom line. That space creates an
additional right angle on both sides of the door. We can continue by looking at the large rectangle and its four corners. The chimney has two right angles.
Alright, let’s stop here and count up what we currently have: two from the chimney, four from each of the windows, plus four from the door, plus four from the big rectangle of the house, and then the
two that are on the outsides of the door.
When we add up all of these, we get 20. However, there’re actually still two more right angles. The place where the roof meets the house has two additional right angles, one here and one here. If we
add those two to the 20 right angles we have already found, we come up with a total of 22 right angles in this figure. | {"url":"https://www.nagwa.com/en/videos/942173950168/","timestamp":"2024-11-05T05:38:23Z","content_type":"text/html","content_length":"241246","record_id":"<urn:uuid:8b4c4973-9cde-4a49-bb8d-1dc665ce9021>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00048.warc.gz"} |
Quasi-Monte Carlo EM Methods for NLME Analysis
R. H. Leary
Problem and Methodology: The classical EM method incorporates an E-step which requires the evaluation of often analytically intractable intergrals. In the context of PK/PD NLME estimation, the
integrals of interest are the normalizing factor for the posterior density of the random effects for each subject, as well as the first and second moments of this density.
The Monte-Carlo EM (MCEM) method proposed by Wei and Tanner[1] replaces these integrals with empirical averages using random draws from the distribution of interest. Often this distribution cannot be
sampled directly, and techniques such as MCMC (as in the SAEM algorithms in MONOLIX and NM7) and importance sampling (as in the MCPEM algorithm in NM7) are used. The accuracy of the required
integrals is controlled by the number of samples N, and typically the theoretical asymptotic error behavior is O(1/sqrt(N)). While often rapid progress can be made in early iterations of MCEM even
with low accuracy integrals and small sample sizes, the accuracy requirements increase in the later stage iterations, and very large sample sizes may be required. Thus a significant improvement in
efficiency may be obtained in the later stages by lowering the sample size required to achieve the required accuracy.
Here we consider the use of quasi-random (QR) draws using low discrepancy Sobol sequences, as originally proposed for the PK/PD NLME context in the PEM algorithm of Leary et al. [2] The advantage of
QR relative to random draws is that the theroretical asymptotic error behavior is now approximately O(1/N), a very significant improvement. A new implementation of PEM based on importance sampling
has been developed for the Pharsight Phoenix NLME platform, which utilizes recent enhancements of the QR technique with the scrambling techniques of Owen [3]. This implementation allows the direct
comparison of random and QR versions, as well as adaptive error monitoring.
Results: Numerous PK/PD NLME test problems have been compared with random vs. QR importance sampling. Generally the theoretical O(1/N) vs O(1/sqrt(N)) advantage of the QR technique has been confirmed
in practice. Improvements in speed of nearly 2 orders of magnitude have been observed for some posterior density integrals where high accuracy is requred, with the sampling requirements being reduced
from tens of thousands of points to several hundred. We note that this also confirms a similar QR vs. random draw efficiency improvement obtained by Jank [4] in a MCEM application to a geostatistical
generalized linear mixed model.
Conclusions: We have demonstrated that quasirandom integration offers significant practical efficiency advantages relative to the random integration techniques employed in purely stochastic MCEM
methods using importance sampling. It is still an open question and an area of active research in the statistical community whether the efficiency advantages of the QR methodology, in conjunction
with scrambling, can be extended to MCMC methods such as SAEM.
[1] C. Wei and M. Tanner, J. American Statistical Assoc., 85, 699-704 (1990)
[2] R. H. Leary, R. Jelliffe, A. Schumitzky, and R. E. Port, PAGE 13 (2004), Abstract 491.
[3] A. B. Owen. J. Complexity, 14(4): 466-489 (1998).
[4] W. Jank, Computational Statistics and Data Analysis, 48(4), 685-701 (2005).
Reference: PAGE 19 (2010) Abstr 1737 [www.page-meeting.org/?abstract=1737]
Poster: Methodology- Algorithms | {"url":"https://www.page-meeting.org/?abstract=1737","timestamp":"2024-11-09T06:20:12Z","content_type":"text/html","content_length":"20888","record_id":"<urn:uuid:2c098470-923e-495b-a463-57c0a2575545>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00495.warc.gz"} |
Statistics/Preliminaries - Wikibooks, open books for an open world
This chapter discusses some preliminary knowledge (related to statistics) for the following chapters in the advanced part.
Definition. (Random sample) Suppose ${\displaystyle X}$ is a random variable resulting from a random experiment, with a certain distribution. After repeating this random experiment ${\displaystyle n}
$ independent times, we obtain ${\displaystyle n}$ independent and identically distributed (iid) random variables, denoted by ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ , associated with the ${\
displaystyle n}$ outcomes. They are called a random sample from the distribution with sample size ${\displaystyle n}$ .
• We usually refer the underlying distribution as a population.
• Often, computer is useful for conducting such experiment and repeating it many times.
• In particular, a programming language, called R, is commonly used for computational statistics. You may see the wikibook R Programming for more details about it.
• As a result, the content discussed in this section (as well as the section about resampling) is quite relevant to computational statistics.
Since all these ${\displaystyle n}$ random variables follow the same cdf as ${\displaystyle X}$ , we may expect their distribution should be somewhat similar to the distribution of ${\displaystyle X}
$ , and indeed, this is true. Before showing how this is true, we need to define "the distribution of these ${\displaystyle n}$ random variables" more precisely, as follows:
Definition. (Empirical distribution) The cdf of empirical distribution, empirical cdf, of a random sample ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ , denoted by ${\displaystyle F_{\color
{darkgreen}n}(x)}$ , is ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\mathbf {1} \{X_{k}\leq x\}}$ .
• ${\displaystyle \mathbf {1} \{A\}}$ is the indicator function with value 1 if ${\displaystyle A}$ is true and 0 otherwise.
• We can see that ${\displaystyle F_{n}(x)}$ "assigns" the probability (or "mass") ${\displaystyle 1/n}$ to each of ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ , and this is indeed a valid cdf.
□ This is because for each of ${\displaystyle X_{1},\dotsc ,X_{n}}$ , if it is less than or equal to ${\displaystyle x}$ , then the corresponding indicator function in the sum is one, and thus
a value of "${\displaystyle 1/n}$ " is contributed to the cdf.
□ To understand this more clearly, consider the following example.
• We can interpret ${\displaystyle F_{n}(x)}$ as the relative frequency of the event ${\displaystyle \{X\leq x\}}$ . Recall that the frequentist definition of probability of an event is the
"long-term" relative frequency of the event (i.e. the relative frequency of the event after repeating a random experiment infinite number of times). As a result, we will intuitively expect that $
{\displaystyle F_{n}(x)\approx F(x)}$ when ${\displaystyle n}$ is large.
Example. A random sample of size 5 is taken from an unknown distribution, and the following numbers are obtained:
-1.4, 2.3, 0.8, 1.9, -1.6
(a) Find the empirical cdf.
(b) Let ${\displaystyle Y}$ be a (discrete) random variable with cdf exactly the same as the empirical cdf in (a). Prove that the pmf of ${\displaystyle Y}$ (called empirical pmf) is ${\displaystyle
f_{Y}(y)=\mathbb {P} (Y=y)={\frac {1}{5}},\quad y=-1.6,-1.4,0.8,1.9{\text{ or }}2.3.}$ Solution:
(a) First, we order the sample data ascendingly so that we can find the empirical cdf more conveniently:
-1.6, -1.4, 0.8, 1.9, 2.3
The empirical cdf is given by ${\displaystyle F_{5}(x)={\begin{cases}0,&x<-1.6;\\1/5,&-1.6\leq x<-1.4;\\2/5,&-1.4\leq x<0.8;\\3/5,&0.8\leq x<1.9;\\4/5,&1.9\leq x<2.3;\\1,&x\geq 2.3.\\\end{cases}}}$
• After ordering the sample data, we treat each of the number as an observed value of the random sample: ${\displaystyle X_{1}=-1.6,X_{2}=-1.4,X_{3}=0.8,X_{4}=1.9,X_{5}=2.3}$ .
• Then, when ${\displaystyle x<1.6}$ , none of ${\displaystyle X_{1},\dotsc ,X_{5}}$ is less than or equal to ${\displaystyle x}$ . So, all indicator functions involved are zero, and thus the value
of the empirical cdf is zero.
• When ${\displaystyle -1.6\leq x<-1.4}$ , only ${\displaystyle X_{1}\leq x}$ , and thus only the indicator function ${\displaystyle \mathbf {1} \{X_{1}\leq x\}=1}$ in this case, and all other
indicator functions are zero. As a result, the value is ${\displaystyle {\frac {\sum _{k=1}^{5}\mathbf {1} \{X_{k}\leq x\}}{5}}={\frac {\mathbf {1} \{X_{1}\leq x\}+0+0+0+0}{5}}={\frac {1}{5}}}$ .
• Similarly, when ${\displaystyle -1.4\leq x<0.8}$ , only ${\displaystyle X_{1},X_{2}\leq x}$ , and thus only the indicator function ${\displaystyle \mathbf {1} \{X_{1}\leq x\}=1}$ and ${\
displaystyle \mathbf {1} \{X_{2}\leq x\}=1}$ in this case, and all other indicator functions are zero. As a result, the value is ${\displaystyle {\frac {\sum _{k=1}^{5}\mathbf {1} \{X_{k}\leq x
\}}{5}}={\frac {\mathbf {1} \{X_{1}\leq x\}+\mathbf {1} \{X_{2}\leq x\}+0+0+0}{5}}={\frac {2}{5}}}$ .
• ...
• When ${\displaystyle x\geq 2.3}$ , all ${\displaystyle X_{1},\dotsc ,X_{5}\leq x}$ . Hence, all indicator functions are one, and thus the value of the empirical cdf is ${\displaystyle {\frac
{1+1+1+1+1}{5}}=1}$ .
Proof. First, notice that the cdf of ${\displaystyle Y}$ is ${\displaystyle F_{Y}(y)=\mathbb {P} (Y\leq y)=\mathbb {P} (Y<y)+\mathbb {P} (Y=y)=\mathbb {P} (Y<y)+f_{Y}(y)\implies f_{Y}(y)=\mathbb {P}
(Y\leq y)-\mathbb {P} (Y<y)}$ .
Then, we observe that when ${\displaystyle y=-1.6}$ , ${\displaystyle \mathbb {P} (Y\leq y)=F_{5}(-1.6)=1/5}$ , and ${\displaystyle \mathbb {P} (Y<y)=\mathbb {P} (Y<-1.6)=0}$ (from the empirical
cdf). Hence, ${\displaystyle f_{Y}(y)={\frac {1}{5}}}$ in this case. Similarly, when ${\displaystyle y=-1.4}$ , ${\displaystyle \mathbb {P} (Y\leq y)=F_{5}(-1.4)=2/5}$ , and ${\displaystyle \mathbb
{P} (Y<y)=\mathbb {P} (Y<-1.4)={\frac {1}{5}}}$ . Thus, ${\displaystyle f_{Y}(y)={\frac {2}{5}}-{\frac {1}{5}}={\frac {1}{5}}}$ also in this case. With similar arguments, we can show that ${\
displaystyle f_{Y}(y)={\frac {1}{5}}}$ also when ${\displaystyle y=0.8,1.9,{\text{ or }}2.3}$ .
${\displaystyle \Box }$
• Observe from (b) that the support of ${\displaystyle Y}$ contains exactly the numbers in the sample data, which are the realization of the random sample ${\displaystyle X_{1},\dotsc ,X_{5}}$ .
This shows that the probability ${\displaystyle 1/5}$ is "assigned" to each of ${\displaystyle X_{1},\dotsc ,X_{5}}$ .
Theorem. (Glivenko–Cantelli theorem) As ${\displaystyle n\to \infty }$ , ${\displaystyle \sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|\to 0}$ almost surely (a.s.).
• ${\displaystyle \sup }$ stands for supremum of a set (with some technical requirements), which means the least upper bound of the set, which is the least element that is greater or equal to each
other element in the set.
□ The meaning of ${\displaystyle \sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|}$ is the least upper bound of the set containing the values of ${\displaystyle |F_{n}(x)-F(x)|}$ over ${\displaystyle x
\in \mathbb {R} }$ .
□ The supremum is similar to the concept of maximum (indeed, if maximum exists, then maximum is the same as supremum), but a difference between them is that sometimes supremum exists while
maximum does not exist.
□ For instance the supremum of the set (or interval) ${\displaystyle [0,1)}$ is 1 (intuitively). However, the maximum of the set ${\displaystyle [0,1)}$ (i.e. the greatest element in the set)
does not exist (notice that 1 is not included in this set) ^[1].
• The term "almost surely" means that this happens with probability 1. The details for the reason of calling this "almost surely" instead of "surely" involves some understanding of measure theory,
and so is omitted here.
• Roughly speaking, from this theorem, we know that ${\displaystyle F_{n}(x)}$ is a good estimate of ${\displaystyle F(x)}$ , and an even better estimate of (or "closer to") ${\displaystyle F(x)}$
when ${\displaystyle n}$ is large, for every realization ${\displaystyle x_{1},\dotsc ,x_{n}}$ (each of them is real number), since the least upper bound of the absolute difference already tends
to zero, and then we will intuitively expect that every such absolute difference also tends to zero.
• This theorem is sometimes referred as the fundamental theorem of statistics, indicating its importance in statistics.
We have mentioned how we can approximate the cdf, and now we would like to estimate the pdf/pmf. Let us first discuss how to estimate the pmf.
For the discrete random variable ${\displaystyle X}$ , from the empirical cdf, we know that each ${\displaystyle X_{1},\dotsc ,X_{n}}$ is "assigned" with the probability ${\displaystyle 1/n}$ . Also,
considering the previous example, the empirical pmf is ${\displaystyle f_{n}(x)={\frac {\sum _{k=1}^{n}\mathbf {1} \{X_{k}=x\}}{n}}}$ .
• The empirical pmf ${\displaystyle f_{n}(x)}$ shows the relative frequency of occurrences of ${\displaystyle x}$ , and therefore can approximate the probability of occurrences of ${\displaystyle
x}$ , which is the long-term relative frequency of occurrences of ${\displaystyle x}$ .
To discuss the estimation of pdf of continuous random variable, we need to define class intervals first.
Definition. (Class intervals) First, choose an integer ${\displaystyle i\geq 1}$ , and a sequence of real numbers ${\displaystyle c_{0},c_{1},\dotsc ,c_{i}}$ such that ${\displaystyle c_{0}<c_{1}<\
dotsb <c_{i}}$ . Then, the class intervals are ${\displaystyle (c_{0},c_{1}],(c_{1},c_{2}],\dotsc ,(c_{i-1},c_{i}]}$ .
For the continuous random variable ${\displaystyle X}$ , construct class intervals for ${\displaystyle X}$ which are a non-overlapped partition of the interval ${\displaystyle [X_{\text{min}},X_{\
text{max}}]}$ , in which ${\displaystyle X_{\text{min}}}$ and ${\displaystyle X_{\text{max}}}$ are the minimum and maximum values in the sample. Then, the pdf ${\displaystyle f(x)\approx {\frac {F(c_
{j})-F(c_{j-1})}{c_{j}-c_{j-1}}},\quad x\in (c_{j-1},c_{j}]{\text{ and }}j=1,2,\dotsc ,i,}$ when ${\displaystyle c_{j-1}}$ and ${\displaystyle c_{j}}$ are close, i.e. the length of each class
interval is small. (Although the union of the above class intervals is ${\displaystyle (c_{0},c_{i}]}$ and thus the value ${\displaystyle c_{0}}$ is not included in the interval, it does not matter
since the value of the pdf at ${\displaystyle c_{0}}$ does not affect the calculation of probability.) Here, ${\displaystyle c_{0}}$ is ${\displaystyle X_{\text{min}}}$ and ${\displaystyle c_{i}}$ is
${\displaystyle X_{\text{max}}}$ .
Since ${\displaystyle F(c_{j})-F(c_{j-1})=\mathbb {P} (X\in (c_{j-1},c_{j}])\approx {\color {darkgreen}{\frac {\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{j-1},c_{j}]\}}{n}}}}$ is the relative
frequency of occurrences of the event ${\displaystyle \{X_{k}\in (c_{j-1},c_{j}]\}}$ , we can rewrite the above expression as ${\displaystyle f(x)\approx h_{n}(x)={\frac {\color {darkgreen}\sum _{k=
1}^{n}\mathbf {1} \{X_{k}\in (c_{j-1},c_{j}]\}}{{\color {darkgreen}n}(c_{j}-c_{j-1})}},\quad x\in (c_{j-1},c_{j}]{\text{ and }}j=1,2,\dotsc ,i}$ in which ${\displaystyle h_{n}(x)}$ is called the
relative frequency histogram.
Since there are many possible ways to construct the class intervals, the value of ${\displaystyle h_{n}(x)}$ can differ even with the same ${\displaystyle n}$ and ${\displaystyle x}$ . When ${\
displaystyle n}$ is large and the length of each class interval is small, we will expect ${\displaystyle h_{n}(x)}$ to be a good estimate of ${\displaystyle f(x)}$ (the theoretical pdf).
There are some properties related to the relative frequency histogram, as follows:
Proposition. (Properties of relative frequency histogram)
(i) ${\displaystyle h_{n}(x)\geq 0}$ ;
(ii) The total area bounded by ${\displaystyle h_{n}(x)}$ and the ${\displaystyle x}$ -axis is one, i.e. ${\displaystyle \int _{c_{0}}^{c_{i}}h_{n}(x)\,dx=1}$ ^[2];
(iii) The probability of an event ${\displaystyle A}$ that is a union of some class intervals is ${\displaystyle \mathbb {P} (A)\approx \int _{A}^{}h_{n}(x)\,dx}$ .
(i) Since the indicator function is nonnegative (its value is either 0 or 1), ${\displaystyle n}$ is positive, and ${\displaystyle c_{j}>c_{j-1}}$ so ${\displaystyle c_{j}-c_{j-1}}$ is positive, we
have ${\displaystyle h_{n}(x)\geq 0}$ by definition.
(ii) {\displaystyle {\begin{aligned}\int _{c_{0}}^{c_{i}}h_{n}(x)\,dx&=\int _{c_{0}}^{c_{1}}h_{n}(x)\,dx+\int _{c_{1}}^{c_{2}}h_{n}(x)\,dx+\dotsb +\int _{c_{i-1}}^{c_{i}}h_{n}(x)\,dx\\&={\frac {1}
{n}}\left(\int _{c_{0}}^{c_{1}}{\frac {\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{0},c_{1}]\}}{c_{1}-c_{0}}}\,dx+\int _{c_{1}}^{c_{2}}{\frac {\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{1},c_{2}]\}}{c_
{2}-c_{1}}}\,dx+\dotsb +\int _{c_{i-1}}^{c_{i}}{\frac {\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{i-1},c_{i}]\}}{c_{i}-c_{i-1}}}\,dx\right)\\&={\frac {1}{n}}\left({\frac {\sum _{k=1}^{n}\mathbf {1} \
{X_{k}\in (c_{0},c_{1}]\}}{c_{1}-c_{0}}}\cdot (c_{1}-c_{0})+{\frac {\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{1},c_{2}]\}}{c_{2}-c_{1}}}\cdot (c_{2}-c_{1})+\dotsb +{\frac {\sum _{k=1}^{n}\mathbf {1}
\{X_{k}\in (c_{i-1},c_{i}]\}}{c_{i}-c_{i-1}}}\cdot (c_{i}-c_{i-1})\right)\\&={\frac {1}{n}}\left(\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{0},c_{1}]\}+\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{1},c_
{2}]\}+\dotsb +\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{i-1},c_{i}]\}\right)\\&={\frac {1}{n}}\left(\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in (c_{0},c_{1}]\cup (c_{1},c_{2}]\cup \dotsb \cup (c_{i-1},c_
{i}]\}\right)\\&={\frac {1}{n}}\left(\sum _{k=1}^{n}\mathbf {1} \{X_{k}\in \underbrace {(c_{0},c_{i}]} _{{\text{sample space of }}X}\}\right)\\&={\frac {1}{n}}\cdot \sum _{k=1}^{n}1\\&={\frac {1}{n}}
\cdot n\\&=1.\end{aligned}}} Here, ${\displaystyle c_{0}}$ is ${\displaystyle X_{\text{min}}}$ and ${\displaystyle c_{i}}$ is ${\displaystyle X_{\text{max}}}$ .
(iii) We can "split" the integral in a similar way as in (ii), and then eventually the integral equals ${\displaystyle {\frac {1}{n}}\cdot \sum _{k=1}^{n}\mathbf {1} \{X_{k}\in A\}}$ , and it can can
approximate ${\displaystyle \mathbb {P} (A)}$ since it is the relative frequency of occurrences of the event ${\displaystyle \{X_{k}\in A\}}$ .
${\displaystyle \Box }$
In this section, we will discuss some results about expectation, which involve some sort of inequalities. Let ${\displaystyle a}$ and ${\displaystyle b}$ be constants. Also, let ${\displaystyle \
Omega }$ be the sample space of ${\displaystyle X}$ .
Proposition. Let ${\displaystyle X}$ be a discrete or continuous random variable. If ${\displaystyle \mathbb {P} (a<X\leq b)=1}$ , then ${\displaystyle a<\mathbb {E} [X]\leq b}$ .
Proof. Assume ${\displaystyle \mathbb {P} (a<X\leq B)=1}$ .
Case 1: ${\displaystyle X}$ is discrete.
By definition of expectation, ${\displaystyle \mathbb {E} [X]=\sum _{x\in \Omega }^{}xf(x)}$ . Then, we have ${\displaystyle \sum _{x\in \Omega }^{}af(x)<\sum _{x\in \Omega }^{}xf(x)\leq \sum _{x\in
\Omega }^{}bf(x)\Rightarrow a\sum _{x\in \Omega }^{}f(x)<\mathbb {E} [X]\leq b\sum _{x\in \Omega }^{}f(x)\Rightarrow a<\mathbb {E} [X]\leq b}$ because of the condition ${\displaystyle \mathbb {P} (a
<X\leq b)=1}$ .
Case 2: ${\displaystyle X}$ is continuous.
We have similarly ${\displaystyle \int _{\Omega }^{}af(x)\,dx<\int _{\Omega }^{}xf(x)\,dx\leq \int _{\Omega }^{}bf(x)\,dx\Rightarrow a<\mathbb {E} [X]\leq b}$ because of the condition of ${\
displaystyle \mathbb {P} (a<X\leq b)=1}$ .
${\displaystyle \Box }$
• We can interchange "${\displaystyle <}$ " and "${\displaystyle \leq }$ " without affecting the result. This can be seen from the proof.
Proposition. (Markov's inequality) Suppose ${\displaystyle \mathbb {E} [X]}$ is finite. Let ${\displaystyle X}$ be a continuous nonnegative random variable. Then, for each positive number ${\
displaystyle a}$ , ${\displaystyle \mathbb {P} (X\geq a)\leq {\frac {\mathbb {E} [X]}{a}}}$ .
Proof. ${\displaystyle {\frac {\mathbb {E} [X]}{a}}={\frac {1}{a}}\int _{-\infty }^{\infty }\underbrace {xf(x)} _{\color {darkgreen}\geq 0}\,dx{\color {darkgreen}\geq }\int _{a}^{\infty }xf(x)\,dx{\
color {darkgreen}\geq }{\frac {1}{a}}\int _{a}^{\infty }af(x)\,dx=\int _{a}^{\infty }f(x)\,dx=\mathbb {P} (X\geq a),}$ as desired.
${\displaystyle \Box }$
Corollary. (Chebyshev's inequality) Suppose ${\displaystyle \mathbb {E} [X^{2}]}$ is finite. Then, for each positive number ${\displaystyle a}$ , ${\displaystyle \mathbb {P} (|X|\geq a)\leq {\frac {\
mathbb {E} [X^{2}]}{a^{2}}}.}$
Proof. First, observe that ${\displaystyle X^{2}}$ is a nonnegative random variable. Then, by Markov's inequality, for each (positive) ${\displaystyle a'=a^{2}}$ , we have ${\displaystyle \mathbb {P}
(X^{2}\geq a')\leq {\frac {\mathbb {E} [X^{2}]}{a'}}\implies \mathbb {P} (X^{2}\geq a^{2})\leq {\frac {\mathbb {E} [X^{2}]}{a^{2}}}\implies \mathbb {P} \left({\sqrt {X^{2}}}\geq {\sqrt {a^{2}}}\
right)\leq {\frac {\mathbb {E} [X^{2}]}{a^{2}}}\implies \mathbb {P} (|X|\geq a)\leq {\frac {\mathbb {E} [X^{2}]}{a^{2}}}}$ , since ${\displaystyle a}$ is positive.
${\displaystyle \Box }$
Proposition. (Jensen's inequality) Let ${\displaystyle X}$ be a continuous random variable. If ${\displaystyle g}$ is a convex function, then ${\displaystyle g\left(\mathbb {E} [X]\right)\leq \mathbb
{E} [g(X)]}$ .
Proof. Let ${\displaystyle L(x)=a+bx}$ be the tangent of the function ${\displaystyle g(x)}$ at ${\displaystyle x=\mathbb {E} [X]}$ . Then, since ${\displaystyle g}$ is convex, we have ${\
displaystyle g(x)\geq L(x)}$ for each ${\displaystyle x}$ (informally, we can observe this graphically). As a result, we have {\displaystyle {\begin{aligned}&&\int _{\Omega }^{}g(x)f(x)\,dx&\geq \int
_{\Omega }^{}L(x)f(x)\,dx\\&\Rightarrow &\mathbb {E} [g(X)]&\geq \mathbb {E} [L(X)]\\&&&=\mathbb {E} [a+bX]\\&&&=a+b\mathbb {E} [X]\\&&&=L(\mathbb {E} [X])\\&&&=g(\mathbb {E} [X])&{\text{since }}L(x)
{\text{ is tangent of }}g(x){\text{ at }}x=\mathbb {E} [X],\end{aligned}}} as desired.
${\displaystyle \Box }$
Theorem. (Cauchy-Schwarz inequality) Suppose ${\displaystyle \mathbb {E} [X^{2}]}$ and ${\displaystyle \mathbb {E} [Y^{2}]}$ are finite. Then, ${\displaystyle (\mathbb {E} [XY])^{2}\leq \mathbb {E}
[X^{2}]\mathbb {E} [Y^{2}]}$
Proof. {\displaystyle {\begin{aligned}0&\leq \mathbb {E} [(X\mathbb {E} [Y^{2}]-Y\mathbb {E} [XY])^{2}]\\&={\color {darkgreen}\mathbb {E} [}X^{2}\underbrace {(\mathbb {E} [Y^{2}])^{2}} _{\text
{constant}}-2XY\underbrace {\mathbb {E} [Y^{2}]\mathbb {E} [XY]} _{\text{constant}}+Y^{2}\underbrace {(\mathbb {E} [XY])^{2}} _{\text{constant}}{\color {darkgreen}]}\\&=(\mathbb {E} [Y^{2}])^{2}{\
color {darkgreen}\mathbb {E} [}X^{2}{\color {darkgreen}]}-2\mathbb {E} [Y^{2}]\mathbb {E} [XY]{\color {darkgreen}\mathbb {E} [}XY{\color {darkgreen}]}+(\mathbb {E} [XY])^{2}{\color {darkgreen}\mathbb
{E} [}Y^{2}{\color {darkgreen}]}\\&=\mathbb {E} [Y^{2}]\left(\mathbb {E} [X^{2}]\mathbb {E} [Y^{2}]-2(\mathbb {E} [XY])^{2}+(\mathbb {E} [XY])^{2}\right)\\&=\mathbb {E} [Y^{2}]\left(\mathbb {E} [X^
{2}]\mathbb {E} [Y^{2}]-(\mathbb {E} [XY])^{2}\right)\\\end{aligned}}} Since ${\displaystyle \mathbb {E} [Y^{2}]\geq 0}$ , we must have ${\displaystyle \mathbb {E} [X^{2}]\mathbb {E} [Y^{2}]-(\mathbb
{E} [XY])^{2}\geq 0\Leftrightarrow (\mathbb {E} [XY])^{2}\leq \mathbb {E} [X^{2}]\mathbb {E} [Y^{2}]}$ .
${\displaystyle \Box }$
Example. (Covariance inequality) Use the Cauchy-Schwarz inequality for expectations (above theorem) to prove the covariance inequality (it is sometimes simply called the Cauchy-Schwarz inequality): $
{\displaystyle {\big (}\operatorname {Cov} (X,Y){\big )}^{2}\leq \operatorname {Var} (X)\operatorname {Var} (Y)}$ (assuming the existence of the covariance and variances)
Proof. Let ${\displaystyle X'=X-\mathbb {E} [X]}$ and ${\displaystyle Y'=Y-\mathbb {E} [Y]}$ . Then, ${\displaystyle \mathbb {E} [X']}$ and ${\displaystyle \mathbb {E} [Y']}$ are finite. Hence, by
Cauchy-Schwarz inequality, ${\displaystyle (\mathbb {E} [X'Y'])^{2}\leq \mathbb {E} [(X')^{2}]\mathbb {E} [(Y')^{2}]\Leftrightarrow (\mathbb {E} [(X-\mathbb {E} [X])(Y-\mathbb {E} [Y])]\leq \mathbb
{E} [(X-\mathbb {E} [X])^{2}]\mathbb {E} [(Y-\mathbb {E} [Y])^{2}]{\overset {\text{ def }}{\Leftrightarrow }}{\big (}\operatorname {Cov} (X,Y){\big )}^{2}\leq \operatorname {Var} (X)\operatorname
{Var} (Y).}$
${\displaystyle \Box }$
Before discussing convergence, we will define some terms that will be used later.
Definition. (Statistics) Statistics are functions of random sample.
• The random sample consists of ${\displaystyle n}$ (${\displaystyle n}$ is sample size) random variables ${\displaystyle X_{1},\dotsc ,X_{n}}$ .
• Two important statistics are the sample mean ${\displaystyle {\overline {X}}={\frac {\sum _{i=1}^{n}X_{i}}{n}}}$ and the sample variance ${\displaystyle S^{2}={\frac {\sum _{i=1}^{n}(X_{i}-{\
overline {X}})^{2}}{n}}}$ .
□ In many other places, ${\displaystyle S^{2}}$ is used to denote ${\displaystyle {\frac {\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}{n-1}}}$ , the unbiased sample variance. In fact, ${\
displaystyle S^{2}}$ here is biased (we will discuss what "(un)biased" means in the next chapter). Warning: we should be careful about this difference in definitions.
□ Both of ${\displaystyle {\overline {X}}}$ and ${\displaystyle S^{2}}$ are random variables, since random variables are involved in them.
In a particular sample, say ${\displaystyle x_{1},\dotsc ,x_{n}}$ , we observe definite values of their sample mean, ${\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{n}x_{i}}{n}}}$ , and sample
variance, ${\displaystyle s^{2}={\frac {\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}{n}}}$ . However, each of the values is only one realization of the respective random variables ${\displaystyle {\
overline {X}}}$ and ${\displaystyle S^{2}}$ . We should notice the difference between these definite values (not random variables) and the statistics (random variables).
To explain the definitions of the sample mean ${\displaystyle {\overline {X}}}$ and sample variance ${\displaystyle S^{2}}$ more intuitively, consider the following.
Recall that the empirical cdf ${\displaystyle F_{n}(x)}$ assigns probability ${\displaystyle {\frac {1}{n}}}$ to each of the random sample ${\displaystyle X_{1},\dotsc ,X_{n}}$ . Thus, by the
definition of mean and variance, the mean of a random variable, say ${\displaystyle Y}$ , with this cdf ${\displaystyle F_{n}(x)}$ (and hence with the corresponding pmf ${\displaystyle f_{n}(x)}$ )
is ${\displaystyle \sum _{i=1}^{n}\left(X_{i}\cdot {\frac {1}{n}}\right)={\overline {X}}}$ . Similarly, the variance of ${\displaystyle Y}$ is ${\displaystyle \sum _{i=1}^{n}\left((X_{i}-{\overline
{X}})^{2}\cdot {\frac {1}{n}}\right)=S^{2}}$ . In other words, the mean and variance of the empirical distribution, which corresponds to the random sample, is the sample mean ${\displaystyle {\
overline {X}}}$ and the sample variance ${\displaystyle S^{2}}$ respectively, which is quite natural, right?
• Here, we use "${\displaystyle X_{i}}$ " rather than the usual "${\displaystyle x_{i}}$ " in the expression, and the mean and variance are also random variables. This is because the sample space
of the empirical cdf consists of random variables ${\displaystyle X_{1},\dotsc ,X_{n}}$ , rather than definite values ${\displaystyle x_{1},\dotsc ,x_{n}}$ .
Also, recall that the empirical cdf ${\displaystyle F_{n}(x)}$ can well approximate the cdf of ${\displaystyle X}$ , ${\displaystyle F(x)}$ when ${\displaystyle n}$ is large. Since ${\displaystyle {\
overline {X}}}$ and ${\displaystyle S^{2}}$ are the mean and variance of a random variable with cdf ${\displaystyle F_{n}(x)}$ it is natural to expect that ${\displaystyle {\overline {X}}}$ and ${\
displaystyle S^{2}}$ can well approximate the mean and variance of ${\displaystyle X}$ .
Convergence in probability
Definition. (Convergence in probability) Let ${\displaystyle Z_{1},Z_{2},\dotsc }$ be a sequence of random variables. The sequence converges in probability to a random variable ${\displaystyle Z}$ ,
if for each ${\displaystyle \varepsilon >0}$ , ${\displaystyle \mathbb {P} (|Z_{n}-Z|>\varepsilon )\to 0}$ as ${\displaystyle n\to \infty }$ . If this is the case, we write this as ${\displaystyle Z_
{n}\;{\overset {p}{\to }}\;Z}$ for simplicity.
• We may compare this definition with the definition of convergence of a deterministic sequence ${\displaystyle (a_{n}:n\in \mathbb {N} }$ ):
${\displaystyle a_{n}\to a}$ as ${\displaystyle n\to \infty }$ if for each ${\displaystyle \varepsilon >0}$ , there exists an integer ${\displaystyle N>0}$ (which is a function of ${\displaystyle
\varepsilon }$ ), such that when ${\displaystyle n\geq N}$ , ${\displaystyle |a_{n}-a|<\varepsilon }$ (surely).
• For comparison, we may rewrite the above definition as
${\displaystyle Z_{n}\;{\overset {p}{\to }}\;Z}$ as ${\displaystyle n\to \infty }$ if for each ${\displaystyle \varepsilon >0}$ , there exists an integer ${\displaystyle N>0}$ (which is a
function of ${\displaystyle \varepsilon }$ ), such that when ${\displaystyle n\geq N}$ , the probability for ${\displaystyle |Z_{n}-Z|<\varepsilon }$ is very close to one (but this event does not
happen surely).
• ${\displaystyle \varepsilon }$ specifies the accuracy of the convergence. If higher accuracy is desired, then ${\displaystyle \varepsilon }$ will be set to be a smaller (positive) value. The
probability in the definition is very close to zero (we say that the convergence with a certain accuracy (depending on the value of ${\displaystyle \varepsilon }$ ) is "achieved" in this case)
when ${\displaystyle n}$ is sufficiently large.
The following theorem, namely weak law of large number, is an important theorem which is related to convergence in probability.
Theorem. (Weak law of large number (Weak LLN)) Let ${\displaystyle X_{1},\dotsc ,X_{n}}$ be a sequence of independent random variables with the same finite mean ${\displaystyle \mu }$ and same finite
variance ${\displaystyle \sigma ^{2}}$ . Then, as ${\displaystyle n\to \infty }$ , ${\displaystyle {\overline {X}}\;{\overset {p}{\to }}\;\mu }$ .
Proof. We use ${\displaystyle S_{n}}$ to denote ${\displaystyle \sum _{i=1}^{n}X_{i}}$ .
By definition, ${\displaystyle {\overline {X}}\;{\overset {p}{\to }}\;\mu }$ as ${\displaystyle n\to \infty }$ is equivalent to ${\displaystyle \mathbb {P} \left(\left|{\frac {S_{n}}{n}}-\mu \right|>
\varepsilon \right)\to 0}$ as ${\displaystyle n\to \infty }$ .
By Chebyshov's inequality, we have {\displaystyle {\begin{aligned}\mathbb {P} \left(\left|{\frac {S_{n}}{n}}-\mu \right|>\epsilon \right)&\leq {\frac {1}{\varepsilon ^{2}}}\mathbb {E} \left[\left({\
frac {S_{n}}{n}}-\mu \right)^{2}\right]\\&={\frac {1}{\varepsilon ^{2}}}\mathbb {E} \left[\left({\frac {S_{n}-n\mu }{\color {darkgreen}n}}\right)^{2}\right]\\&={\frac {1}{{\color {darkgreen}n^{2}}\
varepsilon ^{2}}}\mathbb {E} \left[\left(S_{n}-n\mu \right)^{2}\right]\\&={\frac {1}{n^{2}\varepsilon ^{2}}}\mathbb {E} \left[\left(\sum _{i=1}^{n}X_{i}-\mu \right)^{2}\right]\\&={\frac {1}{n^{2}\
varepsilon ^{2}}}\mathbb {E} \left[\sum _{i=1}^{n}\sum _{j=1}^{n}(X_{i}-\mu )(X_{j}-\mu )\right]\\&={\frac {1}{n^{2}\varepsilon ^{2}}}\left(\mathbb {E} \left[\sum _{i=j=1}^{n}(X_{i}-\mu )^{2}\right]+
\mathbb {E} \left[\sum _{i=1}^{n}\sum _{jeq i,j=1}^{n}(X_{i}-\mu )(X_{j}-\mu )\right]\right)\\\end{aligned}}}
Since ${\displaystyle X_{1},X_{2},\dotsc }$ are independent (and hence functions of them are also independent) and the expectation is multiplicative under independence, {\displaystyle {\begin
{aligned}{\frac {1}{n^{2}\varepsilon ^{2}}}\left(\mathbb {E} \left[\sum _{i=j=1}^{n}(X_{i}-\mu )^{2}\right]+\mathbb {E} \left[\sum _{i=1}^{n}\sum _{jeq i,j=1}^{n}(X_{i}-\mu )(X_{j}-\mu )\right]\
right)&={\frac {1}{n^{2}\varepsilon ^{2}}}\left(\mathbb {E} \left[\sum _{i=j=1}^{n}(X_{i}-\mu )^{2}\right]+\sum _{i=1}^{n}\sum _{jeq i,j=1}^{n}\underbrace {\mathbb {E} [X_{i}-\mu ]} _{=\mu -\mu =0}\
underbrace {\mathbb {E} [X_{j}-\mu ]} _{=\mu -\mu =0}\right)\\&={\frac {1}{n^{2}\varepsilon ^{2}}}\cdot \sum _{i=1}^{n}\underbrace {\mathbb {E} \left[(X_{i}-\mu )^{2}\right]} _{=\sigma ^{2}}\\&={\
frac {n\sigma ^{2}}{n^{2}\varepsilon ^{2}}}\\&={\frac {\sigma ^{2}}{n\varepsilon ^{2}}}\\&\to 0&{\text{as }}n\to \infty .\end{aligned}}} So, the probability ${\displaystyle \mathbb {P} \left(\left|{\
frac {S_{n}}{n}}-\mu \right|>\varepsilon \right)}$ is less than or equal to an expression that tends to be 0 as ${\displaystyle n\to \infty }$ . Since the probability is nonnegative (${\displaystyle
\geq 0}$ ), it follows that the probability also tends to be 0 as ${\displaystyle n\to \infty }$ .
${\displaystyle \Box }$
• There is also strong law of large number, which is related to almost sure convergence (which is stronger than probability convergence, i.e. implies probability convergence).
There are also some properties of convergence in probability that help us to determine a complex expression converges to what thing.
Proposition. (Properties of convergence in probability) If ${\displaystyle X_{n}\;{\overset {p}{\to }}\;X}$ and ${\displaystyle Y_{n}\;{\overset {p}{\to }}\;Y}$ , then
• (linearity) ${\displaystyle aX_{n}+bY_{n}\;{\overset {p}{\to }}\;aX+bY}$ where ${\displaystyle a,b}$ are constants;
• (multiplicativity) ${\displaystyle X_{n}Y_{n}\;{\overset {p}{\to }}\;XY}$ ;
• ${\displaystyle X_{n}/Y_{n}\;{\overset {p}{\to }}\;X/Y}$ given that ${\displaystyle Y_{n}eq 0}$ and ${\displaystyle Yeq 0}$ ;
• (continuous mapping theorem) if ${\displaystyle g}$ is a continuous function, then ${\displaystyle g(X_{n})\;{\overset {p}{\to }}\;g(X)}$ (and ${\displaystyle g(Y_{n})\;{\overset {p}{\to }}\;g
(Y)}$ )
Proof. Brief idea: Assume ${\displaystyle X_{n}\;{\overset {p}{\to }}\;X}$ and ${\displaystyle Y_{n}\;{\overset {p}{\to }}\;Y}$ . Continuous mapping theorem is first proven so that we can use it in
the proof of other properties (the proof is omitted here). Also, it can be shown that ${\displaystyle (X_{n},Y_{n})\;{\overset {p}{\to }}\;(X,Y)}$ (joint convergence in probability, the definition is
similar, except that the random variables become ordered pairs, so the interpretation of "${\displaystyle |Z_{n}-Z|}$ " becomes the distance between the two points in Cartesian coordinate system,
which are represented by the ordered pairs)
After that we define ${\displaystyle g(z_{1},z_{2})=az_{1}+bz_{2}}$ , ${\displaystyle g(z_{1},z_{2})=z_{1}z_{2}}$ , and ${\displaystyle g(z_{1}/z_{2})=z_{1}/z_{2}}$ respectively, where each of these
functions is continuous, and ${\displaystyle a,b}$ are constants. Then, applying the continuous mapping theorem using each of these functions gives us the first three results.
${\displaystyle \Box }$
Convergence in distribution
Definition. (Convergence in distribution) Let ${\displaystyle Z_{1},Z_{2},\dotsc }$ be a sequence of random variables. The sequence converges in distribution to a random variable ${\displaystyle Z}$
if as ${\displaystyle n\to \infty }$ , ${\displaystyle G_{n}(x)\to G(x)}$ for each ${\displaystyle x}$ at which ${\displaystyle G(x)}$ is continuous, where ${\displaystyle G_{n}(x)}$ and ${\
displaystyle G(x)}$ are the cdf of ${\displaystyle Z_{n}}$ and ${\displaystyle Z}$ respectively. If this is the case, we write this as ${\displaystyle Z_{n}\;{\overset {d}{\to }}\;Z}$ for simplicity.
• The requirement for ${\displaystyle G(x)}$ to be continuous is added so that the convergence in distribution still holds even if the convergence of cdf's fails at some points at which ${\
displaystyle G(x)}$ is discontinuous.
• We may alternatively express the definition as ${\displaystyle \lim _{n\to \infty }G_{n}(x)=G(x)}$ which has the same meaning as ${\displaystyle G_{n}(x)\to G(x)}$ as ${\displaystyle n\to \infty
}$ .
• It can be shown that convergence in probability implies convergence in distribution. That is, if ${\displaystyle X_{n}\;{\overset {p}{\to }}\;X}$ , then ${\displaystyle X_{n}\;{\overset {d}{\to
}}\;X}$ , but the converse is true only when the limiting "${\displaystyle X}$ " is a constant, i.e. if ${\displaystyle X_{n}\;{\overset {d}{\to }}\;c}$ , then ${\displaystyle X_{n}\;{\overset
{p}{\to }}\;c}$ where ${\displaystyle c}$ is a constant.
A very important theorem in statistics which is related to convergence in distribution is central limit theorem.
Theorem. (Central limit theorem (CLT)) Let ${\displaystyle X_{1},X_{2},\dotsc }$ be a sequence of independent random variables with the same finite mean ${\displaystyle \mu }$ and variance ${\
displaystyle \sigma ^{2}}$ . Then, as ${\displaystyle n\to \infty }$ , ${\displaystyle {\frac {{\overline {X}}-\mathbb {E} [{\overline {X}}]}{\sqrt {\operatorname {Var} ({\overline {X}})}}}={\frac
{{\sqrt {n}}({\overline {X}}-\mu )}{\sigma }}\;{\overset {d}{\to }}\;Z}$ , in which ${\displaystyle Z}$ follows the standard normal distribution, ${\displaystyle {\mathcal {N}}(0,1)}$ .
There are some properties of convergence in distribution, but they are a bit different from the properties of convergence in probability. These properties are given by Slutsky's theorem, and also
continuous mapping theorem.
Theorem. (Continuous mapping theorem) If ${\displaystyle X_{n}\;{\overset {d}{\to }}\;X}$ , then ${\displaystyle g(X_{n})\;{\overset {d}{\to }}\;g(X)}$ given that ${\displaystyle g}$ is a continuous
Proof. Omitted.
${\displaystyle \Box }$
Theorem. (Slutsky's theorem) If ${\displaystyle X_{n}\;{\overset {d}{\to }}\;X}$ and ${\displaystyle Y_{n}\;{\overset {p}{\to }}\;c}$ where ${\displaystyle c}$ is a constant, then
• ${\displaystyle X_{n}+Y_{n}\;{\overset {d}{\to }}\;X+c}$ ;
• ${\displaystyle X_{n}Y_{n}\;{\overset {d}{\to }}\;cX}$ ;
• ${\displaystyle X_{n}/Y_{n}\;{\overset {d}{\to }}\;X/c}$ given that ${\displaystyle ceq 0}$ .
Proof. Brief idea: Assume ${\displaystyle X_{n}\;{\overset {d}{\to }}\;X}$ and ${\displaystyle Y_{n}\;{\overset {p}{\to }}\;c}$ . Then, it can be shown that ${\displaystyle (X_{n},Y_{n})\;{\overset
{d}{\to }}\;(X,c)}$ (joint convergence in distribution, and the definitions of this is similar, except that the cdf's become joint cdf's of ordered pairs). After that, we define ${\displaystyle g(z_
{1},z_{2})=z_{1}+z_{2}}$ ,${\displaystyle g(z_{1},z_{2})=z_{1}z_{2}}$ , and ${\displaystyle g(z_{1},z_{2})=z_{1}/z_{2}}$ respectively, where each of the functions is continuous, and then applying the
continuous mapping theorem using each of these functions gives us the three desired results.
${\displaystyle \Box }$
• Notice that the assumption mentions that ${\displaystyle Y_{n}\;{\overset {\color {darkgreen}p}{\to }}\;c}$ but not ${\displaystyle Y_{n}\;{\overset {\color {darkgreen}d}{\to }}\;c}$ .
By resampling, we mean creating new samples based on an existing sample. Now, let us consider the following for a general overview of the procedure of resampling.
Suppose ${\displaystyle X_{1},\dotsc ,X_{n}}$ is a random sample from a distribution of a random variable ${\displaystyle X}$ with cdf, ${\displaystyle F(x)}$ . Let ${\displaystyle x_{1},\dotsc ,x_
{n}}$ be a corresponding realization of the random sample ${\displaystyle X_{1},\dotsc ,X_{n}}$ . Based on this realization, we have also a realization of the empirical cdf: ${\displaystyle {\frac
{1}{n}}\sum _{k=1}^{n}\mathbf {1} \{x_{k}\leq x\}}$ ^[3]. Since this is a realization of the empirical cdf, by Glivenko-Cantelli theorem, it is a good estimate of the cdf ${\displaystyle F(x)}$ when
${\displaystyle n}$ is large ^[4]. In other words, if we denote the random variable with the same pdf as that realization of the empirical cdf by ${\displaystyle X^{*}}$ , ${\displaystyle X^{*}}$ and
${\displaystyle X}$ have similar distributions when ${\displaystyle n}$ is large.
Notice that a realization of empirical cdf is a discrete cdf (since the support ${\displaystyle x_{1},\dotsc ,x_{n}}$ is countable). We now draw a random sample (called the bootstrap (or resampling)
random sample) with sample size ${\displaystyle B}$ (called the bootstrap sample size) ${\displaystyle X_{1}^{*},\dotsc ,X_{B}^{*}}$ from the distribution of a random variable ${\displaystyle X^{*}}$
(${\displaystyle X^{*}}$ comes from sampling from ${\displaystyle X}$ , so the behaviour of sampling from ${\displaystyle X^{*}}$ is called resampling).
Then, the relative frequency historgram of ${\displaystyle X_{1}^{*},\dotsc ,X_{B}^{*}}$ should be close to that of the corresponding realization of the empirical pmf of ${\displaystyle X^{*}}$
(found from the realization of the empirical cdf of ${\displaystyle X^{*}}$ ), which is close to pdf ${\displaystyle f(x)}$ of ${\displaystyle X}$ . This means the relative frequency historgram of $
{\displaystyle X_{1}^{*},\dotsc ,X_{B}^{*}}$ is close to the pdf ${\displaystyle f(x)}$ of ${\displaystyle X}$ .
In particular, since the cdf of ${\displaystyle X^{*}}$ , ${\displaystyle F_{n}(x)}$ , assigns probability ${\displaystyle 1/n}$ to each of ${\displaystyle X_{1}^{*},\dotsc ,X_{B}^{*}}$ ^[5], the pmf
of ${\displaystyle X^{*}}$ is ${\displaystyle \mathbb {P} (X^{*}=x_{i})={\frac {1}{n}},\quad i=1,2,\dotsc ,n}$ . Notice that this pmf is quite simple, and therefore it can make the related
calculation about it simpler. For example, in the following, we want to know the distribution of ${\displaystyle T^{*}=g(X_{1}^{*},\dotsc ,X_{n}^{*})}$ , and this simple pmf can make the resulting
distribution also quite simple.
Remark. For the things involved in the bootstrap method ("bootstrapped" things), there is usually an additional "*" in each of their notations.
In the following, we will discuss an application of the bootstrap method (or resampling) mentioned above, namely using bootstrap method to approximate the distribution of a statistic ${\displaystyle
T=g(X_{1},X_{2},\dotsc ,X_{n})}$ (the inputs of the functions are random variables and ${\displaystyle g}$ is a function). The reason for approximating, rather than finding the distribution exactly,
is that the latter is usually infeasible (or may be too complicated).
To do this, consider the "bootstrapped statistic" ${\displaystyle T^{*}=g(X_{1}^{*},X_{2}^{*},\dotsc ,X_{n}^{*})}$ and the statistic ${\displaystyle T=g(X_{1},X_{2},\dotsc ,X_{n})}$ . ${\displaystyle
X_{1}^{*},X_{2}^{*},\dotsc ,X_{n}^{*}}$ is the bootstrap random sample (with bootstrap sample size ${\displaystyle n}$ ) from the distribution of ${\displaystyle X^{*}}$ and ${\displaystyle X_{1},X_
{2},\dotsc ,X_{n}}$ is the random sample from the distribution of ${\displaystyle X^{*}}$ . When ${\displaystyle n}$ is large, since the distribution of ${\displaystyle X^{*}}$ is similar to that of
${\displaystyle X}$ , the bootstrap random sample ${\displaystyle X_{1}^{*},X_{2}^{*},\dotsc ,X_{B}^{*}}$ and the random sample ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ are also similar. It
follows that ${\displaystyle T^{*}}$ and ${\displaystyle T}$ are similar as well, or to be more precise, the distributions of ${\displaystyle T^{*}}$ and ${\displaystyle T}$ are close. As a result,
we can utilize the distribution of ${\displaystyle T^{*}}$ (which is easier to find and simpler, since the pmf of ${\displaystyle X^{*}}$ is simple as in above) to approximate the distribution of ${\
displaystyle T}$ . A procedure to do this is as follows:
1. Generate a bootstrapped realization ${\displaystyle x_{1}^{*},x_{2}^{*},\dotsc ,x_{n}^{*}}$ from the bootstrap random sample ${\displaystyle X_{1}^{*},X_{2}^{*},\dotsc ,X_{n}^{*}}$ , which is
from the distribution of ${\displaystyle X^{*}}$ .
2. Calculate a realization of the bootstrapped statistic ${\displaystyle T^{*}}$ , ${\displaystyle t^{*}=g(x_{1}^{*},x_{2}^{*},\dotsc ,x_{n}^{*})}$ .
3. Repeat 1. to 2. ${\displaystyle j}$ times to get a sequence of ${\displaystyle j}$ realizations of ${\displaystyle T^{*}}$ : ${\displaystyle t_{1}^{*},t_{2}^{*},\dotsc ,t_{j}^{*}}$ .
4. Plot the relative frequency historgram of the ${\displaystyle j}$ realizations ${\displaystyle t_{1}^{*},t_{2}^{*},\dotsc ,t_{j}^{*}}$ .
This histogram of the ${\displaystyle j}$ realizations (which are a realization of a random sample from ${\displaystyle T^{*}}$ with sample size ${\displaystyle j}$ ) is close to the pmf of ${\
displaystyle T^{*}}$ ^[6], and thus close to the pmf of ${\displaystyle T}$ .
1. ↑ Intuitively, given a candidate for the maximum, we can always add "a little bit" to it to get a greater candidate. So, there is no "greatest" element in the set.
2. ↑ This is because ${\displaystyle X_{\text{min}}=c_{0}}$ and ${\displaystyle X_{\text{max}}=c_{i}}$ .
3. ↑ This is different from the empirical cdf ${\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\mathbf {1} \{X_{k}\leq x\}}$ .
4. ↑ For Glivenko-Cantelli theorem, the empirical cdf is a good estimate of the cdf ${\displaystyle F(x)}$ , regardless of what the actual values (realization) of the random sample are, i.e. for
each realization of the empirical cdf, it is a good estimate of the cdf ${\displaystyle F(x)}$ , when ${\displaystyle n}$ is large.
5. ↑ That is, for a realization of random sample ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ , say ${\displaystyle x_{1},x_{2},\dotsc ,x_{n}}$ , the probability for ${\displaystyle X^{*}}$ to equal
${\displaystyle x_{1},x_{2},\dotsc ,x_{n}}$ (which corresponds to the realization of ${\displaystyle X_{1},X_{2},\dotsc ,X_{n}}$ respectively), is ${\displaystyle 1/n}$ each.
6. ↑ The reason is mentioned similarly above: the histogram should be close to the pmf of ${\displaystyle T^{*}}$ since the cdf corresponding to the histogram (i.e. the realization of the empirical
cdf of the random sample ${\displaystyle T_{1}^{*},T_{2}^{*},\dotsc ,T_{j}^{*}}$ ) is close to the cdf of ${\displaystyle T^{*}}$ | {"url":"https://en.m.wikibooks.org/wiki/Statistics/Preliminaries","timestamp":"2024-11-06T05:59:06Z","content_type":"text/html","content_length":"817485","record_id":"<urn:uuid:7352572b-93ba-467a-9634-fc8c43ecd094>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00546.warc.gz"} |
python – Converting Hex to RGB value in Python
Sure! Here is an example of Python code that converts a hexadecimal color value to its corresponding RGB value:
def hex_to_rgb(hex_value):
hex_value = hex_value.lstrip('#') # Remove '#' character if present
r = int(hex_value[0:2], 16) # Extract red component
g = int(hex_value[2:4], 16) # Extract green component
b = int(hex_value[4:6], 16) # Extract blue component
return r, g, b
# Examples
hex_values = ['#FF0000', '#00FF00', '#0000FF', '#FFFFFF', '#000000', '#123456', '#ABCDEF', '#AABBCC']
for hex_value in hex_values:
rgb_value = hex_to_rgb(hex_value)
print(f'{hex_value} -> RGB{rgb_value}')
Explanation step by step:
1. The `hex_to_rgb` function takes a hexadecimal color value as input and returns the corresponding RGB value as a tuple.
2. `hex_value.lstrip(‘#’)` removes the ‘#’ character from the input if it exists.
3. `int(hex_value[0:2], 16)` extracts the red component by converting the first two characters of the hexadecimal value to an integer using base 16 (hexadecimal).
4. `int(hex_value[2:4], 16)` extracts the green component by converting the third and fourth characters of the hexadecimal value to an integer using base 16.
5. `int(hex_value[4:6], 16)` extracts the blue component by converting the fifth and sixth characters of the hexadecimal value to an integer using base 16.
6. The `return` statement returns the RGB values as a tuple.
7. The `hex_values` list holds the example hexadecimal color values.
8. The for loop iterates over each hexadecimal value in `hex_values`.
9. `hex_to_rgb(hex_value)` is called to convert each hexadecimal value to its corresponding RGB value.
10. The result is printed using f-string formatting: `print(f'{hex_value} -> RGB{rgb_value}’)`.
Here are the results for the provided examples:
#FF0000 -> RGB(255, 0, 0)
#00FF00 -> RGB(0, 255, 0)
#0000FF -> RGB(0, 0, 255)
#FFFFFF -> RGB(255, 255, 255)
#000000 -> RGB(0, 0, 0)
#123456 -> RGB(18, 52, 86)
#ABCDEF -> RGB(171, 205, 239)
#AABBCC -> RGB(170, 187, 204)
I hope this helps! Let me know if you have any further questions. | {"url":"https://pythonkb.com/python-converting-hex-to-rgb-value-in-python/","timestamp":"2024-11-07T07:07:26Z","content_type":"text/html","content_length":"71487","record_id":"<urn:uuid:1a3c6383-5aa0-4283-a55d-38691dd957ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00865.warc.gz"} |
The equation(s) of the medians of the triangle formed by the po... | Filo
The equation(s) of the medians of the triangle formed by the points and is/are :
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Question Text The equation(s) of the medians of the triangle formed by the points and is/are :
Updated On Feb 9, 2023
Topic Straight Lines
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 2
Upvotes 386
Avg. Video Duration 2 min | {"url":"https://askfilo.com/math-question-answers/the-equations-of-the-medians-of-the-triangle-formed-by-the-points-4832-and-5-6-182555","timestamp":"2024-11-10T22:38:34Z","content_type":"text/html","content_length":"324305","record_id":"<urn:uuid:185e4102-6674-4ca5-b017-74e19c379734>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00393.warc.gz"} |
Identification in closed loop · ControlSystemIdentification Documentation
This example will investigate how different identification algorithms perform on closed-loop data, i.e., when the input to the system is produced by a controller using output feedback.
We will consider a very simple system $G(z) = \dfrac{1}{z - 0.9}$ with colored output noise and various inputs formed by output feedback $u = -Ly + r(t)$, where $r$ will vary between the experiments.
It is well known that in the absence of $r$ and with a simple regulator, identifiability is poor, indeed, if
$$\[y_{k+1} = a y_k + b u_k, \quad u_k = L y_k\]$$
we get the closed-loop system
$$\[y_{k+1} = (a + bL)y_k\]$$
where we can not distinguish $a$ and $b$. The introduction of $r$ resolves this
\[\begin{aligned} y_{k+1} &= a y_k + b u_k \\ u_k &= L y_k + r \\ y_{k+1} &= (a + bL)y_k + b r_k \end{aligned}\]
The very first experiment below will illustrate the problem when there is no excitation through $r$.
We start by defining a model of the true system, a function that simulates some data and adds colored output noise, as well as a function that estimates three different models and plots their
frequency responses. We will consider three estimation methods
1. arx, a prediction-error approach based on a least-squares estimate.
2. A subspace-based method subspaceid, known to be biased in the presence of output feedback.
3. The prediction-error method (PEM) newpem
The ARX and PEM methods are theoretically unbiased in the presence of output feedback, see ^[Ljung], while the subspace-based method is not. (Note: the subspace-based method is used to form the
initial guess for the iterative PEM algorithm)
using ControlSystemsBase, ControlSystemIdentification, Plots
G = tf(1, [1, -0.9], 1) # True system
function generate_data(u; T)
E = c2d(tf(1 / 100, [1, 0.01, 0.1]), 1) # Noise model for colored noise
e = lsim(E, randn(1, T)).y # Noise realization
function u_noise(x, t)
y = x .+ e[t] # Add the measured noise to the state to simulate feedback of measurement noise
u(y, t)
res = lsim(G, u_noise, 1:T, x0 = [1])
d = iddata(res)
d.y .+= e # Add the measurement noise to the output that is stored in the data object
function estimate_and_plot(d, nx=1; title, focus=:prediciton)
Gh1 = arx(d, 1, 1)
sys0 = subspaceid(d, nx; focus)
Gh2, _ = ControlSystemIdentification.newpem(d, nx; sys0, focus)
figb = bodeplot(
[G, Gh1, sys0.sys, Gh2.sys];
ticks = :default,
lab = ["True system" "ARX" "Subspace" "PEM"],
plotphase = false,
figd = plot(d)
plot(figb, figd)
estimate_and_plot (generic function with 2 methods)
In the first experiment, we have no reference excitation, with a small amount of data ($T=80$), we get terrible estimates
L = 0.5 # Feedback gain u = -L*x
u = (x, t) -> -L * x
title = "-Lx"
estimate_and_plot(generate_data(u, T=80), title=title*", T=80")
with a larger amount of data $T=8000$, we get equally terrible estimates
estimate_and_plot(generate_data(u, T=8000), title=title*", T=8000")
This indicates that we can not hope to estimate a model if the system is driven by noise only.
We now consider a simple, periodic excitation $r = \sin(t)$
L = 0.5 # Feedback gain u = -L*x
u = (x, t) -> -L * x .+ 5sin(t)
title = "-Lx + 5sin(t)"
estimate_and_plot(generate_data(u, T=80), title=title*", T=80")
In this case, all but the subspace-based method performs quite well
estimate_and_plot(generate_data(u, T=8000), title=title*", T=8000")
More data does not help the subspace method.
With a more complex excitation (random white-spectrum noise), all methods perform well
L = 0.5 # Feedback gain u = -L*x
u = (x, t) -> -L * x .+ 5randn()
title = "-Lx + 5randn()"
estimate_and_plot(generate_data(u, T=80), title=title*", T=80")
and even slightly better with more data.
estimate_and_plot(generate_data(u, T=8000), title=title*", T=8000")
If the feedback is strong but the excitation is weak, the results are rather poor for all methods, it's thus important to have enough energy in the excitation compared to the feedback path.
L = 1 # Feedback gain u = -L*x
u = (x, t) -> -L * x .+ 0.1randn()
title = "-Lx + 0.1randn()"
estimate_and_plot(generate_data(u, T=80), title=title*", T=80")
In this case, we can try to increase the model order of the PEM and subspace-based methods to see if they are able to learn the noise model (which has two poles)
estimate_and_plot(generate_data(u, T=8000), 3, title=title*", T=8000")
learning the noise model can sometimes work reasonably well, but requires more data. You may extract the learned noise model using noise_model.
It is sometimes possible to detect the presence of feedback in a dataset by looking at the cross-correlation between input and output. For a causal system, there shouldn't be any correlation for
negative lags, but feedback literally feeds outputs back to the input, leading to a reverse causality:
L = 0.5 # Feedback gain u = -L*x
u = (x, t) -> -L * x .+ randn.()
title = "-Lx + 5sin(t)"
crosscorplot(generate_data(u, T=500), -5:10, m=:circle)
Here, the plot clearly has significant correlation for both positive and negative lag, indicating the presence of feedback. The controller used here is a static P-controller, leading to a one-step
correlation backwards in time. With a dynamic controller (like a PI controller), the effect would be more significant.
If we remove the feedback, we get
L = 0.0 # no feedback
u = (x, t) -> -L * x .+ randn.()
title = "-Lx + 5sin(t)"
crosscorplot(generate_data(u, T=500), -5:10, m=:circle)
now, the correlation for negative lags and zero lag is mostly non-significant (below the dashed lines).
• LjungLjung, Lennart. "System identification–-Theory for the user", Ch 13. | {"url":"https://baggepinnen.github.io/ControlSystemIdentification.jl/dev/examples/closed_loop_id/","timestamp":"2024-11-05T18:37:17Z","content_type":"text/html","content_length":"17561","record_id":"<urn:uuid:1b317b12-7adc-4296-a20c-761a7e6e4b98>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00733.warc.gz"} |
Interpreting Image Series - Fitting Functions to Time Series
Cloud-Based Remote Sensing with Google Earth Engine
Fundamentals and Applications
Part F4: Interpreting Image Series
One of the paradigm-changing features of Earth Engine is the ability to access decades of imagery without the previous limitation of needing to download all the data to a local disk for processing.
Because remote-sensing data files can be enormous, this used to limit many projects to viewing two or three images from different periods. With Earth Engine, users can access tens or hundreds of
thousands of images to understand the status of places across decades.
Chapter F4.6: Fitting Functions to Time Series
Andréa Puzzi Nicolau, Karen Dyson, Biplov Bhandari, David Saah, Nicholas Clinton
The purpose of this chapter is to establish a foundation for time-series analysis of remotely sensed data, which is typically arranged as an ordered stack of images. You will be introduced to the
concepts of graphing time series, using linear modeling to detrend time series, and fitting harmonic models to time-series data. At the completion of this chapter, you will be able to perform
analysis of multi-temporal data for determining trend and seasonality on a per-pixel basis.
Learning Outcomes
• Graphing satellite imagery values across a time series.
• Quantifying and potentially removing linear trends in time series.
• Fitting linear and harmonic models to individual pixels in time-series data.
Assumes you know how to:
• Import images and image collections, filter, and visualize (Part F1).
• Perform basic image analysis: select bands, compute indices, create masks (Part F2).
• Create a graph using ui.Chart (Chap. F1.3).
• Use normalizedDifference to calculate vegetation indices (Chap. F2.0).
• Write a function and map it over an ImageCollection (Chap. F4.0).
• Mask cloud, cloud shadow, snow/ice, and other undesired pixels (Chap. F4.3).
Github Code link for all tutorials
This code base is collection of codes that are freely available from different authors for google earth engine.
Introduction to Theory
Many natural and man-made phenomena exhibit important annual, interannual, or longer-term trends that recur—that is, they occur at roughly regular intervals. Examples include seasonality in leaf
patterns in deciduous forests and seasonal crop growth patterns. Over time, indices such as the Normalized Difference Vegetation Index (NDVI) will show regular increases (e.g., leaf-on, crop growth)
and decreases (e.g., leaf-off, crop senescence), and typically have a long-term, if noisy, trend such as a gradual increase in NDVI value as an area recovers from a disturbance.
Earth Engine supports the ability to do complex linear and non-linear regressions of values in each pixel of a study area. Simple linear regressions of indices can reveal linear trends that can span
multiple years. Meanwhile, harmonic terms can be used to fit a sine-wave-like curve. Once you have the ability to fit these functions to time series, you can answer many important questions. For
example, you can define vegetation dynamics over multiple time scales, identify phenology and track changes year to year, and identify deviations from the expected patterns (Bradley et al. 2007,
Bullock et al. 2020). There are multiple applications for these analyses. For example, algorithms to detect deviations from the expected pattern can be used to identify disturbance events, including
deforestation and forest degradation (Bullock et al. 2020).
If you have not already done so, be sure to add the book’s code repository to the Code Editor by entering https://code.earthengine.google.com/?accept_repo=projects/gee-edu/book into your browser. The
book’s scripts will then be available in the script manager panel.
Section 1. Multi-Temporal Data in Earth Engine
As explained in Chaps. F4.0 and F4.1, a time series in Earth Engine is typically represented as an ImageCollection. Because of image overlaps, cloud treatments, and filtering choices, an
ImageCollection can have any of the following complex characteristics:
• At each pixel, there might be a distinct number of observations taken from a unique set of dates.
• The size (length) of the time series can vary across pixels.
• Data may be missing in any pixel at any point in the sequence (e.g., due to cloud masking).
The use of multi-temporal data in Earth Engine introduces two mind-bending concepts, which we will describe below.
Per-pixel curve fitting. As you have likely encountered in many settings, a function can be fit through a series of values. In the most familiar example, a function of the form y=mx + b can represent
a linear trend in data of all kinds. Fitting a straight “curve” with linear regression techniques involves estimating m and b for a set of x and y values. In a time series, x typically represents
time, while y values represent observations at specific times. This chapter introduces how to estimate m and b for computed indices through time to model a potential linear trend in a time series. We
then demonstrate how to fit a sinusoidal wave, which is useful for modeling rising and falling values, such as NDVI over a growing season. What can be particularly mind-bending in this setting is the
fact that when Earth Engine is asked to estimate values across a large area, it will fit a function in every pixel of the study area. Each pixel, then, has its own m and b values, determined by the
number of observations in that pixel, the observed values, and the dates for which they were observed.
Higher-dimension band values: array images. That more complex conception of the potential information contained in a single pixel can be represented in a higher-order Earth Engine structure: the
array image. As you will encounter in this lab, it is possible for a single pixel in a single band of a single image to contain more than one value. If you choose to implement an array image, a
single pixel might contain a one-dimensional vector of numbers, perhaps holding the slope and intercept values resulting from a linear regression, for example. Other examples, outside the scope of
this chapter but used in the next chapter, might employ a two-dimensional matrix of values for each pixel within a single band of an image. Higher-order dimensions are available, as well as array
image manipulations borrowed from the world of matrix algebra. Additionally, there are functions to move between the multidimensional array image structure and the more familiar, more easily
displayed, simple Image type. Some of these array image functions were encountered in Chap. F3.1, but with less explanatory context.
First, we will give some very basic notation (Fig. F4.6.1). A scalar pixel at time t is given by pt, and a pixel vector by pt. A variable with a “hat” represents an estimated value: in this context,
p̂t is the estimated pixel value at time t. A time series is a collection of pixel values, usually sorted chronologically: {pt; t =t0...tN}, where t might be in any units, t0 is the smallest, and tN
is the largest such t in the series.
Fig. F4.6.1 Time series representation of pixel p
Section 2. Data Preparation and Preprocessing
The first step in analysis of time-series data is to import data of interest and plot it at an interesting location. We will work with the USGS Landsat 8 Level 2, Collection 2, Tier 1 ImageCollection
and a cloud-masking function (Chap. F4.3), scale the image values, and add variables of interest to the collection as bands. Copy and paste the code below to filter the Landsat 8 collection to a
point of interest over California (variable roi) and specific dates, and to apply the defined function. The variables of interest added by the function are: (1) NDVI (Chap. F2.0), (2) a time variable
that is the difference between the image’s current year and the year 1970 (a start point), and (3) a constant variable with value 1.
///////////////////// Sections 1 & 2 /////////////////////////////
// Define function to mask clouds, scale, and add variables
// (NDVI, time and a constant) to Landsat 8 imagery.
function maskScaleAndAddVariable(image){
// Bit 0 - Fill
// Bit 1 - Dilated Cloud
// Bit 2 - Cirrus
// Bit 3 - Cloud
// Bit 4 - Cloud Shadow
var qaMask=image.select('QA_PIXEL').bitwiseAnd(parseInt('11111',
var saturationMask=image.select('QA_RADSAT').eq(0);
// Apply the scaling factors to the appropriate bands.
var opticalBands=image.select('SR_B.').multiply(0.0000275).add(-
var thermalBands=image.select('ST_B.*').multiply(0.00341802)
// Replace the original bands with the scaled ones and apply the masks.
var img=image.addBands(opticalBands, null, true)
.addBands(thermalBands, null, true)
var imgScaled=image.addBands(img, null, true);
// Now we start to add variables of interest.
// Compute time in fractional years since the epoch.
var date=ee.Date(image.get('system:time_start'));
var years=date.difference(ee.Date('1970-01-01'), 'year');
// Return the image with the added bands.
return imgScaled
// Add an NDVI band.
.addBands(imgScaled.normalizedDifference(['SR_B5', 'SR_B4'])
// Add a time band.
// Add a constant band.
// Import point of interest over California, USA.
var roi=ee.Geometry.Point([-121.059, 37.9242]);
// Import the USGS Landsat 8 Level 2, Collection 2, Tier 1 image collection),
// filter, mask clouds, scale, and add variables.
var landsat8sr=ee.ImageCollection('LANDSAT/LC08/C02/T1_L2')
.filterDate('2013-01-01', '2018-01-01')
// Set map center over the ROI.
Map.centerObject(roi, 6);
Next, to visualize the NDVI at the point of interest over time, copy and paste the code below to print a chart of the time series (Chap. F1.3) at the location of interest (Fig. F4.6.2).
// Plot a time series of NDVI at a single location.
var landsat8Chart=ui.Chart.image.series(landsat8sr.select('NDVI'), roi)
title: 'Landsat 8 NDVI time series at ROI',
lineWidth: 1,
pointSize: 3,
Fig. F4.6.2 Time series representation of pixel p
We can add a linear trend line to our chart using the trendlines parameters in the setOptions function for image series charts. Copy and paste the code below to print the same chart but with a linear
trend line plotted (Fig. F4.6.3). In the next section, you will learn how to estimate linear trends over time.
// Plot a time series of NDVI with a linear trend line
// at a single location.
var landsat8ChartTL=ui.Chart.image.series(landsat8sr.select('NDVI'), roi)
title: 'Landsat 8 NDVI time series at ROI',
color: 'CC0000'
lineWidth: 1,
pointSize: 3,
Fig. F4.6.3 Time series representation of pixel p with the trend line in red
Now that we have plotted and visualized the data, lots of interesting analyses can be done to the time series by harnessing Earth Engine tools for fitting curves through this data. We will see a
couple of examples in the following sections.
Code Checkpoint F46a. The book’s repository contains a script that shows what your code should look like at this point.
Section 3. Estimating Linear Trend Over Time
Time series datasets may contain not only trends but also seasonality, both of which may need to be removed prior to modeling. Trends and seasonality can result in a varying mean and a varying
variance over time, both of which define a time series as non-stationary. Stationary datasets, on the other hand, have a stable mean and variance, and are therefore much easier to model.
Consider the following linear model, where et is a random error:
pt =β0 + β1t + et (Eq. F4.6.1)
This is the model behind the trend line added to the chart created in the previous section (Fig. F4.6.3). Identifying trends at different scales is a big topic, with many approaches being used (e.g.,
differencing, modeling).
Removing unwanted to uninteresting trends for a given problem is often a first step to understanding complex patterns in time series. There are several approaches to remove trends. Here, we will
remove the linear trend that is evident in the data shown in Fig. F4.6.3 using Earth Engine’s built-in tools for regression modeling. This approach is a useful, straightforward way to detrend data in
time series (Shumway and Stoffer 2019). Here, the goal is to discover the values of the β’s in Eq. F4.6.1 for each pixel.
Copy and paste code below into the Code Editor, adding it to the end of the script from the previous section. Running this code will fit this trend model to the Landsat-based NDVI series using
ordinary least squares, using the linearRegression reducer (Chap. F3.0).
///////////////////// Section 3 /////////////////////////////
// List of the independent variable names
var independents=ee.List(['constant', 't']);
// Name of the dependent variable.
var dependent=ee.String('NDVI');
// Compute a linear trend. This will have two bands: 'residuals' and
// a 2x1 (Array Image) band called 'coefficients'.
// (Columns are for dependent variables)
var trend=landsat8sr.select(independents.add(dependent))
.reduce(ee.Reducer.linearRegression(independents.length(), 1));
Map.addLayer(trend,{}, 'trend array image');
// Flatten the coefficients into a 2-band image.
var coefficients=trend.select('coefficients')
// Get rid of extra dimensions and convert back to a regular image
Map.addLayer(coefficients,{}, 'coefficients image');
If you click over a point using the Inspector tab, you will see the pixel values for the array image (coefficients “t” and “constant”, and residuals) and two-band image (coefficients “t” and
“constant”) (Fig. F4.6.4).
Fig. F4.6.4 Pixel values of array image and coefficients image
Now, copy and paste the code below to use the model to detrend the original NDVI time series and plot the time series chart with the trendlines parameter (Fig. F4.6.5).
// Compute a detrended series.
var detrended=landsat8sr.map(function(image){
return image.select(dependent).subtract(
.copyProperties(image, ['system:time_start']);
// Plot the detrended results.
var detrendedChart=ui.Chart.image.series(detrended, roi, null, 30)
title: 'Detrended Landsat time series at ROI',
lineWidth: 1,
pointSize: 3,
color: 'CC0000'
Fig. F4.6.5 Detrended NDVI time series
Code Checkpoint F46b. The book’s repository contains a script that shows what your code should look like at this point.
Section 4. Estimating Seasonality with a Harmonic Model
A linear trend is one of several possible types of trends in time series. Time series can also present harmonic trends, in which a value goes up and down in a predictable wave pattern. These are of
particular interest and usefulness in the natural world, where harmonic changes in greenness of deciduous vegetation can occur across the spring, summer, and autumn. Now we will return to the initial
time series (landsat8sr) of Fig. F4.6.2 and fit a harmonic pattern through the data. Consider the following harmonic model, where A is amplitude, ω is frequency, φ is phase, and et is a random error.
pt =β0 + β1t + Acos(2πωt - φ) + et
=β0 + β1t + β2cos(2πωt) + β3sin(2πωt) + et (Eq. F4.6.2)
Note that β2 =Acos(φ) and β3 =Asin(φ), implying A =(β22 + β32)½ and φ =atan(β3/β2) (as described in Shumway and Stoffer 2019). To fit this model to an annual time series, set ω =1 (one cycle per
year) and use ordinary least squares regression.
The setup for fitting the model is to first add the harmonic variables (the third and fourth terms of Eq. F4.6.2) to the ImageCollection. Then, fit the model as with the linear trend, using the
linearRegression reducer, which will yield a 4 x 1 array image.
///////////////////// Section 4 /////////////////////////////
// Use these independent variables in the harmonic regression.
var harmonicIndependents=ee.List(['constant', 't', 'cos', 'sin']);
// Add harmonic terms as new image bands.
var harmonicLandsat=landsat8sr.map(function(image){
var timeRadians=image.select('t').multiply(2 * Math.PI);
return image
// Fit the model.
var harmonicTrend=harmonicLandsat
// The output of this reducer is a 4x1 array image.
Now, copy and paste the code below to plug the coefficients into Eq. F4.6.2 in order to get a time series of fitted values and plot the harmonic model time series (Fig. F4.6.6).
// Turn the array image into a multi-band image of coefficients.
var harmonicTrendCoefficients=harmonicTrend.select('coefficients')
// Compute fitted values.
var fittedHarmonic=harmonicLandsat.map(function(image){
return image.addBands(
// Plot the fitted model and the original data at the ROI.
fittedHarmonic.select(['fitted', 'NDVI']), roi, ee.Reducer
.mean(), 30)
.setSeriesNames(['NDVI', 'fitted'])
title: 'Harmonic model: original and fitted values',
lineWidth: 1,
pointSize: 3,
Fig. F4.6.6 Harmonic model of NDVI time series
Returning to the mind-bending nature of curve-fitting, it is worth remembering that the harmonic waves seen in Fig. F4.6.6 are the fit of the data to a single point across the image. Next, we will
map the outcomes of millions of these fits, pixel by pixel, across the entire study area.
We’ll compute and map the phase and amplitude of the estimated harmonic model for each pixel. Phase and amplitude (Fig. F4.6.7) can give us additional information to facilitate remote sensing
applications such as agricultural mapping and land use and land cover monitoring. Agricultural crops with different phenological cycles can be distinguished with phase and amplitude information,
something that perhaps would not be possible with spectral information alone.
Fig. F4.6.7 Example of phase and amplitude in harmonic model
Copy and paste the code below to compute phase and amplitude from the coefficients and add this image to the map (Fig. F4.6.8).
// Compute phase and amplitude.
var phase=harmonicTrendCoefficients.select('sin')
// Scale to [0, 1] from radians.
.unitScale(-Math.PI, Math.PI);
var amplitude=harmonicTrendCoefficients.select('sin')
// Add a scale factor for visualization.
// Compute the mean NDVI.
var meanNdvi=landsat8sr.select('NDVI').mean();
// Use the HSV to RGB transformation to display phase and amplitude.
var rgb=ee.Image.cat([
phase, // hue
amplitude, // saturation (difference from white)
meanNdvi // value (difference from black)
Map.addLayer(rgb,{}, 'phase (hue), amplitude (sat), ndvi (val)');
Fig. F4.6.8 Phase, amplitude, and NDVI concatenated image
The code uses the HSV to RGB transformation hsvToRgb for visualization purposes (Chap. F3.1). We use this transformation to separate color components from intensity for a better visualization.
Without this transformation, we would visualize a very colorful image that would not look as intuitive as the image with the transformation. With this transformation, phase, amplitude, and mean NDVI
are displayed in terms of hue (color), saturation (difference from white), and value (difference from black), respectively. Therefore, darker pixels are areas with low NDVI. For example, water bodies
will appear as black, since NDVI values are zero or negative. The different colors are distinct phase values, and the saturation of the color refers to the amplitude: whiter colors mean amplitude
closer to zero (e.g., forested areas), and the more vivid the colors, the higher the amplitude (e.g., croplands). Note that if you use the Inspector tool to analyze the values of a pixel, you will
not get values of phase, amplitude, and NDVI, but the transformed values into values of blue, green, and red colors.
Code Checkpoint F46c. The book’s repository contains a script that shows what your code should look like at this point.
Section 5. An Application of Curve Fitting
The rich data about the curve fits can be viewed in a multitude of different ways. Add the code below to your script to produce the view in Fig. 4.6.9. The image will be a close-up of the area around
Modesto, California.
///////////////////// Section 5 /////////////////////////////
// Import point of interest over California, USA.
var roi=ee.Geometry.Point([-121.04, 37.641]);
// Set map center over the ROI.
Map.centerObject(roi, 14);
var trend0D=trend.select('coefficients').arrayProject([0])
var anotherView=ee.Image(harmonicTrendCoefficients.select('sin'))
min: -0.03,
max: 0.03
'Another combination of fit characteristics');
Fig. F4.6.9 Two views of the harmonic fits for NDVI for the Modesto, California area
The upper image in Fig. F4.6.9 is a closer view of Fig. F4.6.8, showing an image that transforms the sine and cosine coefficient values, and incorporates information from the mean NDVI. The lower
image draws the sine and cosine in the red and blue bands, and extracts the slope of the linear trend that you calculated earlier in the chapter, placing that in the green band. The two views of the
fit are similarly structured in their spatial pattern—both show fields to the west and the city to the east. But the pixel-by-pixel variability emphasizes a key point of this chapter: that a fit to
the NDVI data is done independently in each pixel in the image. Using different elements of the fit, these two views, like other combinations of the data you might imagine, can reveal the rich
variability of the landscape around Modesto.
Code Checkpoint F46d. The book’s repository contains a script that shows what your code should look like at this point.
Section 6: Higher-Order Harmonic Models
Harmonic models are not limited to fitting a single wave through a set of points. In some situations, there may be more than one cycle within a given year—for example, when an agricultural field is
double-cropped. Modeling multiple waves within a given year can be done by adding more harmonic terms to Eq. F4.6.2. The code at the following checkpoint allows the fitting of any number of cycles
through a given point.
Code Checkpoint F46e. The book’s repository contains a script to use to begin this section. You will need to start with that script and edit the code to produce the charts in this section.
Beginning with the repository script, changing the value of the harmonics variable will change the complexity of the harmonic curve fit by superimposing more or fewer harmonic waves on each other.
While fitting higher-order functions improves the goodness-of-fit of the model to a given set of data, many of the coefficients may be close to zero at higher numbers or harmonic terms. Fig. F4.6.10
shows the fit through the example point using one, two, and three harmonic curves.
Fig. F4.6.10 Fit with harmonic curves of increasing complexity, fitted for data at a given point
Assignment 1. Fit two NDVI harmonic models for a point close to Manaus, Brazil: one prior to a disturbance event and one after the disturbance event (Fig. F4.6.11). You can start with the code
checkpoint below, which gives you the point coordinates and defines the initial functions needed. The disturbance event happened in mid-December 2014, so set filter dates for the first
ImageCollection to '2013-01-01','2014-12-12', and set the filter dates for the second ImageCollection to '2014-12-13','2019-01-01'. Merge both fitted collections and plot both NDVI and fitted values.
The result should look like Fig. F4.6.12.
Code Checkpoint F46s1. The book’s repository contains a script that shows what your code should look like at this point.
Fig. F4.6.11 Landsat 8 images showing the land cover change at a point in Manaus, Brazil; (left) July, 6, 2014, (right) August 8, 2015
Fig. F4.6.12 Fitted harmonic models before and after disturbance events to a given point in the Brazilian Amazon
What do you notice? Think about how the harmonic model would look if you tried to fit the entire period. In this example, you were given the date of the breakpoint between the two conditions of the
land surface within the time series. State-of-the-art land cover change algorithms work by assessing the difference between the modeled and observed pixel values. These algorithms look for
breakpoints in the model, typically flagging changes after a predefined number of consecutive observations.
Code Checkpoint F46s2. The book’s repository contains a script that shows what your code should look like at this point.
In this chapter, we learned how to graph and fit both linear and harmonic functions to time series of remotely sensed data. These skills underpin important tools such as Continuous Change Detection
and Classification (CCDC, Chap. F4.7) and Continuous Degradation Detection (CODED, Chap. A3.4). These approaches are used by many organizations to detect forest degradation and deforestation (e.g.,
Tang et al. 2019, Bullock et al. 2020). These approaches can also be used to identify crops (Chap. A1.1) with high degrees of accuracy (Ghazaryan et al. 2018).
To review this chapter and make suggestions or note any problems, please go now to bit.ly/EEFA-review. You can find summary statistics from past reviews at bit.ly/EEFA-reviews-stats.
Cloud-Based Remote Sensing with Google Earth Engine. (n.d.). CLOUD-BASED REMOTE SENSING WITH GOOGLE EARTH ENGINE. https://www.eefabook.org/
Cloud-Based Remote Sensing with Google Earth Engine. (2024). In Springer eBooks. https://doi.org/10.1007/978-3-031-26588-4
Bradley BA, Jacob RW, Hermance JF, Mustard JF (2007) A curve fitting procedure to derive inter-annual phenologies from time series of noisy satellite NDVI data. Remote Sens Environ 106:137–145.
Bullock EL, Woodcock CE, Olofsson P (2020) Monitoring tropical forest degradation using spectral unmixing and Landsat time series analysis. Remote Sens Environ 238:110968. https://doi.org/10.1016
Ghazaryan G, Dubovyk O, Löw F, et al (2018) A rule-based approach for crop identification using multi-temporal and multi-sensor phenological metrics. Eur J Remote Sens 51:511–524. https://doi.org
Shumway RH, Stoffer DS (2019) Time Series: A Data Analysis Approach Using R. Chapman and Hall/CRC
Tang X, Bullock EL, Olofsson P, et al (2019) Near real-time monitoring of tropical forest disturbance: New algorithms and assessment framework. Remote Sens Environ 224:202–218. https://doi.org/ | {"url":"https://google-earth-engine.com/Interpreting-Image-Series/Fitting-Functions-to-Time-Series/","timestamp":"2024-11-13T15:21:22Z","content_type":"text/html","content_length":"109493","record_id":"<urn:uuid:5fef4b25-5ce6-4640-9bef-5b7fc6595bf8>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00401.warc.gz"} |
Chaos theory, numbers and more
The entropy of the universe causes the universe to be random leading to chaos but chaos is not random it's transformation it is consciousness evolving to a higher state/level it is evolving to become
an infinite fractal.
What is Chaos Theory?
Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic
laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying
patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization.
Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like
gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain
states, and so on. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. Many natural objects exhibit fractal properties, including landscapes,
clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit complex, chaotic behavior. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and
wisdom. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. By understanding that our ecosystems, our social
systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being.
Principles of Chaos
• The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the
butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial
conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. Who knows what the long-term effects of teaching millions of kids about chaos and
fractals will be?
• Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system.
Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc)
in the World, accurate long-range weather prediction will always remain impossible.
• Order / Disorder Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.
• Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Examples: Two neighboring water molecules
may end up in different parts of the ocean or even in different oceans. A group of helium balloons that launch together will eventually land in drastically different places. Mixing is thorough
because turbulence occurs at all scales. It is also nonlinear: fluids cannot be unmixed.
• Feedback: Systems often become chaotic when there is feedback present. A good example is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or
sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.
• Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over
in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns
are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.
Chaos as a spontaneous breakdown of topological supersymmetry
In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry, which is an intrinsic property of evolution operators of all stochastic and
deterministic (partial) differential equations. This picture of dynamical chaos works not only for deterministic models, but also for models with external noise which is an important generalization
from the physical point of view, since in reality, all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated
with chaotic dynamics (e.g., the butterfly effect) is a consequence of the Goldstone's theorem—in the application to the spontaneous topological supersymmetry breaking.
Quantum chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is:
"What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the
limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a
fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in
classical chaos, which must be the correspondence principle limit of quantum mechanics?
In seeking to address the basic question of quantum chaos, several approaches have been employed:
1. Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.
2. Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian (system).
3. Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
4. Direct application of the correspondence principle.
In 1917 Albert Einstein wrote a paper that was completely ignored for 40 years. In it he raised a question that physicists have only, recently begun asking themseves: What would classical chaos,
which lurks everywhere in our work do to quantum mechanics, the theory describing the atomic and subatomic worlds? The effects of classical chaos, of course, have long been observed - Kepler knew
about the motion of the moon around the earth and Newton complained bitterly about the phenomenon. At the end of the 19th century the American astronomer William Hill demonstrated that the
irregularity is the result entirely of the gravitational pull of the sun. So thereafter, the great French mathematician-astronomer-physicist Henri Poincare surmised that the moon's motion is only
mild case of a congenital disease affecting nearly everything. In the long run Poincare realized, most dynamic systems show no discernible regularity or repetitive pattern. The behavior of even a
simple system can depend so sensitively on its initial conditions that the final outcome is uncertain. At about the time of Poincare's seminal work on classical chaos, Max Planck started another
revolution, which would lead to the modern theory of quantum mechanics. The simple systems that Newton had studied were investigated again, but this time on the atomic scale. The quantum analogue of
the humble pendulum is the laser; the flying cannonballs of the atomic world consist of beams of protons or electrons, and the rotating wheel is the spinning electron (the basis of magnetic tapes).
Even the solar system itself is mirrored in each of the atoms found in the periodic table of the elements. Perhaps the single most outstanding feature of the quantum world is its smooth and wavelike
nature. This feature leads to the question of how chaos makes itself felt when moving from the classical world to the quantum world. How can the extremely irregular character of classical chaos be
reconciled with the smooth and wavelike nature of phenomena on the atomic scale? Does chaos exist in the quantum world'? Preliminary work seems to show that it does. Chaos is found in the
distribution of energy levels of certain atomic systems; it even appears to sneak into the wave patterns associated with those levels. Chaos is also found when electrons scatter from small molecules.
I must emphasize, however, that the term 'quantum chaos' serves more to describe a conundrum than to define a well-posed problem.
Considering the following interpretation of the bigger picture may be helpful in coming to grips with quantum chaos. All our theoretical discussions of mechanics can be somewhat artificially divided
into three compartments [see illustration] although nature recognizes none of these divisions. Elementary classical mechanics falls in the first compartment. This box contains all the nice, clean
systems exhibiting simple and regular behavior, and so I shall call it R, for regular. .Also contained in R is an elaborate mathematical tool called perturbation theory which is used to calculate the
effects of small interactions and extraneous disturbances, such as the influence of the sun on the moon's motion around the earth. With the help of perturbation theory, a large part of physics is
understood nowadays as making relatively mild modifications of regular systems. Reality though, is much more complicated; chaotic systems lie outside the range of perturbation theory and they
constitute the second compartment. Since the first detailed analyses of the systems of the second compartment were done by Poincare, I shall name this box P in his honor. It is stuffed with the
chaotic dynamic systems that are the bread and butter of science. Among these systems are all the fundamental problems of mechanics, starting with three, rather than only two bodies interacting with
one another, such as the earth, moon and sun, or the three atoms in the water molecule, or the three quarks in the proton. Quantum mechanics, as it has been practiced for about 90 years, belongs in
the third compartment, called Q. After the pioneering work of Planck, Einstein and Niels Bohr, quantum mechanics was given its definitive form in four short years, starting in 1924. The seminal work
of Louis de Broglie, Werner Heisenberg, Erwin Schrodinger, Max Born, Wolfgang Pauli and Paul Dirac has stood the test of the laboratory without the slightest lapse. Miraculously. it provides physics
with a mathematical framework that, according to Dirac, has yielded a deep understanding of 'most of physics and all of chemistry" Nevertheless, even though most physicists and chemists have learned
how to solve special probleins in quantum mechanics, they have yet to come to terms with the incredible subtleties of the field. These subtleties are quite separate from the difficult, conceptual
issues having to do with the interpretation of quantum mechanics. The three boxes R (classic, simple sytems), P (classic chaotic systems) and Q (quantum systems) are linked by several connections.
The connection between R and Q is known as Bohr's correspondence principle. The correspondence principle claims, quite reasonably, that classical mechanics must be contained in quantum mechanics in
the limit where objects become much larger than the size of atoms. The main connection between R and P is the Kolmogorov-Arnold-Moser (KAM) theorem. The KAM theorem provides a powerful tool for
calculating how much of the structure of a regular system survives when a small perturbation is introduced, and the theorem can thus identify perturbations that cause a regular system to undergo
chaotic behaviour. Quantum chaos is concerned with establishing the relation between boxes P (chaotic systems) and Q (quantum systems). In establishing this relation, it is useful to introduce a
concept called phase space. Quite amazingly this concept, which is now so widely exploited by experts in the field of dynamic systems, dates back to Newton. The notion of phase space can be found in
Newton's mathematical Principles of Natural Philosophy published in 1687. In the second definition of the first chapter, entitled "Definitions", Newton states (as translated from the original Latin
in 1729): "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly." In modern English this means that for every object there is a quantity,
called momentum, which is the product of the mass and velocity of the object. Newton gives his laws of motion in the second chapter, entitled 'Axioms, or Laws of motion.' The second law says that the
change of motion is proprotional to the motive force impressed. Newton relates the force to the change of momentum (not to the acceleration as most textbooks do).
Momentum is actually one of two quantities that, taken together, yield the complete information about a dynamic system at any instant. The other quantity is simply position, which determines the
strength and direction of the force. Newton's insight into the dual nature of momentum and position was put on firmer ground some 130 years later by two mathematicians, William Rowan Hamilton and
Karl Gustav-Jacob Jacobi. The pairing of momentum and position is no longer viewed in the good old Euclidean space or three dimensions; instead it is viewed in phase space, which has six dimensions,
three dimensions for position and three for momentum. The introduction of phase space was a powerful step from a mathematical point of view, but it represents a serious setback from the standpoint of
human intuition. Who can visualize six dimensions? In some cases fortunately phase space can be reduced to three or even better, two dimensions. Such a reduction is possible in examining the behavior
of a hydrogen atom in a strong magnetic field. The hydrogen atom has long been a highly desirable system because of its simplicity. A lone electron moves around a lone proton. And yet the classical
motion of the electron becomes chaotic when the magnetic field is turned on. How can we claim to understand physics if we cannot explain this basic problem?
Under normal conditions, the electron of a hydrogen atom is tightly bound to the proton. The behavior of the atom is governed by quantum mechanics. The atom is not free to take on any arbitrary
energy, it can take on only discrete, or quantized, energies. At low energies, the allowed values are spread relatively far apart. As the energy of the atom is increased, the atom grows bigger,
because the electron moves farther from the proton, and the allowed energies get closer together. At high enough energies (but not too high, or the atom will be stripped of its electron!), the
allowed energies get very close together into what is effectively a continuum, and it now, becomes fair to apply the rules of classical mechanics. Such a highly excited atom is called a Rydberg atom.
Rydberg atoms inhabit the middle ground between the quantum and the classical worlds, and they are therefore ideal candidates for exploring Bohr's correspondence principle which connects boxes Q
(quantum phenomena) and R (classic phenomenal). If a Rydberg atom could be made to exhibit chaotic behavior in the classical sense, it might provide a clue as to the nature of quantum chaos and
thereby shed light on the middle ground between boxes Q and P (chaotic phenomena). A Rydberg atom exhibits chaotic behaviour in a strong magnetic field, but to see this behavior we must reduce the
dimension of the phase space. 'The first step is to note that the applied magnetic field defines an axis of symmetry through the atom. The motion of the electron takes place effectively in a
two-dimensional plane, and the motion around the axis can be separated out; ornly the distances along the axis and from the axis matter. The symmetry of motion reduces the dimension of the phase
space from six to four. Additional help comes from the fact that no outside force does any work on the electron. As a consequence, the total energy does not change with time. By focusing attention on
a particular value of the energy, one can take a three-dimensional slice-called an energy shell-out of the four-dimensional phase space. The energy shell allows one to watch the twists and turns of
the electron, and one can actually see something resembling a tangled wire sculpture. The resulting picture can be simplffied even further through a simple idea that occurred to Poincare. He
suggested taking a fixed two-dimensional plane (called a Poincare section, or a surface of section) through the energy shell and watching the points at which the trajectory intersects the surface.
The Poincare section reduces the tangled wire sculpture to a sequence of points in an ordinary plane. A Poincare section for a highly excited hydrogen atom in a strong magnetic field is shown on the
opposite page. The regions of the phase space where the points are badly scattered indicate chaotic behavior. Such scattering is a clear symptom of classical chaos, and it allows one to separate
systems into either box P or box R.
What does the Rydberg atom reveal about the relation between boxes P and Q? I have mentioned that one of the trademarks of a quantum mechanical system is its quantized energy levels, and in fact the
energy levels are the first place to look for quantum chaos. Chaos does not make itself felt at any particular energy level, however; rather its presence is seen in the spectrum, or distribution, of
the levels. Perhaps somewhat paradoxically in a nonchaotic quantum system the energy levels are distributed randomly and without correlation, whereas the energy levels of a chaotic quantum system
exhibit strong correlations [see illustration]. The levels of the regular system are often close to one another, because a regular system is composed of smaller subsystems that are completely
decoupled. The energy levels of the chaotic system, however, almost seem to be aware of one another and try to keep a safe distance. A chaotic sytem cannot be decomposed; the motion along one
coordinate axis is always coupled to what happens along the other axis.
The spectrum of a chaotic quantum system was first suggested by Eugene P. Wigner, another early master of quantum mechanics. Wigner observed, as had many others, that nuclear physics does not possess
the safe underpinnings of atomic and molecular physics: the origin of the nuclear force is still not clearly understood. He therefore asked whether the statistical properties of nuclear spectra could
be derived from the assumption that many parameters in the problem have definite, but unknown values. This rather vague starting point allowed him to find the most probable formula for the
distribution. Oriol Bohigas and Marie-Joya Giannoni of the Institute of Nuclear Physics in Orsay France, first pointed out that Wigner's distribution happens io be exactly what is found for the
spectrum of a chaotic dynamic system.
Chaos does not seem to limit itself to the distribution of quantum energy levels, however, it even appears to work its way into the wavelike nature of the quantm world. The position of the electron
in the hydrogen atom is described by a wave pattern. The electron cannot be pinpointed in space; it is a cloudlike smear hovering near the proton. Associated with each allowed energy level is a
stationary state, which is a wave pattern that does not change with time. A stationary state corresponds quite closely to the vibrational pattern of a membrane that is stretched over a rigid frame,
such as a drum. The stationary states of a chaotic system have surprisingly interesting structure, as demonstrated in the early 1980s by Eric Heller of the University of Washington. He and his
students calculated a series of stationary states for a two-dimensional cavity in the shape of a stadium. The corresponding problem in classical mechanics was known to be chaotic, for a typical
trajectory quickly covers most of the available ground quite evenly. Such behavior suggests that the stationary states might also look random, as if they had been designed without rhyme or reason. In
contrast. Heller discovered that most stationary states are concentrated around narrow channels that form simple shapes inside the stadium, and he called these channels "scars" [see illustration].
Similar structure can also be found in the stationary states of a hydrogen atom in a strong magnetic field [see illustration] The smoothness of the quantum wave forms is preserved from point to
point, but when one steps back to view the whole picture, the fingerprint of chaos emerges. It is possible to connect the chaotic signature of the energy spectrum to ordinary classical mechanics. A
clue to the prescription is provided in Einstein's 1917 paper, He examined the phase space of a regular system from box R and described it geometrically as filled with surfaces in the shape of a
donut; the motion of the system corresponds to the trajectory of a point over the surface of a particular donut. The trajectory winds its way around the surface of the donut in a regular manner, but
it does not necessarily close on itself. In Einstein's picture, the application of Bohr's correspondence principle to find the energy levels of the analogous quantum mechanical system is simple. The
only trajectories that can occur in nature are those in which the cross section of the donut encloses an area equal to an integral multiple of Planck's constant, h (2pi times the fundamental quantum
of angular momentum having the units of momentum multiplied by length). It tums out that the integral multiple is precisely the number that specifies the corresponding energy level in the quantum
system. Unfortunately as Einstein clearly saw, his method cannot be applied if the system is chaotic, for the trajectory does not lie on a donut and there is no natural area to enclose an integral
multiple of Planck's constant. A new approach must be sought to explain the distribution of quantum mechanical energy levels in terms of the chaotic orbits of classical mechanics. Which features of
the trajectory of classical mechanics help us to understand quantum chaos? Hill's discussion of the moon's irregular orbit because of the presence of the sun provides a clue. His work represented the
first instance where a particular periodic orbit is found to be at the bottom of a difficult mechanical problem. (A periodic orbit is tike a closed track on which the system is made to run: there are
many of them, although they are isolated and unstable.) Inspiration can also be drawn from Poincare, who emphasized the general importance of periodic orbits. In the begining of his three-volume
work, "The New Methods of Celestial Mechanics" which appeared in 1892, he expresses the belief that periodic orbits "offer the only opening through which we might penetrate into the fortress that has
the reputation of being impregnable." Phase space for a chaotic system can be organized, at least partially around periodic orbits, even though they are sometimes quite difficult to find.
In 1970 I discovered a very general way to extract information about the quantum mechanical spectrum from a complete enumeration of the classical periodic orbits. The mathematics of the approach is
too difficult to delve into here, but the main result of the method is a relatively simple expression called a trace formula. The approach has now been used by a number of investigators, including
Michael V. Berry of the University of Bristol, who has used the formula to derive the statistical properties of the spectrum. I have applied the trace formula to compute the lowest two dozen energy
levels for an electron in a semiconductor lattice, near one of the carefully controlled impurities. (the semicondoctor, of course, is the basis of the marvellous devices on which modern life depends;
because of its impurities, the electrical conductivity of the material is half-way between that of an insulator, such as plastic, and that of a conductor, such as copper.) The trajectory of the
electron can be uniquely characterized by a string of symbols, which has a straightforward interpretation. The string is produced by defining an axis through the semiconductor and simply noting when
the trajectory crosses the axis. A crossing to the "positive" side of the axis gets the symbol +, and a crossing to the 'negative" side gets the symbol -. A trajectory then looks exactly like the
record of a coin toss. Even if the past is known in all detail even if all the crossings have been recorded-the future is still wide open. The sequence of crossings can be chosen arbitrarily. Now, a
periodic orbit consists of a binary sequence that repeats itself; the simplest such sequence is (+ -), the next is (+ -), and so on (Two crossings in a row having the same sign indicate that the
electron has been trapped temporarily.) All periodic orbits are thereby enumerated, and it is possible to calculate an appropriate spectrum with the help of the trace formula. In other words, the
quantum mechanical energy levels are obtained in an approximation that relies on quantities from classical mechanics only. The classical periodic orbits and the quantum mechanical spectrum are
closely bound together through the mathematical process called Fourier analyis. The hidden regularities in one set, and the frequencies with which they show up, are exactly given by the other set.
This idea was used by John B. Delos of the College of William and Mary and Dieter Wintgen of the Max Planck Institute for Nuclear Physics in Heidelberg to interpret the spectrum of the hydrogen atom
m a strong magnetic field. Experimental work on such spectra has been done by Karl H. Welge and his colleagues at the University of Bielefeld, who have excited hydrogen atoms nearly to the point of
ionization where the electron tears itself free of the proton. The energies at which the atoms absorb radiation appear to be quite random [see illustration], but a Fourier analysis converts the
jumble of peaks into a set of well-separated peaks. The important feature here is that each of the well-separated peaks corresponds precisely to one of several standard classical periodic orbits.
Poincare's insistence on the importance of periodic orbits now takes on a new meaning. Not only does the classical organization of phase space depend critically on the classical periodic orbits, but
so too does the understanding of a chaotic quantum spectrum.
So far I have talked only about quantum systems in which an S electron is trapped or spatially confined. Chaotic effects are also present in atomic systems where an electron can roam freelly, as it
does when it is scattered from the atoms in a molecule. Here energy is no longer quantized, and the electron can take on any value, but the effectiveness of the scattering depends on the energy.
Chaos shows up in quantum scattering as variations in the amount of time the electron is temporarily caught inside the molecule during the scattering process. For simplicity the problem can be
examined in two dimensions. To the electron, a molecule consisting of four atoms looks like a small maze. When the electron approaches one of the atoms, it has two choices: it can turn left or right.
Each possible trajectory of the electron through the molecule can be recorded as a series of left and right turns around the atom until the particle finally emerges. All of the trajectories are
unstable: even a minute change in the energy or the initial direction of the approach will cause a large change in the direction in which the electron eventually leaves molecule. The chaos in the
scattering process comes from the fact that the number of trajectories increases rapidly with path length. Only an interpretation From the quantum mechanical point of view gives reasonable results; a
purely classical calculation yields nonsensical results. In quantum mechanics each classical trajectory is used to deftne a little wavelet that finds its way through the molecule. The quantum
mechanical result follows from simply adding up all such wavelets. Recently I have done a calculation of the scattering process for a special case in which the sum of the wavelets is exact An
electron of known momentum hits a and emerges with the same momentum. The arrival time for the electron to reach a fixed monitoring station varies as a function of the momentum and the way in which
it varies is so fascinating about this problem. The arrival time fluctuates over small changes in the momentum but over large changes a chaotic imprint emerges which never settles down to any simple
pattern [see illustration].
A particularly tantalizing aspect of the chaotic scattering process is that it may connect the mysteries of quantum chaos with the mysteries of number theory. The calculation of the time delay leads
straight into what is probably the most enigmatic object in mathematics, Riemann's zeta function. Actually it was first emploed by Leonhard Euler in the middle of the 18th century to show the
existence of an infinite number of prime numbers (integers that cannot be divided by any smaller integer other than one). About a century later Bernhard Riemann, one of the founders of modem
mathematics, employed the function to delve into the distribution of the primes. In his only paper on the subject, he called the function by the Greek letter zeta. The zeta function is a function of
two variables, x and y which exist in the complex plane). To understand the distribution of prime numbers, Riemann needed to know when the zeta function has the value of zero. Without giving a valid
argument, he stated that it is zero only when x is set equal to 1/2. Vast calculations have shown that he was right without exception for the first billion zeros, but no mathematician has come even
close to providing a proof. If Riemann's conjecture is correct, all kinds of interesting properties of prime numbers could be proved. The values of y for which the zeta function is zero form a set of
numbers that is much like the spectrum of energies of an atom. Just as one can study the distribution of energy levels in the spectrum so can one study the distribution of zeros for the zeta
function. Here the prime numbers play the same role as the classical closed orbits of the hydrogen atom in a magnetic field: the primes indicate some of the hidden correlations among the zeros of the
zeta function. In the scattering problem the zeros of the zeta function give the values of the momentum where the time delay changes strongly. The chaos of the Riemann zeta function is particularly
apparent in a theorem that has only recently been proved: the zeta function fits locally any smooth function. The theorem suggests that the function maydescribe all the chaotic behavior a quantum
system can exhibit. If the mathematics of quantum mechanics could be handled more skilfully, many examples of locally smooth, yet globally chaotic, phenomena might be found.
TWICE in 20th-century physics, the notion of unpredictability has shaken scientists' view of the Universe. The first time was the development of quantum mechanics, the theory that describes the
behaviour of matter on an atomic scale. The second came with the classical phenomenon of chaos In both areas unpredictable features changed scientists understanding of matter in ways that were
totally unforeseen. How ironic then, that these two fields, which have something so fundamental in common, should end up as antagonists when combined. For by rights, chaos should not exist at all in
quantum systems- the laws of quantum mechanics actually forbid it. Yet recent experiments seem to show the footprints of quantum chaos in remarkable swirling patterns of atomic disorder. These
intriguing patterns could illuminate one of the darkest corners of modern physics: the twilight zone where the quantum and classical worlds meet. The quantum theory is one of the most successful
theories in modern science.
Developed in the 1920s, it accounts for a vast range of phenomena from the nature of chemical bonds to the behaviour of subatomic particles, making predictions that have been tested to unprecedented
levels of accuracy. But at its core there are troublesome features: Prominent among them is Heisenberg's uncertainty principle-if you know the speed of a quantum particle, for instance, you can never
know its exact location. The notion that some aspects of nature are simply unknowable has caused sleepless nights for more than a few physicists. Chaos is a younger discipline. Although some of its
conceptual elements had already been appreciated by Leibnitz in the 17th century and Poincare in the 19th century, chaos theory did not become fashionable until the 1980s when scientists began to
realize that the phenomenon is widespread in the natural world. It arises when a system is unusually sensitive to its initial conditions so that a small perturbation of the system changes its
subsequent behaviour in a way that grows exponentially with time. Chaos has been observed in, among other things, pendulums, the growth of populations, planetary dynamics, and weather systems.
Probably the most famous example of chaos is the so-called "butterfly effect" in which, in theory, the tiny air disturbance from the flapping of a butterfly's wings can ultimately lead to a dramatic
storm. of course, although both these theories place fundamental limits on what we can know about the world, the unpredictabilities in quantum theory and chaos are different in kind. But the
particular problem with quantum chaos is that in quantum mechanics small perturbations generally only lead to small perturbations in subsequent states. Without the exponential divergence in
evolutionary paths, it is difficult to see how there can be any chaos. This behaviour of quantum systems is often attributed to a special property of the quantlani equations: their linearity.
An everyday example of linearity can be seen in a rubber band. When it is stretched a little the extension is proportional to the force. Nonlinearity steps in when you pull too far and the band
reaches its limit of elasticity. Stretch even further and it snaps. Because nonlinearity is known to be a crucial ingredient in chaotic systems. it is often said that quantum mechanics cannot be
chaotic because it is linear. But according to Michael Berry, a leading theorist in the study of quantum chaos at the University of Bristol, this issue of linearity is a red herring. "This is one of
the biggest misconceptions in the business," he says. Berry's preferred explanation for the difference between what happens in classical and quantum systems as they edge towards chaos is that quantum
uncertainty imposes a fundamental limit on the sharpness of the dynamics. The amount of uncertainty is quantified in Heisenberg's uncertainty principle by a fixed value known as Planck's constant.
"In classical mechanics, objects can move along infinitely many trajectories," says Berry. "This makes it easy to set up complicated dynamics in which an object will never retrace its path - the sort
of beliaviour that leads to chaos. But in quantum mechanics, Planck's constant blurs out the fine detail, smoothing away the chaos."
This raises some interesting questions. What happens if you scale down a classically chaotic system to atomic size? Do you still get chaos or does quantum regularity suddenly prevail? Or does
someting entirely new happen? And why is it that macroscopic systems can be chaotic, given that ultimately everything is made out of atoms and therefore quantum in nature? These questions have been
the subject of intense debate for more than a decade. But now a number of experimental approaches have begun to offer answers.
Scrambled spectra
One of the earliest clues came from investigations of atomic absorption spectra. If an atom absorbs a photon of light it is possible for one of its electrons to be kicked into a higher energy state.
Normally, an atom's energy levels are spaced at mathematically regular intervals, accounted for by an empirical formula given 19th century physicist Johannes Rydberg. If an atom absorbs photons with
different energies, electrons are kicked into different levels, and the result is a nice tidy absorbtion spectrum whose details are characteristic of the chemical element involved. But when the atom
is subjected to a magnetic field the line structure of the spectrum becomes distorted. When the field is sufficiently intense the spectrum becomes so scrambled it looks pretty much random at higher
energies. The phenomenon is easier to understand in classical rather than quantum mechanical terms. Viewed classically, atomic electrons move in orbits around the nucleus rather like planets round
the Sun. A magnetic field, though, introduces an additional force which causes the electrons to swerve from their normal trajectories. It's rather like a stray star encroaching upon the Solar System.
If it got sufficiently close the gravitational pull would at some point become comparable to the pull between the Earth and our sun. At this moment the earth would find itself in a tug-of-war between
the sun and the interloping star. Such a system would very probably be unstable, with the Earth switching critically between orbits around the sun and the other star. The result would be a chaotic
orbit. In the case of excited atoms, for small fields and lower energy states. The electromagnetic swerving is small compared with the electrostatic pull towards the nucleus and the electron
continues to follow a stable orbit. But for strong fields and highly excited states where the electron is on average very much further away from the nucleus, the swerving force becomes comparable to
the inward pull of the nucleus In this situation, according to vclassical predictions, the motion ought to be chaotic. The effect was first studied back in 1969 by two astronomers Garton and Tonkins
of Imperial College, London, who wanted to find out how the spectra of stars would be affected by their powerful magnetic fields. Their experiments on barium atoms produced one of the first
surprisesbecause their resulting spectrum still displayed considerable regularity. A group at the University of Bielefield in Germany repeated the experiments in the 1980s using higher resolution
equipment. Although the randomness was more apparent in their spectra, it was still clear that quantum mechanics was in some strange way superimposing its own order on the chaos.
Quantum billiards
More recently, signs of quantum suppression of chaos have come from anotheianother experimental approach to quantum chaos: quantum billiards. On a conventional billiard table it is quite common for a
player to pot a ball by bouncing the cue ball off the cushion first. In the hands of a skilled player, such shots are often quite repeatable. But if you were to try the saine shot on a rounded,
stadium-shaped table, the results are far less predictible: the slightest change in starting position alters the ball's trajectory drastically. So what you get if you play stadium billiards is chaos.
In 1992 at Boston's Northeastern University, Srinivas Sridhar and colleagues substituted microwaves for billiard balls and a shallow stadium-shaped copper cavity for the table. Sridhar's team then
observed how the microwaves settled down inside the cavity. Although their apparatus is not of atomic proportions (a cavity typically measures several millimetres across) the experiment exploits the
precise similarity between the wave equations of quantum mechanics and the equations of the electromagnetic waves in this two-dimensional situation. If microwaves behaved like billiard balls, you
would not expect to see any regular patterns. The experiments, however, reveal structures known its "scars" that suggest the waves concentrate along particular paths. But where do these paths come
from? One answer is provided by theoretical work carried out back in the 1970s by Martin Gutzwiller of of the IBM Thomas Watson Center in Yorktown Heights near New York. He produced a key formula
that showed how classical chaos might relate to quantum chaos. Basically it indicates that the quantum regularities are related to a very limited range of classical orbits. These orbits are ones that
are periodic in the classical system. If, for example, you placed a ball on the stadium table and hit it along exactly the right path, you could get it to retrace its path after only a few bounces
off the cushions. However, because the system is chaotic these orbits are unstable. You only need a minuscule error and the ball will move off course within a few bounces. So classically you would
not expect to see these orbits stand out. But thanks to the uncertainty in quantum mechanics, which "frizzes" the trajectories of the balls, tiny errors become less significant and the periodic
orbits are reinforced in some strange way so that they predominate. Sridhar's millimetre-sized stadium was a good analogy for quantum behaviour, but would the same effects occur in a truly
quantum-sized system? This question was answered recently by Laurence Eaves from the University of Nottingham, and his colleagues at Nottingham and at Tokyo University. Eaves conducted his game of
quantum billiards inside an elaborate semiconductor "sandwich". He used electrons for balls, and for cushions he used a combination of quantum barriers and magnetic fields. The quantum barriers are
formed by the outer layers of the sandwich, which gives the electrons a couple of straight edges to bounce back and forth between, The other edges of the table are created by the restraining effect
of the magnetic field, which curves the electron motion in a complicated way. As in Sridhar's stadium cavity, the resulting dynamics ought to be chaotic.
Number crunching
To do the exeriments, Eaves needed ultra-intense magnetic fields, so he took his device to the High Magnetic Field Laboratory at the University of Tokyo, which is equipped with some of the most
powerful sources of pulsed magnetic fields in the world. Meanwhile his colleagues in Noitingham, Paul Wilkinson, Mark Fromhold and Fred Sheard, squared up to a heroic series of calculations, deducing
from purely quantum mechanical principles what the results should look like. In a spectacular pape that made the cover of Nature last month, the team produced the first definitive evidence for
quantum scarring, and precisely confirmed the quantum mechanical predictions. Sure enough, the current flowing through the device was predominantly carried by electrons moving in certain 'scarred'
paths. Quantum regularity was lingering in the chaos rather like the smile of the Cheshire cat in Alice's adventures in wonderland. In case these ideas seem academic it is worth noting that quantum
chaos could play an important role in the design of future seniiconductor devices. At the moment, transistor devices on silicon chips are still large enough for the electrons to move through them
diffusively like molecules in a gas. But as chip manufacturers squeeze ever more logic gates onto silicon, says Eaves, in the next is years transistors may become so small that electrons will instead
flow through them more like quantum billiard balls. "At this point, we may well need the principles of quantum chaos to understand how these devices will work," he says. But where does that leave the
problem of how quantum mechanics turns into the classical world on larger scales? One way of looking at the problem is to investigate how a quantum chaos system actually evolves with time. Last
December, Mark Raizen and his colleagues at. the University of Texas managed to do just that, using an experimental version of a quantum kicked rotor. The idea is to couple two oscillating systems to
produce chaos. Imagine pushing a child's swing. If you time your pushes in rhythm with the swing, then it simply rises higher and higher. if you push at a different frequency, the swing will
sometimes be given a boost and sometimes slowed down. if this is done too vigorously, the oscillations become chaotic. In Raizen's quantum version, ultra-cold sodium atoms were subjected to a special
kind of pulsed laser light. The laser beam was bounced between mirrors to set up a short-lived standing wave - a periodic lattice of light that remains motionless in space rather like the acoustic
nodes on a violin string. Depending on their precise location in the standing waves, the sodium atoms are pushed around by the magnetic fields in the lattice. According to classical calculations, the
result is that the atoms should be kicked chaotically along an increasingly energetic random walk. Raizen's results confirmed a long-standing prediction of the quantum theoretical descriptions of
these systems. The atoms did indeed move in a chaotic way to begin with. But after around 100 microseconds (which corresponds to around 50 kicks) the build-up in energy reached a plateau.
Break time
In other words. quantum mechanics does suppress the chaos but only after a certain amount of time known as the 'quantum break time'. This turns out to be the crucial feature that distinguishes
between quantum and classical predictions of chaotic systems. Before the break time, quantum systems are able to mimic the behaviour of classical systems by looking essentially random. But after the
break time, the system simply retraces its path, it is no longer random, but akin to a repeating loop, albeit of considerable complexity. But if this is right, how can classical systems exhibit
chaos? Macroscopic objects such as pendulums and planets are, after all, made out of atoms and are therefore, ultimately, quantum systems. it turns out that classical systems are in fact behaving
exactly like quantum systems. The only difference is that for classical systems, the quantum break times of macroscopic systems are extraordinarily long-far longer than the age of the Universe. If we
could study a classical system for longer than its quantum break time, we would see that the behaviour was not chaotic but quasi-periodic instead. Thus, quantum and classical realities can be
reconciled, with the classical world naturally embedded in a larger quantum reality. Or, as physicist Dan Kleppner of ttie Massachusetts Institute of Technology puts it, "Anything classical mechanics
can do, quantum mechanics can do better". Since much of the experimental work on quantum chaos has agreed with theoretical predictions, it could be tempting to say "So what?". We already knew that
quantum theory was right. Well, research on quantum chaos does hold out the promise of some remarkable discoveries. Berry is excited by what appears to be a deep connection between the problem of
finding the energy levels of a quantum system that is classically chaotic and one of the biggest unsolved mysteries in mathematics: the Riemann hypothesis. This concerns the distribution of prime
numbers. If you choose a number n and ask how many prime numbers there are less than n it turns out that the answer closely approximates the formula: n/log n. The formula is not exact, though:
sometimes it is a little high and sometimes it is a little low. Riemann looked at these deviations and saw that they contained periodicities. Berry likens these to musical harmonies: "The question is
what are the harmonies in the music of the primes? Amazingly, these harmonies or magic numbers behave exactly like the energy levels in quantum systems that classically would be chaotic."
Deep connection
This correspondence emerges from statistical correlations between the spacing of the Riemann numbers and the spacing of the energy levels. Berry and his collaborator Jon Keating used them to show how
techniques in number theory can be applied to problems in quantum chaos and vice versa. In itself such a connection is very tantalising. Although sonictimes described as the Queen of mathematics,
number theory is often thought of as pretty useless, so this deep connection with physics is quite astonishing. Berry is also convinced that there must be a particular chaotic system which when
quantised would have energy levels that exactly duplicate the Riemann numbers. "Finding this system could be the discovery of the century," he says. it would become a model system for describing
chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The
search for this model system could be the holy grail of chaos. Until we cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to
totally new physics. It is a tantalising thought. Out there is a physical structure waiting to be discovered. if we find it, the remarkable experiments that we have recently witnessed in this
discipline would be crowned by an experimental apparatus that could do more than anything to unlock the secrets of quantum chaos.
Prime numbers
Prime 7
The following information is from thejaxis:
The multiversal language of mathematics is literally composed of an endless matrix of possible combinations within numbers and equations. All of these possibilities share connection to the whole, as
is the Law of One.
Using fourth dimensional mathematics (seemingly breaking laws of math although patterns appear), any numerical form can be broken down by using prime 7. This occurs through scale-symmetry of
overlapped digits within a numerical form. (See my highlights for numerical symmetries). Prime 7 accomplishes this by appearing within multiples, 6-digit or less, in every complex numerical form or
Prime 7 acts as a mathematical key in which to use 4D math to begin breaking any number down to one. Like color to light, we have 7 becoming 1. We literally use 7 as a key to the mathematical matrix,
which is funny because major and minor keyes in music are composed of 7 notes.
Light and color
The decimal form of 1/7 = mathematical light refraction, just as one light becomes seven colors. Light and color cannot be defined by length, width, nor depth as why I believe them to be
fourth-dimensional. Matter, as we perceive it in it's physicality is 3D, but all matter has some sort of 4D color which overlaps the 3D shape of the matter.
Padovan-Fibonacci spiral forms dimensions and prime 7
Both the Fibonacci and Padovan Sequence create the same spiral mathematically. The Fibonacci Sequence uses squares, and the Padovan Sequence uses equilateral triangles. I believe these sequences are
giving us insights into how higher dimensions behave and exist.
In 2D, we see a hexagon of 120°angles, or triangles of 60° angles. In 3D, we can see cubic shapes of 90° angles, as well as equilateral pyramids of 60° angles. Depending on our perspective of the
shape(s) these angles can alternate and exist simultaneously.
We can see how the Fibonacci and Padovan sequences are composed of 6-digit or less multiples of 7, using scale-symmetry to overlap these multiples as they exist simultaneously. I believe this is a 4D
expression of geometry using arithemetic, which is supported by the math mimicking symmetry. The 6-digit or less multiples represent the six cardinal directions of 3D geometry based on axes, and
prime 7 is the seventh direction of within as concerned with 4D geometry, where multiple perspectives all exist within one shape.
I believe that the 4th dimension does not continue to grow outwards as the first three dimensions, but rather uses the already created space and fills it with infinite possible perspectives. We
already understand this 4th dimensional nature in our understanding of tesseracts. A tesseract is a cube which is mathematically and geometrically perfectly balanced within the lines and angles of
its outer surrounding cube. The cube uses the first 3Ds to exist, but the tesseract uses the space within the cube to find balance.
When finding the volume of a shape which lies within, we need all measurements of the 3Ds to create the 4D measure of volume or space within. This is what makes the 7th direction of 4D geometry to be
within, towards the center. Higher dimensions are not beyond us, but amongst us as they overlap the 3D matter of our reality. This 7th direction of possible geometry behaves the same as gravity, a 4D
measure that is measured through and within the center.
The breaking of symmetry
The breaking down of numbers by using prime 7 reminds me of quantum atom theory which says the breaking of symmetry forms sacred geometry and now that I think about it this could link into chaos as a
spontaneous breakdown of topological supersymmetry. Prime 7 could therefore be the hexagonal yod/cubistic matrix. 7 appears within multiples, 6-digits or less, this could explain how 7 expresses
itself which is why it forms the 6-fold flower of life.
Prime numbers are like musical chords
This is an interesting quote from this PDF: "Prime numbers are a lot like musical chords, Berry explains. A chord is a combination of notes played simultaneously. Each note is a particular frequency
of sound created by a process of resonance in a physical system, say a saxophone. Put together, notes can make a wide variety of music. In number theory, zeroes of the zeta-function are the notes,
prime numbers are the chords, and theorems are the symphonies."
quasicrystals connection to prime numbers
About a year ago, the theoretical chemist Salvatore Torquato met with the number theorist Matthew de Courcy-Ireland to explain that he had done something highly unorthodox with prime numbers, those
positive integers that are divisible only by 1 and themselves.
A professor of chemistry at Princeton University, Torquato normally studies patterns in the structure of physical systems, such as the arrangement of particles in crystals, colloids and even, in one
of his better-known results, a pack of M&Ms. In his field, a standard way to deduce structure is to diffract X-rays off things. When hit with X-rays, disorderly molecules in liquids or glass scatter
them every which way, creating no discernible pattern. But the symmetrically arranged atoms in a crystal reflect light waves in sync, producing periodic bright spots where reflected waves
constructively interfere. The spacing of these bright spots, known as “Bragg peaks” after the father-and-son crystallographers who pioneered diffraction in the 1910s, reveals the organization of the
scattering objects.
Torquato told de Courcy-Ireland, a final-year graduate student at Princeton who had been recommended by another mathematician, that a year before, on a hunch, he had performed diffraction on
sequences of prime numbers. Hoping to highlight the elusive order in the distribution of the primes, he and his student Ge Zhang had modeled them as a one-dimensional sequence of particles —
essentially, little spheres that can scatter light. In computer experiments, they bounced light off long prime sequences, such as the million-or-so primes starting from 10,000,000,019. (They found
that this “Goldilocks interval” contains enough primes to produce a strong signal without their getting too sparse to reveal an interference pattern.)
It wasn’t clear what kind of pattern would emerge or if there would be one at all. Primes, the indivisible building blocks of all natural numbers, skitter erratically up the number line like the
bounces of a skipping rock, stirring up deep questions in their wake. “They are in many ways pretty hard to tell apart from a random sequence of numbers,” de Courcy-Ireland said. Although
mathematicians have uncovered many rules over the centuries about the primes’ spacings, “it’s very difficult to find any clear pattern, so we just think of them as ‘something like random.’”
But in three new papers — one by Torquato, Zhang and the computational chemist Fausto Martelli that was published in the Journal of Physics A in February, and two others co-authored with de
Courcy-Ireland that have not yet been peer-reviewed — the researchers report that the primes, like crystals and unlike liquids, produce a diffraction pattern.
“What’s beautiful about this is it gives us a crystallographer’s view of what the primes look like,” said Henry Cohn, a mathematician at Microsoft Research New England and the Massachusetts Institute
of Technology.
The resulting pattern of Bragg peaks is not quite like anything seen before, implying that the primes, as a physical system, “are a completely new category of structures,” Torquato said. The
Princeton researchers have dubbed the fractal-like pattern “effective limit-periodicity.”
It consists of a periodic sequence of bright peaks, which reflect the most common spacings of primes: All of them (except 2) are at odd-integer positions on the number line, multiples of two apart.
Those brightest bright peaks are interspersed at regular intervals with less bright peaks, reflecting primes that are separated by multiples of six on the number line. These have dimmer peaks between
them corresponding to farther-apart pairs of primes, and so on in an infinitely dense nesting of Bragg peaks.
Dense Bragg peaks have been seen before, in the diffraction patterns of quasicrystals, those strange materials discovered in the 1980s with symmetric but nonrepeating atomic arrangements. In the
primes’ case, though, distances between peaks are fractions of one another, unlike quasicrystals’ irrationally spaced Bragg peaks. “The primes are actually suggesting a completely different state of
particle positions that are like quasicrystals but are not like quasicrystals,” Torquato said.
According to numerous number theorists interviewed, there’s no reason to expect the Princeton team’s findings to trigger advances in number theory. Most of the relevant mathematics has been seen
before in other guises. Indeed, when Torquato showed his plots and formulas to de Courcy-Ireland last spring (at the suggestion of Cohn), the young mathematician quickly saw that the prime
diffraction pattern “can be explained in terms of almost universally accepted conjectures in number theory.”
It was the first of many meetings between the two at the Institute for Advanced Study in Princeton, N.J., where Torquato was spending a sabbatical. The chemist told de Courcy-Ireland that he could
use his formula to predict the frequency of “twin primes,” which are pairs of primes separated by two, like 17 and 19. The mathematician replied that Torquato could in fact predict all other
separations as well. The formula for the Bragg peaks was mathematically equivalent to the Hardy-Littlewood k-tuple conjecture, a powerful statement made by the English mathematicians Godfrey Hardy
and John Littlewood in 1923 about which “constellations” of primes can exist. One rule forbids three consecutive odd-numbered primes after {3, 5, 7}, since one in the set will always be divisible by
three, as in {7, 9, 11}. This rule illustrates why the second-brightest peaks in the primes’ diffraction pattern come from pairs of primes separated by six, rather than four.
Hardy and Littlewood’s conjecture further specified how often all the allowed prime constellations will occur along the number line. Even the simplest case of Hardy-Littlewood, the “twin primes
conjecture,” although it has seen a burst of modern progress, remains unproved. Because prime diffraction essentially reformulates it, experts say it’s highly unlikely to lead to a proof of
Hardy-Littlewood, or for that matter the famous Riemann hypothesis, an 1859 formula linking the primes’ distribution to the “critical zeros” of the Riemann zeta function.
The findings resonate, however, in a relatively young research area called “aperiodic order,” essentially the study of nonrepeating patterns, which lies at the intersection of crystallography,
dynamical systems, harmonic analysis and discrete geometry, and grew after the discovery of quasicrystals. “Techniques that were originally developed for understanding crystals … became vastly
diversified with the discovery of quasicrystals,” said Marjorie Senechal, a mathematical crystallographer at Smith College. “People began to realize they suddenly had to understand much, much more
than just the simple straightforward periodic diffraction,” she said, “and this has become a whole field, aperiodic order. Uniting this with number theory is just extremely exciting.”
The primes’ pattern resembles a kind of aperiodic order known since at least the 1950s called limit periodicity, “while adding a surprising twist,” Cohn said. In true limit-periodic systems, periodic
spacings are nested in an infinite hierarchy, so that within any interval, the system contains parts of patterns that repeat only in a larger interval. An example is the tessellation of a strange,
multipronged shape called the Taylor-Socolar tile, discovered by the Australian amateur mathematician Joan Taylor in the 1990s, and analyzed in detail with Joshua Socolar of Duke University in 2010.
According to Socolar, computer experiments indicate that limit-periodic phases of matter should be able to form in nature, and calculations suggest such systems might have unusual properties. No one
guessed a connection to the primes. They are “effectively” limit periodic — a new kind of order — because the synchronicities in their spacings only hold statistically across the whole system.
For his part, de Courcy-Ireland wants to better understand the “Goldilocks” scale at which effective limit-periodicity emerges in the primes. In 1976, Patrick Gallagher of Columbia University showed
that the primes’ spacings look random over short intervals; longer strips are needed for their pattern to emerge. In the new diffraction studies, de Courcy-Ireland and his chemist collaborators
analyzed a quantity called an “order metric” that controls the presence of the limit-periodic pattern. “You can identify how long the interval has to be before you start seeing this quantity grow,”
he said. He is intrigued that this same interval length also shows up in a different prime number rule called Maier’s theorem. But it’s too soon to tell whether this thread will lead anywhere.
The main advantage of the prime diffraction pattern, said Jonathan Keating of the University of Bristol, is that “it is evocative” and “makes a connection with different ways of thinking.” But the
esteemed number theorist Andrew Granville of the University of Montreal called Torquato and company’s work “pretentious” and “just a regurgitation of known ideas.”
Torquato isn’t especially concerned about how his work will be perceived by number theorists. He has found a way to glimpse the pattern of the primes. “I actually think it’s stunning,” he said. “It’s
a shock.”
Quasicrystal metamaterials interact with spacetime and light(as I will go over in a future post which will be linked here) and this could link into the rhombic hexeconta geometry of fractal fields
and this could possibly link into Rhombic triacontahedron and this all suggest spacetimes structure is a fractal what fractal? the E8 tetrahedron grid fractal which I have already proved and it shows
that quasicrystal geometry is encoded in these fractals. These crystals squeeze light causing negative mass fields and this could link into negative mass fields and imaginary mass fields.
Big Question About Primes Proved in Small Number Systems
On September 7, two mathematicians posted a proof of a version of one of the most famous open problems in mathematics. The result opens a new front in the study of the “twin primes conjecture,” which
has bedeviled mathematicians for more than a century and has implications for some of the deepest features of arithmetic.
“We’ve been stuck and running out of ideas on the problem for a long time, so it’s automatically exciting when anyone comes up with new insights,” said James Maynard, a mathematician at the
University of Oxford.
The twin primes conjecture concerns pairs of prime numbers with a difference of 2. The numbers 5 and 7 are twin primes. So are 17 and 19. The conjecture predicts that there are infinitely many such
pairs among the counting numbers, or integers. Mathematicians made a burst of progress on the problem in the last decade, but they remain far from solving it.
The new proof, by Will Sawin of Columbia University and Mark Shusterman of the University of Wisconsin, Madison, solves the twin primes conjecture in a smaller but still salient mathematical world.
They prove the conjecture is true in the setting of finite number systems, in which you might only have a handful of numbers to work with.
These number systems are called “finite fields.” Despite their small size, they retain many of the mathematical properties found in the endless integers. Mathematicians try to answer arithmetic
questions over finite fields, and then hope to translate the results to the integers.
“The ultimate dream, which is maybe a bit naive, is if you understand the finite field world well enough, this might shed light on the integer world,” Maynard said.
In addition to proving the twin primes conjecture, Sawin and Shusterman have found an even more sweeping result about the behavior of primes in small number systems. They proved exactly how
frequently twin primes appear over shorter intervals — a result that establishes tremendously precise control over the phenomenon of twin primes. Mathematicians dream of achieving similar results for
the ordinary numbers; they’ll scour the new proof for insights they could apply to primes on the number line.
A New Kind of Prime
The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are
infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want.
Alphonse de Polignac posed the conjecture in its current form in 1849. Mathematicians made little progress on it for the next 160 years. But in 2013 the dam broke, or at least sprung major leaks.
That year Yitang Zhang proved that there are infinitely many prime pairs with a gap of no more than 70 million. Over the next year other mathematicians, including Maynard and Terry Tao, closed the
prime gap considerably. The current state of the art is a proof that there are infinitely many prime pairs with a difference of at most 246.
But progress on the twin primes conjecture has stalled. Mathematicians understand they’ll need a wholly new idea in order to solve the problem completely. Finite number systems are a good place to
look for one.
To construct a finite field, start by extracting a finite subset of numbers from the counting numbers. You could take the first five numbers, for instance (or any prime number’s worth). Rather than
visualizing the numbers along a number line the way we usually do, visualize this new number system around the face of a clock.
Arithmetic then proceeds, as you might intuit it, by wrapping around the clock face. What’s 4 + 3 in the finite number system with five elements? Start at 4, count three spaces around the clock face,
and you’ll arrive at 2. Subtraction, multiplication and division work similarly.
Only there’s a catch. The typical notion of a prime number doesn’t make sense for finite fields. In a finite field, every number is divisible by every other number. For example, 7 isn’t ordinarily
divisible by 3. But in a finite field with five elements, it is. That’s because in this finite field, 7 is the same number as 12 — they both land at 2 on the clock face. So 7 divided by 3 is the same
as 12 divided by 3, and 12 divided by 3 is 4.
Because of this, the twin primes conjecture for finite fields is about prime polynomials — mathematical expressions such as x^2 + 1.
For example, let’s say your finite field contains the numbers 1, 2 and 3. An polynomial in this finite field would have those numbers as coefficients, and a “prime” polynomial would be one that can’t
be factored into smaller polynomials. So x^2 + x + 2 is prime because it cannot be factored, but x^2 − 1 is not prime: It’s the product of (x + 1) and (x − 1).
Once you have the notion of prime polynomials, it’s natural to ask about twin prime polynomials — a pair of polynomials that are both prime and that differ by a fixed gap. For example, the polynomial
x^2 + x + 2 is prime, as is x^2 + 2x + 2. The two differ by the polynomial x (add x to the first to get the second).
The twin primes conjecture for finite fields predicts that there are infinitely many pairs of twin prime polynomials that differ not just by x, but by any gap you want.
Geometries of primes
Finite fields and prime polynomials might seem contrived, of little use in learning about numbers in general. But they’re analogous to a hurricane simulator — a self-contained universe that provides
insights about phenomena in the wider world.
“There is an ancient analogy between integers and polynomials, which allows you to transform problems about integers, which are potentially very difficult, into problems about polynomials, which are
also potentially difficult, but possibly more tractable,” Shusterman said.
Finite fields burst into prominence in the 1940s, when André Weil devised a precise way of translating arithmetic in small number systems to arithmetic in the integers. Weil used this connection to
spectacular effect. He proved arguably the most important problem in mathematics — the Riemann hypothesis — as interpreted in the setting of curves over finite fields (a problem known as the
geometric Riemann hypothesis). That proof, along with a series of additional conjectures that Weil made — the Weil conjectures — established finite fields as a rich landscape for mathematical
Weil’s key insight was that in the setting of finite fields, techniques from geometry can be used with real force to answer questions about numbers. “This is part of the thing that’s special to
finite fields. Many problems you want to solve, you can rephrase them geometrically,” Shusterman said.
To see how geometry arises in such a setting, imagine each polynomial as a point in space. The polynomial’s coefficients serve as the coordinates that define where the polynomial is located. Going
back to our finite field of 1, 2 and 3, the polynomial 2x + 3 would be located at the point (2, 3) in two-dimensional space.
But even the simplest finite field has an infinite number of polynomials. You can construct more elaborate polynomials by increasing the size of the largest exponent, or degree, of the expression. In
our case, the polynomial x^2 − 3x − 1 would be represented by a point in three-dimensional space. The polynomial 3x^7 + 2x^6 + 2x^5 − 2x^4 − 3x^3 + x^2 − 2x + 3 would be represented by a point in
eight-dimensional space.
In the new work, this geometric space represents all polynomials of a given degree for a given finite field. The question then becomes: Is there a way to isolate all the points representing prime
Sawin and Shusterman’s strategy is to divide the space into two parts. One of the parts will have all the points corresponding to polynomials with an even number of factors. The other part will have
all the points corresponding to polynomials with an odd number of factors.
Already this makes the problem simpler. The twin primes conjecture for finite fields concerns polynomials with just one factor (just as a prime number has a single factor — itself). And since 1 is
odd, you can discard the part of the space with the even factors entirely.
The trick is in the dividing. In the case of a two-dimensional object, such as the surface of a sphere, the thing that cuts it in two is a one-dimensional curve, just as the equator cuts the surface
of the Earth in half. A higher-dimensional space can always be cut with an object that has one fewer dimension.
Yet the lower-dimensional shapes that divide the space of polynomials are not nearly as elegant as the equator. They are sketched by a mathematical formula called the Möbius function, which takes a
polynomial as an input and outputs 1 if the polynomial has an even number of prime factors, −1 if it has an odd number of prime factors, and 0 if it has only a repeated factor (the way 16 can be
factored into 2 × 2 × 2 × 2).
The curves drawn by the Möbius function twist and turn wildly, crossing themselves in many places. The places where they cross — called singularities — are especially difficult to analyze (and they
correspond to polynomials with a repeated prime factor).
Sawin and Shusterman’s principal innovation was in finding a precise way to slice the lower-dimensional loops into shorter segments. The segments were easier to study than the complete loops.
Once they cataloged polynomials with an odd number of prime factors — the hardest step — Sawin and Shusterman had to determine which of them were prime, and which were twin primes. To do this, they
applied several formulas that mathematicians use to study primes among the regular numbers.
Sawin and Shusterman used their technique to prove two major results about prime polynomials in certain finite fields.
First, the twin primes conjecture for finite fields is true: There are infinitely many pairs of twin prime polynomials separated by any gap you choose.
Second, and even more consequentially, the work provides a precise count of the number of twin prime polynomials you can expect to find among polynomials of a given degree. It’s analogous to knowing
how many twin primes fall within any sufficiently long interval on the number line — a kind of dream result for mathematicians.
“This is the first work that gives a quantitative analogue of what is expected to be true over the integers, and that is something that really stands out,” said Zeev Rudnick of Tel Aviv University.
“There hasn’t been anything like this until now.”
Sawin and Shusterman’s proof shows how nearly 80 years after André Weil proved the Riemann hypothesis in curves over finite fields, mathematicians are still energetically following his lead.
Mathematicians pursuing the twin primes conjecture will now turn to Sawin and Shusterman’s work and hope that it, too, will provide a deep well of inspiration.
Pascals triangle, Prime numbers, Music and Gas
A musical note travels through space in the form of longitudinal waves, the result of billions of collisions of gas particles. Plichta demonstrates that the mechanism of control of the transmission
of a musical note through space "could only be caused by the sequence of the reciprocal prime numbers". The sequence of the reciprocals of the prime numbers obeys a certain pattern that we find in
fractal geometry. How does fractal geometry relate to the transmission of a musical note in space? Plichta applies probability theory to gas filled space and showing that the collisions of particles
in a gas medium creates a distribution which follows a pattern that conforms to the numbers in Pascal’s triangle. The pattern emerges as a result of billions of collisions of particles in a gas
medium which reduce to yes/no decisions.
Base 10 problem
Everything is based on grids and in our modern world the common grid system is based upon the geometry of the square and this square-based system forms the cartesian coordinate system and this system
works with the numbers 0-9 and higher, this counting system governs everything we order and therefore the way we perceive reality.
This system underlies all of science since we have been using it, this system is great at separating things but as we shift into a quantum paradime we must break out of the box, from the quantum
perceptive the limits of the cartesian coordinate system becomes painfully obvious the cartesian coordinate system is the biggest problem in physics and limits physics, this system dominates all of
maths and physics.
The cartesian coordinate system is very linear and as a result, it governs us and our entire way we think into a box it limits our ability to think holographically and to see the whole organism(The
whole universe as one).
"Physical law should refer primarily to an order of undivided wholeness similar to that indicated by the hologram. Rather than to an order of analysis into separate parts." - David Bohm
This system is more about static systems then our living dynamic organic and interconnected reality. This system is the relationship between our consciousness and spacetime and is deeply embedded in
Pythagorean numerology
The system originates from the Pythagorean numerology which does have some true magick and some aspects of the Pythagorean model(based on the tetractys) are good but the use of the numbers 0-9 keeps
us stuck in a limited way of being because 9 is an ending. The numbers 0-9 are important as archetypes, they are on the tree of life they are eminations from source each number is a ray of creation
they were never meant to be used as a counting system.
reforming base 10 and using base 9
A better system to use is the Chaldean numerology system which uses numbers 0-9 but the counting system only uses 1-8 it is more musical and this system is seen in the octagon, the I Ching and it is
also seen in the 3D tree of life when it forms an octagon when it rotates.
168(1.68) and 1.618(phi)
An accidental discovery when I was looking at numbers in base 9, 1.68 which is Robert Grant's constant, 168 is an important number in Stephen M. Philips work, is 1.61(phi) in base 9!
Chaos theory and numbers in Xen Qabbalah
8(4+4 fold) and 5
When placing the Fibonacci spiral over the 10/12 tree of life(Kathara grid) the spiral goes through the 5th point and curves around a circle contained within a square where the top point of the
square is the 8th point in the 10/12 grid.
8 and 5 both form 13 when added together and the number 13 is important because there are 13 spheres that make up the 3D seed of life which is 12 surrounding one which would be the 12 points of the
10/12 grid and the 13th being the center of the grid AKA the zero-point.
The circle joining the 5 and 8 points lines up with the center circle making up the 2D seed of life which is the hexagonal yod.
There are 15 pathways and 12 sephirot making up the 10/12 tree of life and 15+12=27 so the 10/12 grid corresponds to three to the power of three, the tetractys/seed of life corresponds to three to
the power of two.
These diagrams also show the Krystal Fibonacci spiral(Doubling sequence+Fibonacci sequence) where the spiral is more circular encodes pi therefore encoding the tripling sequence.
The Hunab Ku is a 4 fold geometry which forms the 441 cube matrix another 4 fold geometry. The Hunab Ku is the view of the 3D tree of life from above and corresponds to the 4 walls around Yantras
such as the Sri Yantra which corresponds to the I Ching(64 tetrahedron grid) which corresponds to the Ying Yang of the Hunab Ku. The I Ching also corresponds to binary/doubling sequence which
corresponds to the Kathara grid spiral(Also something random to say 4 Kathara grids can be placed into a cube).
The Chaldean numerology system uses numbers 0-9 but the counting system only uses 1-8 it is more musical and this system is seen in the octagon, the I Ching and it is also seen in the 3D tree of life
when it forms an octagon when it rotates.
Plato's lambda and trinity sequences
The tetrahedron grid fractal which is formed out of the doubling sequence forms the cosmic tree of life in the form of the powers of 3 showing base 3 comes out of base 2. If we place the doubling
sequence and powers of 3 on the outside of a triangle we form Plato's lambda.
Pascals triangle tetractys
The 10 spacetime dimensions can be repersented in the form of the pascals triangle tetractys to form the 4 large spacetime dimensions and the hexagonal yod(Cube) encoding the 12 vibrational
dimensions. Pascals triangle is a membrane that originated from the zero-point field which is beyond the 12/15 vibrational dimensions and is pure energy which is also known as the God worlds.
Plato's lambda tetractys trinity
The Zero-point is formed out of 3 sequences as shown on the diagram and two of them form Plato's lambda.
• The doubling sequence corresponds to the number E which is related to growth and this is therefore related to the growth of light.
• The Fibonacci sequence is related to phi as we already know.
• The tripling/powers of 3 sequence would therefore correspond to pi talked about below.
Trinity Fibonacci expansion of light
If we repeat this fractal to infinity we would have all, infinite, Fibonacci numbers and as shown the Fibonacci numbers repeat and I said they double but how does that work? The amount of each
Fibonacci number is one of the numbers in the doubling sequence which corresponds this infinite fractal to the infinite doubling vesica pices.
multidimensional polygons and tetractys
Pascals triangle and tetractys levels both encode multidimensional polygons which makes sense because multidimensional polygons correspond to numbers and vibrational dimensions.
Plato's lambda
In Timaeus, Plato’s treatise on Pythagorean cosmology, the central character, Timaeus of Locri (possibly a real person), describes how the Demiurge divided the World Soul into harmonic intervals.
Having blended the three ingredients of the World Soul — Sameness, Difference and Existence — into a kind of malleable stuff, the Demiurge took a strip of it and divided its length into portions
measured by the numbers forming two geometrical series of four terms each: 1, 2, 4, 8 and 1, 3, 9, 27, generated by multiplying 1 by 2 and 3 (Fig. 1). This became known as "Plato’s Lambda" because of
its resemblance to Λ, the Greek letter lambda. Then, according to Timaeus:
“he went on to fill up both the double and the triple intervals, cutting off yet more parts from the original mixture and placing them between the terms, so that within each interval there were two
means, the one (harmonic)* exceeding the one extreme and being exceeded by the other by the same fraction of the extremes, the other (arithmetic) exceeding the one extreme by the same number whereby
it was exceeded by the other. These linksgave rise to intervals of 3/2 and 4/3 and 9/8 within the original intervals. And he went on to fill up all the intervals of 4/3 (i.e., fourths) with the
interval of 9/8 (the tone), leaving over in each a fraction. This remaining interval of the fraction had its terms in the numerical proportion of 256 to 243 (semitone). By this time, the mixture from
which he was cutting off these portions was all used up.”
These numbers line but two sides of a tetractys array of ten numbers from whose relative proportions the musicians of ancient Greece worked out the frequencies of the notes of the now defunct
Pythagorean musical scale.
The three numbers missing from the Lambda are shown in red in Figure 2. The sum of the 10 integers is 90 and the sum of the integers 1, 8 & 27 at the corners of the tetractys is 36. The seven
integers at the centre and corners of the grey hexagon shown in Figure 2 with dashed edges add up to 54. Hence, the 36:54 division displayed by the Lambda Tetractys differentiates between its
corners, which correspond to the Supernal Triad of the Tree of Life, and its seven hexagonal yods, which correspond to the seven Sephiroth of Construction. Historians of science and musical scholars
have focussed upon the integers as "number weights" whose significance is that their ratios are the tone ratios of the notes of the Pythagorean musical scale. They paid no attention to the magnitude
of the numbers themselves because, influenced by the connotation given by Plato that their ratios determined the more important tone ratios, they seemed to have no relevance to music or anything
else. The author's research into various sacred geometries and holistic systems indicates that this is untrue. The number 90 is always present in such systems as a defining parameter, as this section
will demonstrate. The Lambda Tetractys has been regarded as nothing more than a heuristic device for generating the tone ratios of musical notes. Instead, it has profound, fundamental meaning, for —
together with its tetrahedral generalisation to be described shortly — it characterizes the very nature of holistic systems. It is the arithmetic counterpart of the holistic pattern that sacred
geometries embody.
Binary-Trinary Fibonacci Platonic Lambda
Regarding the Platonic Lambda in theTimaeus, Plato states that God created the Cosmic Soul using two mathematical strips of 1, 2, 4, 8 and 1, 3, 9, 27. These two strips follow the shape of an
inverted “V” or the “Platonic Lambda” since it resembles the shape of the 11th letter of the Greek alphabet “Lambda”.
• 1,2,4,8-Doubling Sequence.
• 1,3,9,27-Tripling Sequence.
The 1,2,4,8 follows the same 1,2,4,8,7,5 patterning of the doubling circuits in our device created from the 24 reduced Fibonacci numbers.
The 1,3,9,27 tripling sequence also brings to mind the 3,6,9 vector generated by the 24 reduced Fibonacci numbers.
Plato states; “Now God did not make the soul after the body, although we are speaking of them in this order; for having brought them together he would never have allowed that the elder should be
ruled by the younger… First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as
much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and
a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. After this he filled up the double intervals [i.e. between 1, 2, 4, 8] and the triple
[i.e. between 1, 3, 9, 27] cutting off yet other portions from the mixture and placing them in the intervals”
The Platonic Lambda diagram was first attributed to Crantor of Soli (335-275 BC). It is shown in Cornford’s commentary on the Timaeus, as well as references^. but not in the references of Jowett,
Thomas Taylor and the other commentaries. While the even (double) series of 1, 2, 4, 8, and odd (triple) series of 1, 3, 9, 27 are cited often, none of these commentators mention the sum of the two
series adds up to 55 as shown below:
The Soul of the Universe is the sum of the two series (Timaeus 35b):
• Sum of the double interval series (powers of 2) = 2^0 + 2^1 + 2^2 + 2^3 = 1 + 2 + 4 + 8 = 15
• Sum of the triple interval series (powers of 3) = 3^0 + 3^1 + 3^2 + 3^3 = 1 + 3 + 9 + 27 = 40
Sum of the double & triple interval series (Timaeus) = 15 + 40 = 55
The three laws of resonance
The Real Pythagorean Triangle as I have researched it is a Hermetic/Chinese/Essenic resonance code essential to tha practical application of tha harmonic mathematics transmitted to Pythagoras through
tha Essenes who were then still practicing tha musical secrets of Solomon. We are just now (2600 years later) beginning to understand tha possibilities of using this extremely simple resonance
formula using a Pythagorean Kanon which I believed was secretely used in the Pythagorean Ceremonies and possibly many thousands of years before Pythagoras according to recent information concerning
tha great ancient civilizations that predate Greece. Pythagoras’s musical secrets were not discovered by him but recovered by him- connecting tha still living Hermetic secret traditions of his time.
My personal opinion is that his practical music teaching was at tha source of all his cosmic knowledge. Pythagoras was a sort of musical avatar with a knowledge kept so secret that until now we are
finally ready as an evolving civilization to receive this special gift of quantum resonance or practical string theory.
One of tha undisputed laws of Pythagoras was his Law of Silence. Pythagoreans never wrote anything down. His disciples were required to participate in a a 4 year practice of absolute silence before
being considered to be allowed entrance to tha Inner Temple of Mysteries where I believe tha Universal Laws and Harmonic Principles were demonstrated by Pythagoras to his Inner Temple Disciples; not
utilizing the written word but revealed on the musical instrument known as tha Kanon, or Pythagorean Harp, and since they (the Pythagoreans) never wrote anything down, tha Kanon was in effect their
secret living Bible, expressing tha 3/2 Yin/Yang Cosmic Law of 3 limit mathematical acoustic nature which never changes within an ever changing constellation by altercating the many small individual
movable bridges and by also altercating the tension of each string either up or down in frecuencies matching exact Pythagorean harmonics. Tha result is a radiant cosmic sonic creation used in tha
Pythagorean ceremonies. That is to say tha music spontaneously composed and intuitively created on the Kanon always changes (similar to jazz improvisers and world music virtuosos today) yet uniquely
with the 3/2 spiraling Pythagorean scale where tha underlying unchanging principle of harmony are the 3 laws: tha monadic principle of a vibrating string ,tha law of doubling( the octave), and tha
sacred 3/2 interval which uniquely transforms source energy into an infinite spiral at tha quantum and cosmic level of creation.
After a vow of 4 years of silence in order to join Pythagoras’s inner circle I believe his disciples actually could keep a secret hence the confusion in our contemporary theoretical body of
historical music relating to Pythagoras. Perhaps he actually understood and could practice an ancient Hermetic sonic formula relating to a spiritual science which reveals access for humanity to tha
portal of spherical time, as well being a primary catalyst to practice lucid dreaming, galactic and inter-dimensional travel, experiencing life after death, realizing perfect health, abundance, and
divine miracles.
THE REAL SONIC PYTHAGOREAN TRIANGLE IS EQUALATERAL, REPRESENTING THE TRINITY WHICH DESCRIBES UNIVERSAL CREATION IN MUSICAL TERMS. We now may have found tha First Piece of tha Quantum Puzzle: In The
Beginning was tha Word: “Resonance”
The law of one
The Law of One is referred to by tha Pythagoreans as tha Monad. The Chinese referred to it as tha Tao or the Way of Silence whence the Resonance of Creation or Pythagoras’s idea of tha Universal
Monochord comes from. The Physicists now calls it tha Dark Energy which is mathematical proof of what tha Chinese knew thousands of years ago, that tha nothing in space is actually something (actualy
96% of the energy in our universe): in other words tha Tao or Source Field of energy itself. Tha Monad expresses Tha Cosmic All and Everything. It is most effectively represented by tha Monochord , a
one stringed instrument used by tha Pythagoreans which sonically mirrors tha source field of multi/universal creation. In musical scale terminology it represents tha Tonic or Principal Key Note of
tha Harmonic Pythagorean Modes.
The law of two
Tha Law of 2 Yin (Tha Law of Doubling) or in common musical terms Tha Law of tha Octave. This is tha phenomena we have known about throughout tha ages. Any frequency, for example, tha musical note A
at 222hz ,will be lower in pitch and at the same moment be equal to A 444 and therefore also equal to A 888 and so on (doubling in one direction or halving in the other direction) to infinity: The
psychological and cosmological implications to this concept are vast. It implies 2=1 which our solidly accepted real world perceived by our other non-auditory senses does not compute. 2=1 is tha
musical solution to the enigma of quantum mechanics: ”is tha smallest microcosmic reality a wave? or a particle? or both”. Music does show us the real truth, and if only used correctly can reveal to
us that separation is unity. Applying tha Law of the Octave we can access any supersonic frecuency or even minute waves and particles on the infinite spiral of creation by using the formula 1=2=4=8=
16=32=64=128=256 to infinity and applying that theory backwards from whatever sonic or even super/sonic frequency (or forward from a slower than sound frecuency like an electromagnetic field for
example) , until it returns to us becoming an audible listening frequency that we can apply on tha Pythagorean Harp.
The Duad expresses the number 2, The Yin Principle. It represents the Bridge of the Universal Monochord which divides the monadic string in two equal parts, becoming the twin octaves of the original
monadic key note. These twin vibrations demonstrate the harmonic Law of the Octaves, that tha number 2, the Cosmic Doubler and Separator is actually and paradoxically equal to tha number 1. Therefore
tha proportional points on the monochord which demonstrate tha law of the Duad are 1/1 (as previously explained), 2/1, 4/1, 8/1, and inversely 1/2, 1/4, 1/8. Any of these points intuitively chosen by
setting a bridge on selected strings of a Kanon will provide harmonic grounding when creating a constellation. This sonic phenomena relates biologically to tha negative charge (also charged with dark
energy because of its special cosmic relationship with the One ) and also the grounding end of the axis of our bodies toroidal electromagnetic biofield. We can receive that ground straight from the
exact center of the earth and on through our first 3 chakras and settling into our heart chakra which is located in the exact center of our electromagnetic biofield.
Hence The # 1 and the #2 set the stage for tha multitude of creation to begin.
The law of three
Thus enters the Law of 3 (Yang) ... interacting with Yin and initiating tha Infinite Spiral of tha Sonic Rainbow, mirroring creation: pure yin/yang rainbow of resonance gestated by utilizing a
continuous series of perfect 5ths 3/2 and a continuous series of perfect 4ths ( fifths backwards) 2/3 and so on till infinity. It represents the energy axis of our toroidal electro bio-magnetic
field. It is tha place where we receive light from above through our crown chakra and settling in to the heart chakra where we spontaneously resonate our personal and unique song always in tha moment
– Tha Now, never performing a premeditated melody or harmonic sequence.Tha important thing to understand here is that by simply using this resonant source energy formula utilizing a Pythagorean Kanon
we can access a genuine holistic and balanced creative sonic consciousness, in contrast to the intellectual left brain belief system that music education is currently employing. Harmonic practice
could provide the bridge between Science and Spirituality and potentially reactivate all our 12 human DNA strands if we choose to evolve.
One of our most important limitations today is how contemporary civilization has been thoroughly indoctrinated with our current 12 tone equal temperament tuning system which is non harmonic . By non
harmonic I mean in reference to tha common known fact that our tempered scale tones do not coincide with tha natural overtone and undertone series - including and even beyond tha Pythagorean 3 limit.
Our modern equal temperament tuning system fails to purely activate these Pythagorean frequencies or even any of the natural just harmonic series of our clasical sacred tradition. In fact the
continual indoctrination of tonal control in some form or another as applied to occidental music history dates back at least to tha time of Aristotle.
The True Pythagorean Scale generates a thirteen note scale, and since it actually only uses 12 of the 13 notes it in many ways resembles our 12 note chromatic scale used today. It also may be
reconstructed using whole steps (9/8) obtained by tha formula (3/2x3/2)/2 and half steps at (256/243) obtained by tha formula (1x2x2x2x2x2x2x2x2/1x3x3x3x3x3 in order to construct the Rainbow Modes
which resemble our diatonic scale and the 7 Greek modes with 7 notes each. A Pythagorean 13 note scale is created by spiraling 6 perfect 5ths (3/2 up from the fundamental tonic) and 6 perfect fourths
(2/3 down from tha tonic). Tha two end notes of this spiral cycle of 13 notes (with tha tonic in the center) are our near equivalents to tha augmented 4th and the diminished 5th in our tempered scale
which in this case are no longer tempered and equal. That difference describes an interval which is not normally played and is referred to as tha "Pythagorean Comma" which is 23.46 cents (almost 1/
4th of our equal tempered 1/2 step. By theoretically and symbolically continuing this spiral we create a resonant road to infinity. This illusive number which is precisely 531441/524288 should be
celebrated as the perfect geometric portal to infinity and studied as profoundly or even more so than tha numbers of pi and phi, the golden mean ratio and other special harmonic numbers. There are
many secret and special qualities of this magic ratio which I believe will be revealed to us in time. However almost all musical experts past and present have mistakenly claimed that this number is
and always has been an enigma.They insist that the Pythagoreans were terrorized by this comma. I am sure that neither Pythagoras nor his followers were ever terrorized by this incredible fact of
nature. Tha beauty of this comma is hidden in that very mystery. It contains the secret of our resonance underlying biology, electromagnetism and tha quantum field.This Spiral doesn’t close to make
tha circle of fifths as does our equal tempered chromatic scale. Reaching tha13th interval in the cycle of Pythagorean fifths it instead spirals on to infinity just like nature herself. It is tha
formula of creation itself. The numbers 1, 2 and 3 generate the geometric pattern we call the Flower of Life.
To Recapitulate
The Monad, The Duad, and the Triad then express the Trinity, The Active Yang Principle in Action. The Triad symbolically represents The Many strings of a Harmonic Kanon, a multi stringed instrument
utilizing strings of equal thickness and length, and numerous moveable bridges. This was the musical instrument used in secret by the Pythagoreans that I am now attempting to reactivate today. The
Pythagorean Harp is capable of activating all of tha Pythagorean Rainbow Modes, utilizing exponents of the number 3 Yang as it relates proportionally to tha exponents of tha number 2 Yin (the pure
double of the cosmic 1 Monad) thereby creating a spiraling sonic rainbow with an infinite creative capacity which models the cosmic principles of yin and yang. In other words, the Pythagorean Kanon
is a creative harmonic laboratory mirroring cosmic creation. By programming positive emotional intentions the harmonic points of the strings are activated on the Kanon. Our constellation is created
by positioning the movable bridges on selected strings by intuitively choosing any one the harmonic points: 3/2, 3/1, 6/1, 2/3, 1/3, 1/6 for each chosen string.
The natural tendency for expansion of creation
The natural tendency for expansion of creation is represented by this tenfold Pythagorean triangle of which the secret of the music of the trinity is amplified and reveals the potential harmonics of
cosmic creation.
Every diagonal starting from any of the 10 numbers leaning to the left is yin or the law of doubling. Every diagonal starting from any of the 10 numbers leaning to the right is yang or the law of
tripling. Theoretically it continues in a similar fashion on to infinity.
The Pythagorean Three Limit Lambdoma
The sonic harmonic numbers in actuality expand infinitely. This chart includes all the harmonic intervals which apply to universal creation utilizing the formula 2= yin and 3= yang.
Sacred Geometry, Numerology, and Astrology
Sacred Geometry, Numerology, and Astrology are then gestated by these resonant laws. Water is its conduit. Gravity and electromagnetism describe it. The harmonic laws of tha True Pythagorean Scale
represent these creative kanons and when activated on tha Pythagorean Harp transform into a representative model of tha sonic spiral of tha space/time/multi-dimensional continuum.
All laws and harmonic principles were demonstrated by Pythagoras on tha musical instrument known as tha Kanon, or Pythagorean Harp ,and since they (tha Pythagoreans) were to never write anything
down, the Kanon was in effect their secret Living Bible, expressing tha 3/2 Cosmic Laws of mathematical acoustic nature which never change within an ever changing radiant cosmic sonic creation by
always initiating new constellations. In past times and as reactivated in tha present, tha music spontaneously and intuitively created on the Kanon always changes - yet tha underlying principles of
1,2 and 3,that is tha Tao, Yin and Yang never change. It is tha sacred 3/2 interval which uniquely transforms source energy represented by the number 1 into an infinite spiral at the quantum and
cosmic levels of creation.
Robert Grants Platos Lambda and tetractys
72 = 2^3x3^2......Male to the Power of Female MULTIPLIED by Female to the Power of Male; Square to the Power of Circle EXPANDED by Circle to the Power of Square....Past to the Power of Future TIMES
Future to the Power of Past = ETERNAL NOW. Light to the Power of Dark MAGNIFIED by Dark to the Power of Light = The Universe. Divine Balance through the non-judgement and merger of opposites is
expanded awareness, gratitude, empathy and love....When opposites merge each not only complements but EXPONENTIALLY EXPANDS the other. 432hz Harmonic Tuning is based on the exponential powers of the
numbers 2 and 3—in Perfect “Pythagorean JUST TUNING”. This Grid series determines all TIME numbers, these “Super” Numbers are also the dimensional references of all SpaceTime measurements using the
ancient Imperial systems and are fundamental to the scaling of each celestial body within our Solar System. These include the mile diameter measurements for the Sun, Moon, Earth, all planets,
distances between orbits of each as well as the Great Year’s Precession of Equinox. Each is derived as simple ratios of the number 72. Interestingly, the Square Root of the Slope Angle of the Great
Pyramid of Giza is 7.2° (Slope Angle = 51.84°). All of this latest research comes as a result of the recent decryption of the precise proportions of ONE Circle (=π and Trinary) and ONE Square (e and
Binary). (Incidentally, the base length of the Square DaVinci gave us is 7.2 inches while the top length of the Square is 7.071 inches (= 1/Root2) and represents Euler Expansions, while the base
length is 7.2 inches representing Binary Trinary expansions of 72 at the point where the Vitruvian Man has his feet planted firmly on ‘Terra Firma’. 3^2(2^3) = 72, the perfect mirror symmetry of both
Binary and Trinary expansions infinitely. This Squaring can be depicted infinitely in each direction as multiples of 12 and also as a Cube and Cuboctahedron. 72 is also the ONE number representing
the merger of the Hexagon and Pentagon as 2+3=5 versus 2x3=6; and 72° is the Arc Length of a Pentagon; whereas the Sum of Angles of the Hexagon is 720°. Furthermore, 72 years is precisely ONE Degree
of the Earth’s Precessional Cycle (72 x 360° = 25,920 years). Oh, and you may want to Google just how far our Solar System is from the Center of the Milky Way Galactic “Disc”: “Approximately 26,000
Light Years”?.........what do you want to bet it’s actually 25,920 Light Years? Do you think that the Giza Plateau may actually be ‘The World’s Oldest and Largest Clock’, trying to teach us, in an
age of darkness, TIME’s True nature? The Tetractys.....It’s always been about the Harmonic Series.....now we are understanding that it is about the convergence of both Binary (2^n) and Trinary 3^n)
Numbers.....culminating in the formation (and ultimately of the number 72 and by extension all TIME numbers)
Fibonacci, Mandelbrot set and more
The Mandelbrot set is the set of complex numbers and the julia set is basically something similar which looks just like the dragon curve fractal which is like a fractaling curve superstring membrane
and I suggest that the membrane face of the higher-dimensional shape that our universe is on is like this dragon curve fractal. The fractaling membrane curve that our universe is, is shown above in
the pascals triangle, trinity sequences and Plato's lambda diagram.
Logistic map, Mandelbrot/Mandelbuble, packing of spheres and Fibonacci
The Logistic map is basically a representation of the doubling sequence and it is a fractal and whats interesting is the Logistic map is apart of the Mandelbrot Set. The Mandelbrot Set has a
connection to phi which is shown in the picture and the doubling sequence has a connection to the Fibonacci sequence and there are also connections between the Fibonacci sequence and Mandelbrot Set
which is important to understand for the next bit because the Platonic solids are formed out of the Fibonacci sequence.
The Mandelbulb is a 3D manifestation of the Mandelbrot Set and starts as a sphere. It iterates outward from there to form the Mandelbulb in its various complexities. Interestingly, the Platonic
solids can be found in the 3D Mandelbrot set. You can play connect the dots with the bumps on the shape and get the Platonic solids. Nested Platonic solids fractalize out to create the spheres within
spheres (nested spheres) that create the 3D Mandelbulb. This shows how the Mandelbrot Set actually describes a 3D spherical structure. The Platonic solids as close-packed spheres is the big secret of
the Mandelbrot/Mandelbulb fractal. From the side the Mandelbulb looks like a series of close-packed spheres. Going inside the structure you find the fractalized geometry – the tendrils of the
The Mandelbulb looks very similar to the fractaling flat torus, the torus expands out of the tetrahedron grid fractal and if you place a sphere around each tetrahedron you get the flower of life/
packing of spheres which as shown is formed out of the Mandelbrot/Mandelbulb fractal too. The Platonic Solids faces can fractalize out in infinite iterations until they create perfect spheres. The
Platonic Solids are fractal structures and the Platonic solids and the Mandelbrot set are inextricably connected.
The flat torus is constructed out of a square showing another connection between the torus and one of the faces of the flower of life packing of spheres and this face can correspond to a magic square
and the diagram above in Robert Grants work and it can also correspond to The Theory of Everyone diagram which I will go over in a later post.
The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.
A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4
dimensions using quaternions and bicomplex numbers.
Powers of 3 and Pi
As we know time numbers are multidimensional polygons and they divide numbers into 12 and 12 as a unit of time is related to the circle and 360 degrees and 360 is related to the 8 tetractys which
encodes the 64 tetrahedron grid and the 64 tetrahedron grid forms multidimensional polygons in many ways but we will look into its relation to the 8 tetractys:
• 8 tetractys=1+2+3+4+5+6+7+8=36
• The 8 tetractys encodes the 7 tetractys so:
• 8 tetractys+7 tetractys=64=64 tetrahedron grid
• If each point in the 8 tetractys was a yod then the value of the 8 tetractys would be 360
As we know the 64 tetrahedron grid and tetrahedron grid fractal forms the powers of 3 cosmic tree of life which encodes multidimensional polygons.
There are 6 vortexes as explained in Kathara and vortexes but there are actually 3 since two are one, now these 3 vortexes each correspond to one of the trinity sequences and they are all vesica
pices fractals one of them showed in the first diagram under Kathara and vortexes.
The first 5 powers of 3 form 11 and this relates it to 11:33 as shown in the diagram and this relates it to the trinity and as shown in the diagram this trinity forms the circle and pi.
The pentaflower of life is one of the faces of the dodecahedron which makes up the 120 cell(cubistic matrix) these pentaflowers are vortexes and the pentaflower is made up of 5 Fibonacci spirals this
corresponds to the vibrational dimensions which uses 6 Fibonacci spirals.
1133, Pentagram and dimensions
The connection between the number 11. pi, the trinity, and the pentagram is shown above now we will go deeper into this and look more into the trinity and its connection to 11:33, the 11:33 trinity
forms 1331 the last row of pascals triangle tetractys and this line is equal to 8 and above in the picture, the 3rd line forms 7 and when this is added to the 8 it forms the 7:8 correspondence and
forming the 64 tetrahedron grid.
The pentagram is equal to 11 as shown and therefore it is related to Enlightenment. The number 11 also shows a 5:6 correspondence which is also formed out of the 64 tetrahedron grid and the number
1133 through the 7 star tetrahedron tetractys. This also relates to the 3D seed of life which encodes phi through 8(44 or 4+4).
The bottom right image shows the expansion of the squaring of the circle/star tetrahedron(Hexagram) which encodes the Krystal Fibonacci spiral and this expansion forms the tetrahedron grid fractal
and the left bottom image shows the expansion of the unicursal hexagram in the 64 tetrahedron grid and its relation to the Fibonacci numbers and this relates it to the enneagram and vortex maths
which relates to phi as shown and it also relates it to the 10 spacetime dimension/12 vibrational dimension tree of life which grows based on the Krystal Fibonacci spiral which lines up with the
center of the double seed of life.
All numbers are comprised of prime numbers and all numbers(other than numbers in the 3 times table) reduce to 1, 2, 4, 8, 7 and 5 which is equal to the doubling sequence which can then be reduced to
the number 2 since it is the prime that all the numbers in the doubling sequence can be reduced to, the doubling sequence then corresponds to the (fractaling)tetrahedron/tetrahedron grid fractal and
this corresponds it to different cubistic matrix levels which already corresponded it to the doubling sequence and the basic structure of the cubistic matrix is 4 and 2×2=4 is the first composite
number and it corresponds to the faces of the cubistic matrix and this corresponds it to the 4 points of the prime number cross(PNC).
As shown above 7 forms the basic structure of the cubistic matrix(Hexagonal yod) in 2D and 7 is related to 1/7 and 1/7=0.142857 which is 1, 2, 4, 8, 7 and 5, therefore, encoding the doubling sequence
and 1, 2, 4, 8, 7 and 5 also forms the torus as shown in vortex maths and new layers can be added to the torus making it bigger and at 24 layers it can then encode the PNC and 24 repeating Fibonacci
numbers plus as we know the PNC can be turned into a torus. The torus which is the shape of (electro)magnetic fields forms out of the tetrahedron grid fractal, E8 is a torus and is made out of
tetrahedron in 3D so it is apart of the tetrahedron grid fractal so it is the 64 tetrahedron grid. (more information above: Flower of life and electromagnetism)
We have now explained the hexagonal yod as 1/7 and now lets talk about pascals triangle tetractys. As we know the last line of pascals triangle tetractys forms the 64 tetrahedron grid/231 gates/E8
and each line in pascals triangle is equal to a number in the doubling sequence this shows a correspondence between pascals triangle and the tetrahedron grid fractal(This also shows another way the
tetractys corresponds to the tetrahedron grid fractal) and as we know the tetrahedron grid fractal is made up of black holes(Negative volume ones which shows a connection between -1/12(infinite
tetractys) and the infinite tetrahedron grid because this shows proof and that the infinite tetrahedron grid is a negative volume singularity/black hole) and this shows the numbers in pascals
triangle are black holes so pascals triangle is the tetractys and this is also shown in my X and Y equation.
Extra information
• The tetractys/tetragrammaton/pascals triangle/platos lambda which is formed out of binary-trinary is the triangle which encodes all harmonics which are the vibrational dimensions.
Doubling sequence(e) joining with Tripling sequence(pi) is shown above and the pi and e correspondence to the sequences is also shown above and in this post.
Higher-dimensional numbers
Time numbers encode multidimensional polygons and are lower dimensional forms of multidimensional polygon numbers.
Fractaling universe
"The entropy of the universe causes the universe to be random leading to chaos but chaos is not random it's transformation it is consciousness evolving to a higher state/level it is evolving to
become an infinite fractal."
The entropy leads consciousness to evolve into more complex structures leading to more advance consciousness and the most advance consciousness in the universe is actually the universe because its
structure is like a brain.
• en.wikipedia.org/wiki/Chaos_theory
• en.wikipedia.org/wiki/Quantum_chaos
• empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/quantumchaos.html
• fractalfoundation.org/resources/what-is-chaos-theory/
• www.quantamagazine.org/a-chemist-shines-light-on-a-surprising-prime-number-pattern-20180514/
• www.quantamagazine.org/big-question-about-primes-proved-in-small-number-systems-20190926/
• vismath5.tripod.com/metz/
• Stephen M. Phillips
• tombedlamscabinetofcuriosities.wordpress.com
• thakanon.org/pythagorean-music-theory.html
• en.wikipedia.org/wiki/Mandelbulb | {"url":"https://xenqabbalah.fandom.com/wiki/User_blog:Dimensional_consciousness/Chaos_theory,_numbers_and_more","timestamp":"2024-11-04T08:18:57Z","content_type":"text/html","content_length":"397055","record_id":"<urn:uuid:25451249-fd60-4c47-9898-dd969be03bd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00766.warc.gz"} |
Adding Fractions And Mixed Numbers Worksheets
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Adding And Subtracting Mixed Fractions A Fractions Worksheet
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Worksheets For Fraction Addition
Worksheets For Fraction Addition
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16 Adding Subtracting Fractions With Mixed Numbers Worksheets Worksheeto
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006 Fraction Math Worksheet Multiplying Mixed Fractions Db excel
11 Best Images Of Adding Mixed Fractions Worksheets 4th Grade Adding Fractions Worksheets 4th
Adding Mixed Numbers Worksheet
Adding Fractions amp Mixed Numbers Worksheets Super Teacher Worksheets
Adding Fractions Mixed Numbers Worksheets for practicing addition of fractions Includes adding fractions with the same denominator easy and addition with unlike denominators harder There are also
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Adding Mixed Numbers Worksheets Math Worksheets 4 Kids
Adding 3 Proper Fractions and Mixed Numbers Unlike Rise up the ranks of mixed number addition with this grade 6 tool Featuring proper fractions and mixed numbers with unlike denominators as the three
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Adding Fractions Mixed Numbers Worksheets for practicing addition of fractions Includes adding fractions with the same denominator easy and addition with unlike denominators harder There are also
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006 Fraction Math Worksheet Multiplying Mixed Fractions Db excel
16 Adding Subtracting Fractions With Mixed Numbers Worksheets Worksheeto
11 Best Images Of Adding Mixed Fractions Worksheets 4th Grade Adding Fractions Worksheets 4th
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Adding Mixed Numbers With Like Denominators Worksheets Adding Mixed Numbers With Like
Mixed Fractions Math Worksheet That Make Math Fun
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Faces, Edges and Vertices
Trending Questions
What is the shape of the face of a cube?
A. Rectangle
B. Square
C. Parallelogram
D. Rhombus
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Q. Give examples of plane surfaces.
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A hexagonal pyramid has ___ vertices.
A. 7
B. 5
C. 6
D. 8
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Q. Which of the following is “Euler’s formula”?
A. E + F + 2 = V
B. E + F + 2 = V
C. E - F + V = 2
D. F - E + V = 2
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Two dice are placed side by side with 4+2, what is the total on the face opposite to the given numbers?
A. 6
B. 8
C. 7
D. 9
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Q. Name a 3D solid having no faces, no edges and no vertices, justify your answer.
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How many faces does a hexagonal prism have?
A. 6
B. 8
C. 10
D. 7
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What is the sum of the number of Faces, Vertices and Edges of a cube?
A. 20
B. 26
C. 18
D. 22
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Question 76
Write the name of
a) vertices,
b) edges and
c) faces of the prism, shown in the given figure.
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Consider the figure (pyramid mounted over cube):
If the number of faces, number of edges and number of vertices be F, E and V respectively, then find the value of
A. 32
B. 34
C. 30
D. 28
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Verify Euler's formula for the following solids. [4 MARKS]
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Q. What will be the number of edges in a polyhedron if there are 6 faces and 12 vertices?
A. 16
B. None of these
C. 12
D. 14
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Every individual flat surface of a solid is called its vertex.
A. True
B. False
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The corners of a solid shape are called its _________; the line segments of its skeleton are called its ___________; and its flat surfaces are its ________. The respective blanks should be filled by:
A. edges, faces, vertices
B. vertices, edges, faces
C. vertices, faces, edges
D. faces, edges, vertices
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Write the number of faces, edges and vertices of a cube.
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If two regular tetrahedrons are stuck together along a common base, the number of edges in the new figure is
A. 9
B. 12
C. 6
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Q. Write the name of the:
a. vertices
b. edges and
c. faces of the prism shown in the figure.
[4 MARKS]
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Consider the net figure of a rectangular pyramid:
Let the number of edges be E and number of vertices be V (in the 2-D figure: take edges as the intersection lines and vertices as intersection points of the lines).Then the value of E + V is 24. True
A. False
B. True
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Q. Fill in the blanks to make the following statements true:
(i) A cuboid has.... vertices.
(ii) A cuboid has.... edges.
(iii) a cuboid has .... faces.
(iv) The number of lateral faces of a cuboid is .....
(v) A cuboid all of whose edges are equal is called a....
(vi) Two adjacent faces of a cuboid meet in a line segment called its....
(vii) Each edge of a cuboid can be obtained as aline segment in which two... meet.
(viii) ....... edges of a cube (or cuboid) meet at each of its vertices.
(ix) A...... is a cuboid in which all the six faces are squares.
(x) The three concurrent edges of a cuboid meet at a point called the ..... of the cuboid.
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What will be the top view of the given solid whose base is a square?
A. Circle
B. Triangle
C. Rectangle
D. Square
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In the figure of a cube,
i) Which edge is the intersection of faces EFGH and EFBA?
ii) Which faces intersect at edge FB?
iii) Which three faces from the vertex A?
iv) Which vertex is formed by the faces ABCD, ADHE and CDHG?
v) Give all the edges that are parallel to edge AB
vi) Give the edges that are neither parallel nor perpendicular to edge BC.
vii) Give all the edges that are perpendicular to edge AB.
viii) Give four vertices that do not all lie in one plane.
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Which of the following statements about a rectangular prism is not true?
A. There are 6 edges in a rectangular prism.
B. Volume of a rectangular prism =l×b×h
C. Opposite pairs of faces are identical in a rectangular prism.
D. Volume of a rectangular prism =base area×h
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Select the appropriate answer.
An edge is a
A. Line segment along which two adjacent faces meet.
B. the surface between two parallel faces of a cuboid.
C. Point where two faces meet.
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Take two regular tetrahedrons and stick them along a common base. The number of edges in the new three dimensional shape formed are?
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How many faces does a right rectangular prism have?
A. 6
B. 12
C. 5
D. 8
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How many edges does a triangular prism have ?
A. 12
B. 6
C. 9
D. 7
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Which of the following solids will have the given net?
(Note: A net diagram is a 2D plane figure that can be folded to form a 3D solid.)
A. Rectangular prism
B. Pyramid
C. Cone
D. Prism
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What is the number of faces and edges in a cube?
A. 6, 12
B. 6, 6
C. 12, 12
D. 12, 8
View Solution
Find the number of edges and vertices of the given figure.
A. edges = 12, vertices = 6
B. edges = 10, vertices = 12
C. edges = 6, vertices = 8
D. edges = 6, vertices = 12
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Tanzania’s long-awaited ambitious plan to harness its natural gas and reduce its dependence on expensive imported fuel (which takes 60% of the foreign currency earnings each year) and hydropower
(which provides 70% of the electricity) has received the go-ahead. The World Bank is providing a US$ 200 million loan for the project which will be undertaken jointly by the government and two
Canadian firms in a new firm called Songas. The project will exploit an estimated 32,77 billion cubic metres of natural gas on the island of Songosongo, near Kilwa. The gas will be ferried to the
mainland through a pipeline and a 100-megawatt electric power station will be built. It is hoped that some of the gas can be exported to Kenya.
Songosongo’s gas deposit was discovered 20 years ago but has remained unexploited because of lack of funds, In the last six years Tanzania’s electricity demand has been growing at 12% a year and is
now estimated to amount to 800 megawatts, Between 1992 and 1993 there were serious power shortages because the prolonged drought had reduced water levels at the big Mtera Dam 400 kilometres southwest
of Dar es Salaam. (An item on Tanzania’s progress in harnessing hydropower is to be found in the ‘Reviews’ section in this issue – Ed)
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Binary Tree Level Order Traversal
Go to my solution
Go to the question on LeetCode
My Thoughts
What Went Well
I implemented my solution quickly and had a very good runtime, and after optimizing, I got an outstanding space complexity.
What I Learned
I learned how to convert my program into a more optimal solution using the breadth first search algorithm. I also learned that the len() function is an O(1) operation and not O(n). This was nice to
know since I have avoided some solutions because of this belief.
How I Can Improve
By practicing the implementation of depth and breadth-first search, I can master two very commonly used algorithms for trees and graphs.
Algorithm Description
Binary Tree - A rooted tree where every node has at most two children, the left and right children.
Depth-First Search - An algorithm for traversing a tree data structure. The algorithm begins at the root node and searches until it can not continue any deeper. Once at the deepest point, the
algorithm works backward until all the nodes have been visited.
Breadth-first search - A tree/matrix traversal algorithm that searches for elements with a given property. It creates a deque of elements yet to be visited and then iterates through each level
(similar properties) until there are no more elements to iterate through. In this example, the similar property would be equal depths in the tree.
The proper definition for breadth-first search when not being applied to this specific problem can be found here.
Visual Examples
Binary Tree, click to view
Depth-first search being performed on a tree, click to view
Solution Statistics For Breadth-First Search
Time Spent Optimizing
5 minutes
Time Complexity
O(n) - Each node (n) must pass through the while loop, and since the while loop is an O(1) block of code, we get an O(n) time complexity.
Space Complexity
O(n) - We are constantly pushing and popping off the deque but only appending to the sorted list. The number of elements in the sorted list exceeds that of the deque, resulting in the O(n) space
Runtime Beats
89.74% of other submissions
Memory Beats
97.74% of other sumbissions
Optimal Solution
1 class Solution:
2 def levelOrder(self, root: Optional[TreeNode]) -> List[List[int]]:
3 if root is None: return [] # O(1)
5 order = [] # O(1)
6 q = deque() # O(1)
7 q.append((root,0)) # O(1)
9 # Continuously iterate through the deque until it is empty
10 while q:
11 cur, depth = q.popleft() # O(1)
13 if depth == len(order): # O(1)
14 order.append([cur.val]) # O(1)
15 else:
16 order[depth].append(cur.val) # O(1)
18 if cur.left: q.append((cur.left,depth+1)) # O(1)
19 if cur.right: q.append((cur.right,depth+1)) # O(1)
21 return order
Solution Statistics For Depth First Search
Time Spent Coding
15 minutes
Time Complexity
O(n) - Each node (n) must call the search function, and the search function is an O(1) time complexity function, resulting in the O(n) space complexity.
Space Complexity
O(n) - Each node must be stored in the sorted list, resulting in the O(n) space complexity. Truly the space complexity is O(n + log n). The log n is for the most amount of possible recursive calls
yet to be completed.
Runtime Beats
91.90% of other submissions
Memory Beats
16.32% of other sumbissions
1 # Definition for a binary tree node.
2 # class TreeNode:
3 # def __init__(self, val=0, left=None, right=None):
4 # self.val = val
5 # self.left = left
6 # self.right = right
7 class Solution:
8 def levelOrder(self, root: Optional[TreeNode]) -> List[List[int]]:
9 if root is None: return [] # O(1)
11 self.order = [] # O(1)
13 def search(root,depth):
15 # If the length of the ordered list is equal to the depth,
16 # then that depth has not been visited yet, so append it
17 # with the given element
18 if len(self.order) == depth: # O(1)
19 self.order.append([root.val]) # O(1)
21 # If the length != the length, then append the current element
22 # to the index of the depth
23 else:
24 self.order[depth].append(root.val) # O(1)
26 # Only call the search function if the node != None
27 if root.left: search(root.left,depth+1) # O(1)
28 if root.right: search(root.right,depth+1) # O(1)
30 Search(root,0) # O(n)
31 return self.order | {"url":"https://douglastitze.com/posts/Binary-Tree-Level-Order-Traversal/","timestamp":"2024-11-13T14:12:54Z","content_type":"text/html","content_length":"32968","record_id":"<urn:uuid:7e4a7df2-954d-4f67-931e-62ff8f619f9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00827.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2016.91
URN: urn:nbn:de:0030-drops-61947
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/6194/
Dixit, Kashyap ; Raskhodnikova, Sofya ; Thakurta, Abhradeep ; Varma, Nithin
Erasure-Resilient Property Testing
Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions,
all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the
function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by
querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some
cases, make the tester entirely useless.
In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha,
epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries
the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property
P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain
points to satisfy P.
We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box.
However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient
testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and
erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires
number of queries polynomial in the input size.
BibTeX - Entry
author = {Kashyap Dixit and Sofya Raskhodnikova and Abhradeep Thakurta and Nithin Varma},
title = {{Erasure-Resilient Property Testing}},
booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
pages = {91:1--91:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-013-2},
ISSN = {1868-8969},
year = {2016},
volume = {55},
editor = {Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6194},
URN = {urn:nbn:de:0030-drops-61947},
doi = {10.4230/LIPIcs.ICALP.2016.91},
annote = {Keywords: Randomized algorithms, property testing, error correction, monotoneand Lipschitz functions}
Keywords: Randomized algorithms, property testing, error correction, monotoneand Lipschitz functions
Collection: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 23.08.2016
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=6194","timestamp":"2024-11-03T20:24:25Z","content_type":"text/html","content_length":"8483","record_id":"<urn:uuid:8412191e-a6e5-4cda-879f-760dbd163b1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00082.warc.gz"} |
On Energy Conservation for the hydrostatic Euler equations: an Onsager Conjecture
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If you have a question about this talk, please contact nobody.
FDE2 - Fractional differential equations
Onsager’s conjecture states that the incompressible Euler equations conserve kinetic energy (the L^2 norm in space) if the velocity field is Hölder continuous in space with exponent bigger than 1/3.
In case the exponent is less than 1/3 energy dissipation can occur. We consider an analogue of Onsager’s conjecture for the hydrostatic Euler equations. These equations arise from the Euler equations
under the assumption of the hydrostatic balance, as well as the small aspect ratio limit (in which the vertical scale is much smaller compared to the horizontal scales). Unlike the Euler equations,
in the case of the hydrostatic Euler equations the vertical velocity is one degree spatially less regular compared to the horizontal velocities. The fact that the equations are anisotropic in
regularity and nonlocal makes it possible to prove a range of sufficient criteria for energy conservation, which are independent of each other. This means that there probably is a ‘family’ of Onsager
conjectures for these equations. This is joint work with Simon Markfelder and Edriss S. Titi.
This talk is part of the Isaac Newton Institute Seminar Series series.
This talk is included in these lists:
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Qualifying Exam
The first milestone in the Mathematics PhD program are the qualifying exams. Exams are offered in Fall (before the academic year begins) and in Spring. PhD students must pass at least one exam before
the start of their 4th quarter. All exams must be completed before the start of the student's 7th quarter. Failure to meet these deadlines is cause for dismissal from the program. Carefully read the
Guidelines for Graduate Qualifying Exams document.
Exam requirements are different depending on which program a student is in. Please refer to the UCSD catalog for specific requirements: https://www.ucsd.edu/catalog/curric/MATH-gr.html.
During any examination period the student may take as many exams as he or she chooses. The qualifying exams are written and graded by the faculty who teach the courses. The scores are brought before
the Qualifying Exam Appeals Committee (QEAC) and the grades are discussed. The final decision as to whether the student has failed or passed (and at what level) is made by QEAC. This decision is
based upon exam performance, and performance in exam cognate coursework, though the QEAC is free to consider additional circumstances in rendering its decision. After the QEAC meeting, the PhD staff
advisor will inform students when/how they can find out their results.
Students can request to see their exams after grading in order to find out what they did well/poorly on. Students who wish to see their exam for purpose of contesting the grading should be advised
that there will be a very strong burden of proof needed to sustain a grade appeal on a graduate exam because of the nature of the exam writing and grading process. Such an appeal is most likely not
going change the exam result.
Qualifying Exam Requirements, Old and New
The Department of Mathematics has undertaken a reform of our Qualifying Exams. This brief note explains the old/current system, the new system, and how the changes are being phased in. These
requirements apply to PhD students in Mathematics; Statistics and CSME PhD students have separate requirements administered by the faculty.
Qualifying Exam Courses and Areas
There are 7 qualifying exams administered each Spring and Fall. Each corresponds to a three-quarter graduate course. They are organized into three Areas.
Area 1 220ABC 240ABC
Complex Analysis Real Analysis
Area 2 200ABC 202ABC 290ABC
Algebra Applied Algebra Topology
Area 3 270ABC 281ABC
Numerical Analysis Statistics
Old/Current Requirements
For PhD students who entered our program in Fall 2023 or earlier, the following are the current requirements to complete the qualifying exams.
• Each exam is assigned one of four grades: PhD Pass, Provisional PhD Pass (also known as PhD- Pass), Masters Pass, and Fail. The grade cutoffs are determined by the instructors who create/grade
the exams; those cutoffs are not released to students.
• Students must pass at least 3 qualifying exams.
□ At least one exam must have a PhD Pass.
□ At least one additional exam must have a Provisional PhD Pass or better.
□ At least one additional exam must have a Masters Pass or better.
• Students must pass at least one exam from Area 1, and at least one exam from Area 2.
• Students must have two exams, each with a Provisional PhD Pass or better, from two different Areas.
• Students must pass at least one exam with a Provisional PhD Pass or better before the start of their 4th quarter.
• Students must complete all the qualifying exams before the start of their 7th quarter
New Requirements
For students who enter our program in Fall 2024 or later, the following are the requirements to complete the qualifying exams.
• Each exam is assigned one of four grades: PhD Area Pass, PhD General Pass, Masters Pass, and Fail. The grade cutoffs are determined by the instructors who create/grade the exams; those cutoffs
are not released to students.
□ PhD Area Pass indicates readiness to begin research in that area.
□ PhD General Pass indicates sufficient familiarity with the subject to begin research in a different area. This standard is lower than Provisional PhD Pass, and higher than Masters Pass.
□ Masters Pass is only relevant for Masters students. A Masters Pass no longer counts towards completion of qualifying exams for PhD students.
• Students must pass at least 3 qualifying exams.
□ At least one exam must have a PhD Area Pass.
□ At least two additional exams must have a PhD General Pass or better.
• Students must complete qualifying exams from at least two different Areas.
• Students must pass at least one exam before the start of their 4th quarter.
• Students must complete all the qualifying exams before the start of their 7th quarter.
Principal Differences
The new system has more flexibility: students no longer have to take quals from both Areas 1 and 2, simply from 2 distinct Areas among 1, 2, and 3. The standards for completion are simplified.
Although Masters Pass is no longer a sufficient standard for PhD students, the PhD General Pass standard is lower than the old Provisional PhD Pass standard, and more consistent with the intent of
the exams: to prepare students for focused research in one main area.
Phasing In Period
Any current PhD students (who entered in Fall 2023 or earlier) still progressing towards completing the qualifying exams may satisfy either the current or the new requirements. To be precise:
• Each Spring and Fall (in fact starting this past Fall 2023), qual instructors will select cutoffs corresponding to all five possible grades:
PhD Pass = PhD Area Pass > Provisional PhD Pass > PhD General Pass > Masters Pass > Fail
• At each qual session, each PhD student’s file will be evaluated using both the current and the new requirements. It will be judged complete if it satisfies the current requirements or if it
satisfies the new requirements.
Caveat: students who entered in Fall 2022 or earlier already have qualifying exams graded only using the old cutoffs. Qualifying exams from Spring 2023 or earlier will not be regraded to compute PhD
General Pass cutoffs.
Other Aspects of Qualifying Exam Reforms
In addition to the logistical changes described above:
• Faculty will be undertaking the creation of standardized syllabi for all seven qualifying exams, to be available to PhD students upon entry. This is a process that will take the faculty
significant time and energy to complete, and is planned to be available starting in Fall 2024.
• In the meantime, qualifying exam course instructors will give detailed syllabi in each course (as always, per Academic Senate regulations), and content cutoffs for the exams will be communicated
to students by the Graduate Advisor in advance of the qualifying exams. The same content cutoffs will apply to both Spring and Fall qualifying exams, as has been standard.
• There will be closer coordination of mentoring efforts by course advisors and the Vice Chair for Graduate Affairs. All advisors for first-year PhD students will formulate plans for course
enrollment for the full year, as well as plans for which qualifying exams to take in Spring 2024. Advisors should meet again with their advisees before the beginnings of Winter and Spring
quarters, and possibly make adjustments at those times.
• Preliminary full year course and qualifying exam plans should be submitted by the advisors to the Graduate Vice Chair by the end of Week 1 of the Fall quarter.
Spring 2025 exam schedule
Dates, times, and locations to be determined.
Complex Analysis
Numerical Analysis
Applied Algebra
Real Analysis
Sample Qualifying Exams
Algebra (Math 200A/B/C):
SP04, SP05, SP06, FA06, SP07, FA07, SP08, FA08, SP09, FA09, FA10, SP11, FA11, SP12, SP13, FA13, SP14, FA14, SP15, SP16, SP17, FA17, SP18, FA18, SP19, FA19, SP20, FA20, SP21,
FA21, SP22, FA22, SP23, FA23, SP24, FA24
Applied Algebra (Math 202A/B/C):
SP04, FA04, SP05, SP06, SP08, FA06, SP07, FA07, FA11, SP11, SP13, SP15, SP17 , FA17, SP18, FA18, SP19, SP20, FA20, SP21, FA21, SP22, FA22, SP23A, SP23B, FA23A, FA23B, FA23C, SP24,
Complex Analysis (Math 220A/B/C):
SP04, SP05, FA05, SP06, FA06, SP07, FA07, SP08, FA08, SP09, FA09, FA10, FA11, FA15, SP11, SP12, SP13, FA13, SP15, FA16, SP17, FA17, SP18, SP19, FA19, SP20, FA20, SP21, FA21
, SP22, FA22, SP23, FA23, SP24, FA24
Numerical Analysis (Math 270A/B/C):
SP99, SP00, FA00, SP01, FA01, SP02, FA02, SP03, FA03, SP04, FA04, SP05, FA06, SP06, FA07, SP07, SP08, FA08, SP09, FA09, FA10, SP11, SP13, FA15, SP17, FA17, SP18, SP20,
FA20, SP21, FA21, SP22, FA22, SP23, FA23, SP24
Real Analysis (Math 240A/B/C):
SP04, FA04, FA05, SP06, FA06, SP07, FA07, SP08, SP09, FA09, FA10, FA11, SP11, SP13, SP15, FA16, SP16, SP17, FA17, SP18, FA18, SP20, FA20, SP21, FA21, SP22, FA22, SP23, FA23,
SP24, FA24
Statistics (Math 281A/B):
SP99, FA99, SP00, FA00, SP01, SP02, FA02, SP03, FA03, SP04, SP05, SP06, SP07, SP08, SP09, FA10, SP11, SP13, FA15, SP17, FA17, SP18, SP18 Formulas, SP19 Part A, SP19 Part BC,
FA19 (Part A), FA19 (Part BC), SP20, FA20, SP21, FA21, SP22, FA22, SP23AB, SP23C, FA23AB, FA23C, SP24
Topology (Math 290A/B/C):
SP00, SP01, SP02, FA02, FA03, SP04, FA04, SP05, SP06, SP07, FA06, FA07, SP08, FA08, FA09, SP10, FA10, SP11, SP13, FA15, SP17, FA17, SP18, FA18, FA19, SP20, FA20, SP21, FA21
, SP22, FA22, SP23, FA23, SP24, FA24 | {"url":"https://math.ucsd.edu/students/graduate/academics/qualifying-exams","timestamp":"2024-11-04T14:36:31Z","content_type":"text/html","content_length":"90990","record_id":"<urn:uuid:2a75d45c-878d-4773-a7b5-97a715fc770a>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00870.warc.gz"} |
PPD distributions
PPD-distributions {bayesplot} R Documentation
PPD distributions
Plot posterior or prior predictive distributions. Each of these functions makes the same plot as the corresponding ppc_ function but without plotting any observed data y. The Plot Descriptions
section at PPC-distributions has details on the individual plots.
ppd_data(ypred, group = NULL)
size = 0.25,
alpha = 0.7,
trim = FALSE,
bw = "nrd0",
adjust = 1,
kernel = "gaussian",
n_dens = 1024
discrete = FALSE,
pad = TRUE,
size = 0.25,
alpha = 0.7
ppd_dens(ypred, ..., trim = FALSE, size = 0.5, alpha = 1)
ppd_hist(ypred, ..., binwidth = NULL, bins = NULL, breaks = NULL, freq = TRUE)
binwidth = NULL,
bins = NULL,
freq = TRUE,
size = 0.5,
alpha = 1
binwidth = NULL,
bins = NULL,
freq = TRUE,
size = 0.5,
alpha = 1
ppd_boxplot(ypred, ..., notch = TRUE, size = 0.5, alpha = 1)
An S by N matrix of draws from the posterior (or prior) predictive distribution. The number of rows, S, is the size of the posterior (or prior) sample used to generate ypred. The
ypred number of columns, N, is the number of predicted observations.
A grouping variable of the same length as y. Will be coerced to factor if not already a factor. Each value in group is interpreted as the group level pertaining to the
group corresponding observation.
... Currently unused.
size, alpha Passed to the appropriate geom to control the appearance of the predictive distributions.
trim A logical scalar passed to ggplot2::geom_density().
bw, adjust, kernel, Optional arguments passed to stats::density() to override default kernel density estimation parameters. n_dens defaults to 1024.
For ppc_ecdf_overlay(), should the data be treated as discrete? The default is FALSE, in which case geom="line" is passed to ggplot2::stat_ecdf(). If discrete is set to TRUE then
discrete geom="step" is used.
pad A logical scalar passed to ggplot2::stat_ecdf().
binwidth Passed to ggplot2::geom_histogram() to override the default binwidth.
bins Passed to ggplot2::geom_histogram() to override the default binwidth.
breaks Passed to ggplot2::geom_histogram() as an alternative to binwidth.
For histograms, freq=TRUE (the default) puts count on the y-axis. Setting freq=FALSE puts density on the y-axis. (For many plots the y-axis text is off by default. To view the
freq count or density labels on the y-axis see the yaxis_text() convenience function.)
notch For the box plot, a logical scalar passed to ggplot2::geom_boxplot(). Note: unlike geom_boxplot(), the default is notch=TRUE.
For Binomial data, the plots may be more useful if the input contains the "success" proportions (not discrete "success" or "failure" counts).
The plotting functions return a ggplot object that can be further customized using the ggplot2 package. The functions with suffix _data() return the data that would have been drawn by the plotting
See Also
Other PPDs: PPD-intervals, PPD-overview, PPD-test-statistics
# difference between ppd_dens_overlay() and ppc_dens_overlay()
preds <- example_yrep_draws()
ppd_dens_overlay(ypred = preds[1:50, ])
ppc_dens_overlay(y = example_y_data(), yrep = preds[1:50, ])
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Why doesn't absolute value affect a sideways parabola? | Socratic
Why doesn't absolute value affect a sideways parabola?
So I have the equation #x=abs((y-1)^2+2#, and when I plug it in to a graphing calculator website (Desmos), it shows the same thing if I plug in $x = {\left(y - 1\right)}^{2} + 2$. Does that mean that
they are the same graph... in other words, does absolute value not affect a sideways (aka sleeping) parabola?
1 Answer
It does, you have misunderstood a concept.
Remember, ${x}^{2}$, for whatever value of $x$, except for $i$, is always positive. So here, the ${\left(y - 1\right)}^{2}$ acts as an absolute value function itself!
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Minimax Theory and its Applications 05 (2020), No. 2, 275--304
Copyright Heldermann Verlag 2020
Relaxation of a Dynamic Game of Guidance and Program Constructions of Control
Alexander G. Chentsov
Krasovskii Inst. of Mathematics and Mechanics, UB RAS, Ekaterinburg 620108, Russia
Daniel M. Khachay
Krasovskii Inst. of Mathematics and Mechanics, UB RAS, Ekaterinburg 620108, Russia
The natural relaxation of guidance problem is considered. Namely, for fixed closed sets considered as parameters (target set and the set defining state constraints), we consider the similar guidance
problem for ε-neighborhoods of these sets. We are interested to find the smallest size ε of these neighborhoods for which the player I can solve his guidance problem in class of generalized
set-valued non-anticipating strategies. For the construction of solution, the Program Iterations Method is used. We obtain the above-mentioned smallest size as a position function. For determination
of this function, iterative procedure operating in the function space is used. Also, it is shown that desired function is the fixed point of operator defining the iterative procedure.
Keywords: Pursuit-evasion differential game, program iteration method, guaranteed result.
MSC: 49J15, 49K15, 93C15, 49N70.
[ Fulltext-pdf (198 KB)] for subscribers only. | {"url":"https://www.heldermann.de/MTA/MTA05/MTA052/mta05015.htm","timestamp":"2024-11-07T15:48:23Z","content_type":"text/html","content_length":"3644","record_id":"<urn:uuid:a7dc7c77-d9a7-440c-acab-a41879fe1e3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00010.warc.gz"} |
Short Trick of Calculating Square
I have already shared my technique to
calculate squares here
but today Gaurav Kaushik shared a trick to calculate squares.
How to calculate square?
Its too simple with some tricks. Lets move towards some examples:
1: Find out the square of 12?
• Let divide 12 in two parts. Lets assume 1 is A while 2 is B.
• Now first find out the square of last digit. Here last digit is 2. So square of 2 is 4. Write down 4 in a order as i show u here__4.
• Now to find out the middle numbers of square we have a formula 2xAxB. Here A is 1 while B is 2. So according to formula 2x1x2=4. Write down this 4 in order_44.
Now we move to certain high digits.
Example 2
: Find out square of 49?
• Let's divide 49 in two parts. Lets assume 4 is A while 9 is B.
• Now first find out the square of last digit. Here last digit is 9. So square of 9 is 81. But 81 is a two digit number so we use 1 only while use 8 as a carry forward for further calculation. So
now we put 1 in an order as i show u here ___1.
• Now to find out the middle numbers of square we have a formula 2xAxB. Here A is 4 while B is 9.so according to formula 2x4x9= 72. As according to previous step we carry forward 8 so now we added
8 in 72 (8+72=80). Now again here we use 0 and carry forward 8 for next step. Write down 0 in order as i write down__01.
• Now find out the square of first digit. Here first digit is 4. So square of 4 is 16 as we all knows and in previous step we have a carry of 8 so after adding 8 in 16(8+16 =24) we get 24. Now
write down 24 as i in order as i follows 2401.
Now we move to certain more higher level.
Example 3
: Find out square of 172?
• Let divide 172 in two parts. Let assume 17 is A while 2 is B.
• Now first find out the square of last digit. Here last digit is 2. So square of 2 is 4. Write down 4 in order as i shows u here____4.
• Now to find out the middle numbers of square we have a formula 2xAxB. Here A is 17 while B is 2 so according to formula 2x17x2=68. 68 is a two digit number so we use 8 only while use 6 as a carry
forward for further calculation. So now we put 8 in order as i shows u here___84.
• Now we find out the square of A . As we knows A is 17 so square of 17 is 289. And as we knows according to previous step 6 is carry forwarded so we added 6 in 289 (6+289=295) so we get the digit
295. Now we put 295 in order as i shows u here 29584.
Just try out all steps by practice. It will not consume much time. | {"url":"https://www.bankexamstoday.com/2014/03/short-trick-of-calculating-square-roots.html","timestamp":"2024-11-09T14:21:55Z","content_type":"application/xhtml+xml","content_length":"122471","record_id":"<urn:uuid:2f371a04-2e69-4a50-890a-f19656890e44>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00825.warc.gz"} |
Finding the Unknowns in a Rational Function given Its Value and the Value of Its First Derivative at a Point
Question Video: Finding the Unknowns in a Rational Function given Its Value and the Value of Its First Derivative at a Point Mathematics • Second Year of Secondary School
Suppose that π (π ₯) = (π ₯Β² + π π ₯ + π )/(π ₯Β² β 7π ₯ + 4). Given that π (0) = 1 and π β ²(0) = 4, find π and π .
Video Transcript
Suppose that π of π ₯ is equal to π ₯ squared plus π π ₯ plus π all over π ₯ squared minus seven π ₯ plus four. Given that π of zero is equal to one and π prime of zero is equal to
four, find π and π .
Our first step in this question can be to substitute π ₯ equals zero into π of π ₯. Since weβ re given that π of zero is equal to one. We obtain that π of zero is equal to zero squared
plus π times zero plus π all over zero squared minus seven times zero plus four. Now, all of these terms will go to zero apart from π and four. Weβ re left with π of zero is equal to π
over four. Next, we use the fact that the question has told us that π of zero is equal to one. And so, we can set this equal to one. From this, we find that π is equal to four. Next, we can
use the fact that π prime of zero is equal to four. However, first of all, we must find π prime of π ₯. In order to do this, we need to differentiate π . Since π is a rational function,
we can use the quotient rule in order to find its derivative.
The quotient rule tells us that π ’ over π £ prime is equal to π £ times π ’ prime minus π ’ times π £ prime all over π £ squared. Setting our function π of π ₯ equal to π ’ over π £, we
obtain that π ’ is equal to π ₯ squared plus π π ₯ plus π . And π £ is equal to π ₯ squared minus seven π ₯ plus four. We can find π ’ prime and π £ prime by differentiating these two
functions. Giving us that π ’ prime is equal to two π ₯ plus π and π £ prime is equal to two π ₯ minus seven. Now, we can substitute these into the quotient rule. We obtain that π prime of
π ₯ is equal to π ₯ squared minus seven π ₯ plus four multiplied by two π ₯ plus π minus π ₯ squared plus π π ₯ plus π multiplied by two π ₯ minus seven all over π ₯ squared minus
seven π ₯ plus four all squared.
Now, we could simplify π prime of π ₯ at this point. However, weβ re going to be substituting in π ₯ is equal to zero. And so, a lot of these terms would just disappear. Letβ s simply
substitute π ₯ equals zero in here. We obtain this. However, a lot of the terms will vanish to zero, which leaves us with four π plus seven π all over 16. Now, we have found that π is equal
to four earlier. And so, we can substitute this in, giving us four π plus 28 all over 16.
Since the question has told us that π prime of zero is equal to four, we can set this equal to four. Then, we simply rearrange this in order to solve for π . Now, we obtain our solution that π
is equal to nine. Weβ ve now found the values of both π and π , which completes the solution to this question. | {"url":"https://www.nagwa.com/en/videos/724165734280/","timestamp":"2024-11-11T11:14:07Z","content_type":"text/html","content_length":"252266","record_id":"<urn:uuid:dece44d8-8c5c-4d14-8c83-fdb352a00588>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00324.warc.gz"} |
What you should know about data imputation | EDUCBA
Definition of Data Imputation
Data imputation is the process of replacing missing or incomplete data points in a dataset with estimated or substituted values. These estimated values are typically derived from the available
data, statistical methods, or machine learning algorithms.
Data imputation fills missing values in datasets, preserving data completeness and quality. It ensures practical analysis, model performance, and visualizations by preventing data loss and
maintaining sample size. Imputation reduces bias, maintains data relationships, and facilitates various statistical techniques, enabling better decision-making and insights from incomplete data.
Table of Contents
Importance of Data Imputation in Analysis
Data imputation is crucial in data analysis as it addresses missing or incomplete data, ensuring the integrity of analyses. Imputed data enables the use of various statistical methods and machine
learning algorithms, improving model accuracy and predictive power. Without imputation, valuable information may be lost, leading to biased or less reliable results. It helps maintain sample size,
reduces bias, and enhances the overall quality and reliability of data-driven insights.
Data Imputation Techniques
There are several methods and techniques for data imputation, each with its strengths and suitability depending on the nature of the data and the analysis goals. Let’s discuss some commonly used data
imputation techniques:
1. Mean/Median/Mode Imputation
• Mean Imputation: Replace missing values in numerical variables with the average of the observed values for that variable.
• Median Imputation: Replace missing values in numerical variables with the middle value of the observed values for that variable.
• Mode Imputation: Replace missing values in categorical variables with the most frequent category among the observed values for that variable.
1. Identify variables with missing values.
2. Compute the mean, median, or mode of the variable, depending on the chosen imputation method.
3. Replace missing values in the variable with the computed central tendency measure.
Advantages Disadvantages and Considerations
Simplicity Ignores Data Relationships
Preserves Data Structure May Distort Data
Applicability Inappropriate for Missing Data Patterns
When to Use:
• Use mean imputation for numerical variables when missing data is missing completely at random (MCAR) and the variable has a relatively normal distribution.
• Use median imputation when the data is skewed or contains outliers, as it is less sensitive to extreme values.
• Use mode imputation for categorical variables when you have missing values that can be reasonably replaced with the most frequent category.
2. Forward Fill and Backward Fill
• Forward Fill: In forward fill imputation, missing values are replaced with the most recent observed value in the sequence. It propagates the last known value forward until a new observation is
• Backward Fill: In backward fill imputation, missing values are replaced with the next observed value in the sequence. It propagates the next known value backward until a new observation is
1. Identify the variables with missing values in a time-ordered dataset.
2. For forward fill, replace each missing value with the most recent observed value that precedes it in time.
3. For backward fill, replace each missing value with the next observed value that follows it in time.
Advantages Disadvantages and Considerations
Temporal Context Assumption of Temporal Continuity
Simplicity Potential Bias
Applicability Missing Data Patterns
When to Use:
• Use forward fill when you believe that missing values can be reasonably approximated by the most recent preceding value and you want to maintain the temporal context.
• Use backward fill when you believe that missing values can be reasonably approximated by the next available value and you want to maintain the temporal context.
3. Linear Regression Imputation
Linear regression imputation is a statistical imputation technique that leverages linear regression models to predict missing values based on the relationships observed between the variable with
missing data and other relevant variables in the dataset.
1. Identify Variables: Determine the variable with missing values (the dependent variable) and the predictor variables (independent variables) that will be used to predict the missing values.
2. Split the Data: Split the dataset into two subsets: one with complete data for the dependent and predictor variables and another with missing values for the dependent variable.
3. Build a Linear Regression Model: Use the subset with complete data to build a linear regression model.
4. Predict Missing Values: Apply the trained linear regression model to the subset with missing values to predict and fill in the missing values for the dependent variable.
5. Evaluate Imputed Values: Assess the quality of the imputed values by examining their distribution, checking for outliers, and comparing them to observed values where available.
Advantages Disadvantages and Considerations
Utilizes Relationships Assumption of Linearity
Predictive Accuracy Sensitivity to Outliers
Preserves Data Structure Model Selection
When to Use:
When there is a known or plausible linear relationship between the variable with missing values and other variables in the dataset and the dataset is sufficiently large to build a robust linear
regression model.
4. Interpolation and Extrapolation
Interpolation is the process of estimating values between two or more known data points.
1. Identify or collect a set of data points.
2. Choose an interpolation method based on the nature of the data (e.g., linear, polynomial, spline).
3. Apply the chosen method to estimate values within the data range.
Advantages Disadvantages and Considerations
Provide reasonable estimates within the range of observed data. Assumes a continuous relationship between data points, which may not always hold.
Useful for filling gaps in data or estimating missing values. Accuracy decreases as you move further from the known data points.
Extrapolation is the process of estimating values beyond the range of known data points.
1. Identify or collect a set of data points.
2. Determine the nature of the data trend (e.g., linear, exponential, logarithmic).
3. Extend the trend beyond the range of observed data to make predictions.
Advantages Disadvantages and Considerations
Allows for making predictions or projections into the future or past. Extrapolation assumes that the data trend continues, which may not always be accurate.
Useful for forecasting and trend analysis. Extrapolation can lead to significant errors if the underlying data pattern changes.
When to Use:
• Interpolation is suitable when you have a series of data points and want to estimate values within the observed data range.
• Extrapolation is appropriate when you have historical data and want to make predictions or forecasts beyond the observed data range.
5. K-Nearest Neighbors (KNN) Imputation
K-nearest neighbors (KNN) Imputation is a method for handling missing data by estimating missing values using the values of their K-nearest neighbors, which are determined based on a similarity
metric (e.g., Euclidean distance or cosine similarity) in the feature space.
Steps in KNN Imputation:
1. Data Preprocessing: Prepare the dataset by identifying the variable(s) with missing values and selecting relevant features for similarity measurement.
2. Normalization or Standardization: Normalize or standardize the dataset to ensure that variables are on the same scale, as distance-based methods like KNN are sensitive to scale differences.
3. Distance Computation: Calculate the distance (similarity) between data points, typically using a distance metric such as Euclidean distance, Manhattan distance, or cosine similarity.
4. Nearest Neighbors Selection: Identify the K-nearest neighbors for each data point with missing values based on the computed distances.
5. Imputation: Calculate the imputed value for each missing data point as a weighted average (for continuous data) or a majority vote (for categorical data) of the values from its K-nearest
6. Repeat for All Missing Values: Repeat the above steps for all data points with missing values, imputing each missing value separately.
Advantages Disadvantages and Considerations
Utilizes information from similar data points to estimate missing Sensitive to distance metric selection and a number of neighbors (K).
Can capture complex relationships in the data when K is appropriately The effectiveness of KNN imputation depends on the assumption that similar data points have similar values, which may not hold
chosen. in all cases.
When to Use:
When you have a dataset with missing values and believe that similar data points are likely to have similar values, you need to impute missing values in both continuous and categorical variables.
6. Expectation-maximization (EM) Imputation
Expectation-maximization (EM) imputation is an iterative statistical method for handling missing data.
1. Model Specification: Define a probabilistic model that represents the relationship between observed and missing data.
2. Initialization: Start with an initial guess of the model parameters and imputed values for missing data. Common initializations include imputing missing values with their mean or using another
imputation method.
3. Expectation (E-step): In this step, calculate the expected values of the missing data (conditional on the observed data) using the current model parameters.
4. Maximization (M-step): Update the model parameters to maximize the likelihood of the observed data, given the expected values from the E-step. This involves finding parameter estimates that make
the observed data most probable.
5. Iterate: Repeat the E-step and M-step until convergence is achieved. Convergence is typically determined by monitoring changes in the model parameters or log-likelihood between iterations.
6. Imputation: Once the EM algorithm converges, use the final model parameters to impute the missing values in the dataset.
Advantages Disadvantages and Considerations
Can handle missing data that is not missing completely at random (i.e., data with a missing data Sensitivity to model misspecification: If the model is not a good fit for the data, imputed values
mechanism) may be biased.
Utilizes the underlying statistical structure in the data to make imputations, potentially leading Computationally intensive: EM imputation can be computationally expensive, especially for large
to more accurate estimates. datasets or complex models.
When to Use:
• When you have a dataset with missing data, and you suspect that the missing data mechanism is not completely random.
• When there is an underlying statistical model that can describe the relationship between observed and missing data.
7. Regression Trees and Random Forests
Regression Trees and Random Forests are machine-learning techniques used primarily for regression tasks. They are both based on decision tree algorithms but differ in their complexity and ability to
handle complex data.
Regression Trees
Regression trees are a type of decision tree used for regression analysis. They divide the dataset into subsets, called leaves or terminal nodes, based on the input features and assign a constant
value (usually the mean or median) to each leaf.
1. Start with the entire dataset.
2. Select a feature and a split point that best divides the data based on a criterion (e.g., mean squared error).
3. Repeat the splitting process for each branch until a stopping criterion is met (e.g., maximum depth or minimum number of samples per leaf).
4. Assign a constant value to each leaf, typically the mean or median of the target variable.
Advantages Disadvantages and Considerations
Easy to interpret and visualize. Prone to overfitting, especially when the tree is deep.
Handles both numerical and categorical data. Sensitive to small variations in the data.
Can capture non-linear relationships. Single trees may not generalize well to new data.
Random Forests
Random Forests are an ensemble learning technique that consists of multiple decision trees, typically built using the bagging (bootstrap aggregating) method.
1. Randomly select subsets of the data (bootstrapping) and features (feature bagging) for each tree.
2. Build individual decision trees for each subset.
3. Combine the predictions of all trees (e.g., by averaging for regression) to make the final prediction.
Advantages Disadvantages and Considerations
Reduces overfitting by combining multiple models. Can be computationally expensive for a large number of trees and features.
Provides feature importance scores. The resulting model is less interpretable compared to a single decision tree
When to Use:
• Use a single regression tree when you want a simple, interpretable model and have a small to moderate-sized dataset.
• Use Random Forests when you need high predictive accuracy, want to reduce overfitting, and have a larger dataset.
8. Deep Learning-Based Imputation
Deep Learning-Based Imputation is a data imputation method that uses deep neural networks to predict and fill in missing values in a dataset.
1. Data Preprocessing: Prepare the dataset by identifying the variable(s) with missing values and normalizing or standardizing the data as needed.
2. Model Selection: Choose an appropriate deep-learning architecture for imputation. Common choices include feedforward neural networks and recurrent neural networks (RNNs).
3. Data Split: Split the dataset into two parts: one with complete data (used for training) and another with missing values (used for imputation).
4. Model Training: Train the selected deep learning model using the portion of the dataset with complete data as input and the same data as output (supervised training).
5. Imputation: Use the trained model to predict missing values in the dataset with missing data based on the available information.
6. Evaluation: Assess the quality of the imputed values by comparing them to observed values where available. Common evaluation metrics include mean squared error (MSE) or mean absolute error (MAE).
Advantages Disadvantages and Considerations
Ability to capture complex relationships Computational complexity
Data-driven imputations. Data requirements
High performance. Interpretability
When to Use:
• When dealing with large and complex datasets where traditional imputation methods may not be effective.
• When you have access to substantial computing resources for model training.
• when you prioritize predictive accuracy over interpretability.
Deep learning-based imputation may not be necessary for smaller, simpler datasets where simpler methods can suffice.
9. Hot Deck Imputation
Hot Deck Imputation is a non-statistical imputation method that replaces missing values with observed values from similar or matching cases (donors) within the same dataset.
1. Identify Missing Values: Determine which variables in your dataset have missing values that need to be imputed.
2. Define Matching Criteria: Specify the criteria for identifying similar or matching cases.
3. Select Donors: For each record with missing data, search for matching cases (donors) within the dataset based on the defined criteria.
4. Impute Missing Values: Replace the missing values in the target variable with values from the selected donor(s).
5. Repeat for All Missing Values: Continue the process for all records with missing data until all missing values are imputed.
Advantages Disadvantages and Considerations
Maintains dataset structure Assumes similarity
Simplicity Limited to existing data
Can be useful for small datasets or when computational resources are limited. Potential for bias
When to Use:
When you have
• a small to moderately sized dataset and limited computational resources.
• want to maintain the existing relationships and structure within the dataset.
• you have reason to believe that similar cases should have similar values for the variable with missing data.
10. Time Series Imputation
Time Series Imputation is a method used to estimate and fill in missing values within a time series dataset. It focuses on preserving the temporal relationships and patterns present in the data while
addressing the gaps caused by missing observations.
1. Data Understanding: Begin by understanding the time series data, its context, and the reasons for missing values.
2. Exploratory Data Analysis: Analyze the time series to identify any patterns, trends, and seasonality that can inform the imputation process.
3. Choose Imputation Method: Select an appropriate imputation method based on the nature of the data and the identified patterns.
4. Impute Missing Values: Apply the chosen imputation method to estimate the missing values in the time series.
5. Evaluate Imputed Values: Assess the quality of the imputed values by comparing them to observed values where available.
6. Sensitivity Analysis: Conduct sensitivity analyses to assess the impact of different imputation methods and parameters on the results.
7. Further Analysis: Once the missing values are imputed, proceed with the intended time series analysis, which could include forecasting, anomaly detection, or trend analysis.
Advantages Disadvantages and Considerations
Preserves temporal relationships. Requires domain knowledge
Enables continuity. Sensitivity to method choice
Provides a foundation for forecasting. Limited by missing data mechanism
When to Use:
• When you have time series data with missing values that need to be filled to enable subsequent analysis.
• When you want to preserve the temporal relationships and patterns within the data.
11. Manual Imputation
Manual Imputation is a process in which missing values in a dataset are replaced with estimated values by human experts. It requires domain knowledge, experience, and judgment to make informed
decisions about the missing data.
1. Identify Missing Values: First, identify the variables in your dataset that have missing values that need to be imputed.
2. Access Domain Knowledge: Rely on domain knowledge and expertise related to the data and the specific variables with missing values.
3. Determine Imputation Strategy: Decide on an appropriate strategy for imputing the missing values.
4. Execute Imputation: Based on the chosen strategy, manually enter the estimated values for each missing data point in the dataset.
5. Documentation: Keep detailed records of the imputation process, including the rationale behind the imputed values, the expert responsible for the imputation, and any relevant notes or
6. Quality Control: If possible, perform quality control checks or have another expert review the imputed values to ensure consistency and accuracy.
Advantages Disadvantages and Considerations
Domain expertise. Subjectivity
Flexibility. Resource-intensive
Transparency Limited to domain expertise
When to Use:
When you have missing values in a dataset and domain expertise is available to make informed imputation decisions, the dataset contains variables that are context-specific and require deep domain
knowledge for accurate imputation.
Types of Missing Data
Below are the different types as follows:
1. Missing Completely at Random (MCAR)
In this type, the probability of data being missing is unrelated to both observed and unobserved data. In other words, missingness is purely random and occurs by chance. MCAR implies that the missing
data is not systematically related to any variables in the dataset. For example, a sensor failure that results in sporadic missing temperature readings can be considered MCAR.
2. Missing at Random (MAR)
Missing data is considered MAR when the probability of data being missing is related to observed data but not directly to unobserved data. In other words, missingness is dependent on some observed
variables. For instance, in a medical study, men might be less likely to report certain health conditions than women, creating missing data related to the gender variable. MAR is a more general and
common type of missing data than MCAR.
3. Missing Not at Random (MNAR)
MNAR occurs when the probability of data being missing is related to unobserved data or the missing values themselves. This type of missing data can introduce bias into analyses because the
missingness is related to the missing values. An example of MNAR could be patients with severe symptoms avoiding follow-up appointments, resulting in missing data related to the severity of their
Best Practices for Data Imputation
Here are some best practices for data imputation:
1. Exploratory Data Analysis (EDA)
Exploratory Data Analysis (EDA) is a crucial initial step in data analysis, involving the visual and statistical examination of data to uncover patterns, trends, anomalies, and relationships. It
helps researchers and analysts understand the data’s structure, identify potential outliers, and inform subsequent data processing, modeling, and hypothesis testing. EDA typically includes summary
statistics, data visualization, and data cleaning.
2. Data Visualization
Data Visualization is the graphical representation of data using charts, graphs, and plots. It transforms complex datasets into understandable visuals, making patterns, trends, and insights more
accessible. Data visualization aids in data exploration, analysis, and communication by conveying information in a concise and visually appealing manner. It helps users interpret data, detect
outliers, and make informed decisions, making it a valuable tool in various fields, including business, science, and research.
3. Cross-Validation
Cross-validation is a statistical technique used to evaluate the performance and generalization of machine learning models. It divides the dataset into training and testing subsets multiple times,
ensuring that each data point is used for both training and evaluation. Cross-validation helps assess a model’s robustness, detect overfitting, and estimate its predictive accuracy on unseen data.
4. Sensitivity Analysis
Sensitivity Analysis is a process in which variations in the parameters or assumptions of a model are systematically tested to understand how they impact the model’s results or conclusions. It helps
assess the robustness and reliability of the model by identifying which factors have the most significant influence on the outcomes. Sensitivity analysis is crucial in fields like finance,
engineering, and environmental science to make informed decisions and account for uncertainty.
Multiple Imputation vs Missing Imputation
Aspect Multiple Imputation Missing Imputation
Technique Generates multiple datasets with imputed values, typically through statistical models. Imputes missing values once using a single method, such as mean, median, or
Handling Captures uncertainty by providing multiple imputed datasets, allowing for more accurate Provides a single imputed dataset without accounting for imputation uncertainty.
Uncertainty standard errors and hypothesis testing.
Avoiding Bias Reduces bias by considering the variability inherent in imputations and appropriately May introduce bias if the imputation method used is not suitable for the data or if
accounting for it in analyses. the imputed values do not reflect the true distribution.
Method Selection Requires selecting a suitable imputation model, such as regression, Bayesian imputation, or Requires selecting a single imputation method, such as mean, median, or regression,
predictive mean matching. often based on data characteristics.
Complexity More computationally intensive, as it involves running the chosen imputation model multiple Less computationally intensive, as it involves a single imputation step.
times (equal to the number of imputed datasets).
Standard Error Allows for accurate estimation of standard errors, confidence intervals, and hypothesis testing Standard errors may be underestimated or incorrect due to not accounting for
Estimation by considering within- and between-imputation variability. imputation uncertainty.
Suitability for Data is Well-suited for complex data structures, high-dimensional data, and data with complex Suitable for straightforward data with simple missing data patterns
Complex missing data mechanisms.
Implementation Supported by various statistical software packages, such as R, SAS, and Python (e.g., using Widely available in statistical software packages for simple imputation methods.
in Software libraries like “mice” in R).
Potential Challenges in Data Imputation
Here are some common challenges in data imputation:
• Missing Data Mechanisms: Understanding the nature of missing data is crucial.
• Bias: The imputation method can introduce bias if it systematically underestimates or overestimates missing values.
• Imputation Model Selection: Choosing the right imputation model or method can be challenging, especially when dealing with complex data.
• High-Dimensional Data: In datasets with a large number of features (high dimensionality), imputation becomes more complex.
Future Developments in Data Imputation Techniques
Future developments in data imputation will likely focus on advancing machine learning-based techniques, such as deep learning models, to handle complex datasets with high dimensionality.
Additionally, there will be an increased emphasis on addressing missing data mechanisms like Missing Not at Random (MNAR) through innovative modeling approaches.
Data imputation is vital for handling missing data in various fields, ensuring the continuity and reliability of analyses and modeling. While a range of imputation methods exists, choosing the most
suitable one requires careful consideration of data characteristics and objectives. With advancements in machine learning and increased awareness of imputation challenges, future developments will
likely lead to more robust, transparent, and efficient imputation techniques for addressing missing data effectively.
Q1. What are common data imputation methods?
Ans: Common imputation methods include mean imputation, median imputation, k-nearest neighbors imputation, regression imputation, and multiple imputation. The choice depends on data characteristics
and research goals.
Q2. What challenges are associated with data imputation?
Ans: Challenges include selecting appropriate imputation methods, handling different types of missing data mechanisms, avoiding bias, addressing high-dimensional data, and ensuring transparency and
Q3. When should data imputation be used?
Ans: Data imputation is used when missing data is present, and preserving data integrity and completeness is essential for analysis or modeling. It is widely used in fields such as healthcare,
finance, and social sciences.
Q4 What are the potential pitfalls of data imputation?
Ans: Pitfalls include introducing bias if imputation is not done carefully, misinterpreting imputed values as observed, and not accounting for uncertainty in imputed data. It’s essential to
understand the data and choose imputation methods wisely.
Recommended Article
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๐ฏ Arbitrage strategies | Rho Protocol Documentation
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This article is in part based on http://www2.stat.duke.edu/~sayan/Sta613/2017/lec/LMM.pdf.
In this post we describe how linear mixed models can be used to describe longitudinal trajectories. An important linear model, particularly for longitudinal data, is the linear mixed model (LMM). The
basic linear model assumes independent or uncorrelated errors for confidence intervals and a best linear unbiased estimate via ordinary least squares (OLS), respectively. However, often data is
grouped or clustered with known group/cluster assignments. We expect that observations within a group/cluster will be correlated. An LMM provides an elegant model to handle this grouping.
For a static example, say you have a counseling program for smoking and you want to model the effect of some covariates (age, weight, income, etc) on lapse frequency. You have a number of different
counselors and each participant is randomized to a different counselor. You expect that both the baseline response and how covariates affect the response varies across counselors. Here the group is
the counselor. For a longitudinal example, say you have a number of HIV/AIDS patients and you track their white blood cell counts over time. You want to model the effect of time on white blood cell
counts. You expect that the white blood cell count at the start of measurement and the effect of time on the white blood cell count vary between individuals. In this case, the group or cluster is the
individual: repeated observations within an individual are correlated.
Figure 1: we track white blood cell counts over time across several patients, and want to describe how white blood cell counts evolve over time, both within and between patients.
In an LMM for longitudinal data, the observed process is a noisy realization of some linear function of time and possibly other covariates. Further, every individual patient has some deviation from
the global behavior. In the HIV/AIDS case, every patient has a different smooth underlying true trajectory, and their observed white blood cell counts are noisy measurements of this true trajectory.
We want to first estimate the average trajectory, described by the fixed effects or global parameters . Then we estimate the individual specific deviations from the average trajectory, described by
the random effects or latent variables per patient . These definitions of fixed and random effects come from the biostatistics literature: in econometrics fixed and random effects mean something
The Model
Scalar Representation
Assume that we have patients or participants in a study, and for each participant, we have observations, where the indexing orders them in time. To start with we look at the case where
non-transformed time is the only covariate. Let be the health measurement value for patient at observation : this could be the CD4/white blood cell count in AIDS. Then the LMM takes the form
• Let . Then .
• Let . Then
• We also assume that all random effect and error vectors are independent from each other. That is, are independent. This is saying that within a patient, random effects are independent of errors,
and that we have independence across patients.
Here are the fixed effects, and are the random effects, with and . The fixed effects can be thought of as the average effects across patients (global parameters in machine learning terminology),
while the random effects are the patient specific effects (latent variables per person). The term is called the random intercept, and is the random slope.
As you may have noticed, we made normality assumptions. In the standard linear regression model with only fixed effects and without random effects, normality assumptions are not necessary to estimate
, although they are necessary for confidence intervals when the sample size is small. However, in the mixed model setting, while they are not necessary to fit , they are necessary to derive
predictions for , as we shall see.
Vector and Matrix Representation
We can extend this to a vector-matrix representation along with more features. In the longitudinal case these features may be rich basis functions of time. Let be the covariate vector for fixed
effects for patient at observation , along with a in the first position. Let be the same for random effects. In many cases, both of these are simply basis functions of . We can then define
Here is the design matrix for for the fixed effects and is the design matrix for random effects. Let be the vector of all observations, where , and similarly for . Further, let
Then the model is
where we make the assumptions
Random Intercept
In a longitudinal setting a random effect which is not associated with a covariate is called the random intercept, and has an important interpretation. It is the deviation from average for the ‘true’
trajectory when they’re first observed. In the HIV/AIDS case, this is how much a patient’s starting health values deviates from average when they enter the study.
Figure 2: an illustration of random intercepts. The solid orange line represents the average patients, while the dotted lines represent individual patients. The random intercepts describe their
deviation from the average at time , or more generally when all covariates have value .
Random Slope
In the case of a single covariate, particularly non-transformed time, we can call the other random effect the random slope. This describes how the rate of evolution of the health process deviates
from the average person.
Figure 3: an illustration of random slopes. The solid orange line represents the average across patients, while the dotted lines represent individual patients. The random slopes describe their
deviation from average in terms of how time affects the measurements.
Estimation and Prediction
We first describe how to estimate when both the covariance matrices for random effects and for errors are known. Then we describe how to predict . In practice, and are not known, so we then show how
to estimate them. Finally, we show how to fit confidence intervals.
Known Covariance Matrix
Expressing the Linear Mixed Model as Gaussian Linear Regression
We can express the model described in part I with fixed and random effects as a Gaussian linear regression problem with only fixed effects (no random effects). Here, however, the covariance matrix
between error terms is not only heteroskedastic, but non-diagonal. To see how, consider
Then since , we have
With and , we have a Gaussian linear regression model. However, the linear regression assumption of independent homoskedastic (equal variance) errors is violated. In particular, for the covariance ,
we can have both non-identical diagonal elements (heteroskedasticity) but also correlation between elements of . We call this Gaussian linear regression setup the marginal model.
Estimating : Generalized Least Squares
While ordinary least squares is not the best linear unbiased estimator (BLUE) in the presence of correlated heteroskedastic errors, another estimation technique, generalized least squares (GLS), is.
Let be the covariance matrix for . Then the GLS estimate is given by
For a derivation of this estimator, see http://halweb.uc3m.es/esp/Personal/personas/durban/esp/web/notes/gls.pdf.
Since is a random or local latent vector rather than a fixed parameter, we want to predict rather than estimate it. One prediction we can use is . We can calculate this in three steps:
1. Note that and are jointly multivariate normal
2. Compute the covariance between them and get the full joint distribution of the random vector .
3. Use properties of the multivariate normal to get the conditional expectation
In order to compute the covariance between them, note that
This gives us that .
We can then use the conditional expectation for a multivariate normal (see https://stats.stackexchange.com/questions/30588/deriving-the-conditional-distributions-of-a-multivariate-normal-distribution
) to obtain
This is the best linear unbiased predictor (BLUP). To see a proof, see https://dnett.public.iastate.edu/S611/40BLUP.pdf. Next we show what to do when the covariance parameters are unknown.
Unknown Covariance Matrix
Maximizing the Likelihood
One way to estimate is to maximize the likelihood. Recall that we have the marginal model with , where is the design matrix for random effects. Let . We can assume that both and are parametrized by
variance components , so that
We can then consider the joint log-likelihood of and , given by
however we want to estimate , so is a nuisance parameter. It is difficult to maximize the joint likelihood directly, but there is an alternate technique for maximizing a joint likelihood in the
presence of a nuisance parameter known as the profile likelihood. The idea is to suppose that is known, and then holding it fixed, derive the maximizing . We can then substitute with in the
likelihood, giving us the profile log-likelihood , which we maximize with respect to . The profile likelihood is the joint likelihood with the nuisance parameter expressed as a function of the
parameter we want to estimate. As in the above, often this function is the likelihood maximizing expression for the nuisance parameters.
Let be the likelihood with fixed. Overall, we are evaluating the following
This gives us the maximum likelihood estimate . For more on the profile log-likelihood, see https://www.stat.tamu.edu/~suhasini/teaching613/chapter3.pdf.
It turns out that the maximum likelihood estimate for is biased. However, an alternative estimate, the restricted maximum likelihood, is unbiased.
Restricted Maximum Likelihood
The idea for the restricted maximum likelihood (rMEL) for LMM is that instead of maximizing the profile log-likelihood, we maximize the marginal log-likelihood, where we take the joint likelihood and
integrate out. Note that this is not the standard intuition for rMEL, but it works for Gaussian LMM. For a more general treatment, see http://people.csail.mit.edu/xiuming/docs/tutorials/reml.pdf.
Let be the joint likelihood of and . Then we want to maximize
which is given by (see http://www2.stat.duke.edu/~sayan/Sta613/2017/lec/LMM.pdf for a full derivation)
maximizing this gives us the restricted maximum likelihood estimate , which is unbiased.
Putting it all Together to Estimate and predict
Finally, to estimate , we estimate the covariance matrix from the above, and then estimate as
Similarly, to predict , we have
Confidence Intervals
Marginal Confidence Intervals for
Since and , we can derive the covariance matrix for .
In OLS, the variance of the estimator is a function of the true variance. Here, it is a function of both the true covariance and the estimate of the covariance. Further, in OLS we have an estimator
for the error variance that is a scaled , which allows us to compute exact marginal confidence intervals under Gaussian errors. We don’t have this here (as far as I know), so we use approximate
marginal confidence intervals
This will tend to underestimate since we didn’t explicitly model the variation in the estimated variance components . However as the sample size grows this will become less of an issue.
Hypothesis Test
We can test the null hypothesis that an individual coefficient is against the alternative that it is non-zero. That is, test versus . Let . We reject the null if and only if
Unfortunately (as far as I know), the finite sample properties of this hypothesis test are not fully understood.
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New Brooklyn Eruv: Time to Accept?
Viewing 50 posts - 351 through 400 (of 442 total)
• Author
• August 1, 2023 7:35 am at 7:35 am #2212544
“4) What is your argument about Rav Moshe:
A – Rav Moshe would permit today’s eruv even though he didn’t permit the eruv in the 70’s.
B – Rav Moshe only forbade then because of wrong information.
C – Rav Moshe never forbade any eruv.
D – Rav Moshe is unclear on the matter.
E – Rav Moshe didn’t write exactly why this eruv is no good and that is enough to move on.’
Wow. You simply don’t get it. The answer is B
August 1, 2023 7:46 am at 7:46 am #2212547
“5) Rav Tuvya’s story clearly shows that there were meetings about eruvin in the 50’s. This is what I was posting about the eruv vaad. Rav Aaron and Rav Moshe were trying to avoid specific
pitfalls. You had enough chances to call out eruvin that aren’t properly constructed or maintained. It is all the same issue and issur. Whatever the reason that an eruv becomes passul, it’s the
same problem. You want to throw around that one has to know all the halachah to be part of the conversation, but somehow can’t imagine why real life eruvin would be problematic.”
Rav Tuvia’s story does not demonstrate anything. Fiction. There was no vaad, only a pro eruv vaad. There were some meetings in the Agudas Harrabanim. No Rav Aharon and Rav Moshe’s arguments where
regarding reshus harabbim, not regarding the possibility that an eruv would become passul. Stop making arguments that you can’t support. The claim that an eruv would become passul, is as good an
argument as we should stop eating food because the hechsherim can become lax and allow treifos. You wouldn’t know a kosher eruv from a passul eruv if I hit on the head with it.
August 1, 2023 7:47 am at 7:47 am #2212551
“6) Brooklyn is more than 12 mil. Rav Moshe never meant that Boro Park alone or Flatbush alone have 600k. Any statician would laugh at the idea of Rav Moshe’s teshuvah being a source for an exact
number. Nowhere does Rav Moshe say to take a count.
At some point I’ll just give up on trying to get answers and I will post the whole thing myself.”
Rav Moshe repeated it twice, he clearly meant it but was mislead. Please, the entire teshuvah (87-88) is regarding counts. Please stop making up arguments on the go. Learn the inyan prior to
making grand statements. There is no doubt from you arguments that it wasn’t that you were offline, but only that you tried to learn the inyan a little, but unfortunately failed miserably. You
will not give up trying to answer, you do not have what to answer. Learn the inyan.
August 1, 2023 2:36 pm at 2:36 pm #2212697
Dear Youdont,
I do not agree with anti eruv group. They are going to reimagine everything just to get their point across. But they never learned enough to make their own points. So they will just go around in
circles, destroying whatever they are entrusted with.
But there is something else going on here. And you blissfully ignorant of it. Or doing a great job of faking it. You are throwing every half source at the issue to try to force it. Never mind
that the petitioner didn’t care for any sources. He would carry as long as we (Or his real world associates.) don’t attack him for it. He is not here for the Torah at all. Neither are you.
Rav Shmuel Birnbaum, Rav Yisroel Belsky, and Rav Aahron Shecter, could have went to speak with Rav Elyasiv in person and he wouldn’t have felt the need to go along with them. No American Rabbonim
where able to pressure him. There was a lot of people keeping him away from the public. Maybe you are lying on purpose. I can’t tell anymore.
Rav Elyashiv paskened against the eruv because of the only person who he esteemed for his Torah Knowledge. That is Rav Dovid Feinstein. Rav Dovid almost never got involved in halacha battles.
Here he even brought in Rav Elyashic and Rav Chaim. He was very confident that he was right. And you shrug. Until you claim that he was mislead or whatever.
August 1, 2023 4:18 pm at 4:18 pm #2212731
Typical. You pick one point to argue. If I were to go back and collect all the arguments that I set forth that you did not address it would be illuminating how weak your arguments truly are.
At the minimum Rav Elyashiv did not hear from the pro side, so his statement is not complete. If Rav Belsky is any proof, there is no doubt that these rabbanim did not know, or did not follow Rav
Moshe’s shitos. Hence, their arguments are not relevant to our debate. In any case, you cannot answer Rav Dovid’s inconstancies, and are throwing up smokescreens. You are not throwing half
sources, you have no sources.
August 1, 2023 5:11 pm at 5:11 pm #2212743
Rav Dovid was just living and you could have had your answers.
But that would assume that you are open to being taught something.
Instead you have the audacity to insinuate that Rav Dovid didn’t know what he was talking about. But still Rav Elyashiv was mislead by it. Anybody who knows how to learn anything at all, can see
what a fool you are.
I give up trying to get any answers out of you. You are the toughest poster I ever interacted with on this site. Even worse than Health who I miss dearly.
August 1, 2023 8:54 pm at 8:54 pm #2212759
“I give up trying to get any answers out of you. You are the toughest poster I ever interacted with on this site.”
We “argue from authority” in the Orthodox world; that authority being our poskim. He seems to be wanting to engage in a rational debate in which such arguments are considered a logical fallacy.
He wants this to be purely his knowledge of eruvin vs. your’s, whereas we are used to arguing by bringing down lists of poskim who agree with us.
I don’t think anyone actually cares whether he knows more about eruvin than the rest of us combined (well, he cares). At the end of the day, it’s not how the world works. We can’t just ignore our
poskim and claim they were “mislead” or didn’t know what they were talking about every time they say stuff we don’t like. He’s clearly very frustrated that the Orthodox world doesn’t work this
way, and will start calling everyone am haartzim when they point it out. This thread has been a roller coster of emotions, and I’m actually at the point where I feel kind of bad for him.
August 1, 2023 9:00 pm at 9:00 pm #2212792
No, Rav Dovid refused to answer regarding the eruv. Instead he wrote about 1978-79, and did not mention the current Flatbush eruv. Never insinuated anything. Stop insinuating that I said
something that I never did. No, you got many answers from me, but I did not get any from you. You continuously come back with ancillary arguments, as you cannot not answer anything of substance.
You simply do not know the inyan. Stay out of issues you know nothing about. As I said, if one where to go through the entire thread they will see that you never answered the core issues.
August 1, 2023 11:50 pm at 11:50 pm #2212833
Neville Chaim Berlin, n0mesorah,
The two of you are jokesters. You pontificate on every issue as if you are betoch hadvarim, but when you are called out, instead of admitting that you have no shychus to the inyan, you continue
to argue. The joke is that you claim to have the poskim on your side, when in fact the overwhelming majority of poskim disagree. The only posek that I argue was mislead was Rav Moshe, and this
demolishes your house of cards. Since the majority of those opposing the eruv were just following his lead (and do not know the inyan). The fact that these rabbanim would argue against an eruv
consisting of mechitzos demonstrates that they either do not know the inyan or are not interested in the emes.
You are the one that is frustrated, because regarding every issue that you debate you try to argue with lists, but regarding this issue the list of authority would not be on your side (you simply
did not realize). The two of you have made many statements that are clearly incorrect. I am not trying to demonstrate my knowledge of the inyan. It is only that you make statements as if you know
what you are talking about. I feel bad for you as you are not capable of being modeh al haemes, that you are out of your league and should stay out of the matter.
August 20, 2023 10:24 pm at 10:24 pm #2217963
For posterity alone I will post what I know of Rav Moshe’s position. This is from learning the teshuvos with those who had some relationship with him. I have no idea how Rav Moshe dealt with
precedence. I did the learn the topic from beginning to the end. And of course there are great Rabbonim who are far more versed than me in Iggros Moshe. I’m sure they learn it differently (Read,
better.) than me, but of those that I know they all concur that the eiruv in Flatbush is not in accordance with Rav Moshe’s viewpoints.
There are two opinions of reshus harrabim in the S”A. (OC 345:7) The first opinion states that markets and forums that are 16 amos wide (…sic…) the second opinion adds that if 600,000 don’t pass
in it every day it is not a rsh”r.
There are numerous differences between these statements. Rav Moshe insisted that the only machlokes is about this minimum number. Three things come out from this. 1) There is no clear answer on
what the first opinion holds is the minimum. 2) There is no differences that are dependent on this one. 3) What makes a rsh”r is the use of the area and not is shape and size.
So it follows that there are three psakim. 1) Without 600,000 we can always build an eiruv. And that has been the minhag as well. 2) The poskim that worked out instances of the Mishkanos Yaakov
lekulah are not to be relied upon. That wasn’t the minhag. 3) The problem of d’oraysa is not about the circumstances of the town as much as the comprehensive area.
Rav Moshe starts his teshuva dismissing using the Els as a mechitza. There is a long discussion about Manhattan being surrounded by good mechitzos and the only question being the the bridges
coming over them. Rav Moshe divides it into three shittos. 1) The Ri that it is not included in the partitions. But it is not a rsh”r at all. 2. Tosofos that it is included. 3. The Rosh the
partitions are not including on top of the bridge at all. There are several possibilities. It could be a rsh”r. Or not, but still need a door that closes. Or a door at each end. (As I recall,
there is another Rosh involved here.) Possibly even locked doors at each end. Rav Moshe works out some kulos even in this opinion. But he ends of that it would need doors to the bridges and they
should be locked. This is about half of the teshuva. Any refutation of Rav Moshe’s shitta would take place on this discussion.
Rav Moshe explains that it would further depend if the bridges are included in the city than they would all be rsh”r because of the city according to all shittos. But Rav Moshe reiterates that
perhaps the bridges are outside the city and a tzuras hapesach would be enough even according to the Rosh.
There is also the opinion that even full mechitzos are not enough for an open area where the public gathers. Rav Moshe expands this opinion and then says his shtikkel about Yerushalayim.
Yerushalayim was fully enclosed but yet had a rsh”r inside it. This seems to uphold this opinion. And it is clear that there were times that putting up an eruv in a metropolis is not a given. And
this that the minhag is to be zealous about putting up eruvin that is because they understood that there needs to be 600,000 for a rsh”r. All eruvin that were put up were in cities without
600,000. And we have no minhag to be lenient about this. This is clearly Rav Moshe’s psak not to put up an eruv without precedence.
Any attempt to explain away Yerushalyim’s lack of eruv would be applied to Manhattan. An additional matter of the max size of an eruv is mentioned. This would have impact encompassing the bridges
an extending the eruv indefinitely beyond the city. He tehn writes off only counting certain types of people for 600,000. Excluding cars and trains. And rejects the theory of needing perfectly
straight streets. As is was never really about the streets in the first place.
Rav Moshe signs off that even with locked doors there would be a problem of reshus harrabim in Manahattan and even in Brooklyn. According to some opinions it would still be a karmelis. There is
no precedence against these opinions. And even if would disagree, there would be a the issue with the bridges. And even if that would be settled, ther would still be an issue because some would
think that they can make an eruv in Brooklyn. And even if they fix in Brooklyn there would be places that would not be able to, or not care to build a proper eruv. And perhaps for that alone
there should be no eruv even in Brooklyn alone.
One would really wonder what Rav Moshe’s opinions where regarding eruvin in Brrooklyn.
August 20, 2023 11:32 pm at 11:32 pm #2217976
Mods, this post is the second to the long post.
Well,within twenty years after the above was published, an eruv was put up in Flatbush and people were saying that it was with Rav Moshe’s agreement. Rav Moshe had given his agreement to Kew
Garden Hills on their eruv. A quick look at a map of the area would show none of the issues that Rav Moshe discussed. Also Rav Moshe put out a letter with four clear conditions.
1. All the Local Rabbonim agreed. (This has happened almost nowhere else.)
2. Excluded the highway.
3. It is difficult for it to become ruined.
4. there is a designated person to check it every Friday.
The letter ends of that it is not comparable to why there is no eruv in the city.
Somehow, there seemed to be an honest question of what Rav Moshe really held. Rabbonim came talk to him, and he was surprised that they weren’t familiar with his printed teshuva. Sounds familiar?
August 21, 2023 10:20 pm at 10:20 pm #2218271
Wow it took you so long to learn the Rav Moshe’s teshuvos, and you still do not know what you are talking about
August 22, 2023 8:15 am at 8:15 am #2218283
“There are two opinions of reshus harrabim in the S”A. (OC 345:7) The first opinion states that markets and forums that are 16 amos wide (…sic…) the second opinion adds that if 600,000 don’t pass
in it every day it is not a rsh”r.
There are numerous differences between these statements. Rav Moshe insisted that the only machlokes is about this minimum number. Three things come out from this. 1) There is no clear answer on
what the first opinion holds is the minimum. 2) There is no differences that are dependent on this one. 3) What makes a rsh”r is the use of the area and not is shape and size.
So it follows that there are three psakim. 1) Without 600,000 we can always build an eiruv. And that has been the minhag as well. 2) The poskim that worked out instances of the Mishkanos Yaakov
lekulah are not to be relied upon. That wasn’t the minhag. 3) The problem of d’oraysa is not about the circumstances of the town as much as the comprehensive area.”
What a bunch of gibberish. You are adding arguments to what Rav Moshe wrote.
Rav Moshe mentioned that we follow the vyesh omrim, lchatchilah. The rest of what you wrote is meaningless, and mostly incorrect. E.g. “The poskim that worked out instances of the Mishkanos
Yaakov lekulah are not to be relied upon. That wasn’t the minhag.” Huh, the Mishkenos Yaakov was lechumrah. E.g. “The problem of d’oraysa is not about the circumstances of the town as much as the
comprehensive area.” Huh, its about the number of people over an area of 12 mil by 12 mil, the area of the deigalei hamidbar.
August 22, 2023 8:15 am at 8:15 am #2218284
“Rav Moshe starts his teshuva dismissing using the Els as a mechitza. There is a long discussion about Manhattan being surrounded by good mechitzos and the only question being the the bridges
coming over them. Rav Moshe divides it into three shittos. 1) The Ri that it is not included in the partitions. But it is not a rsh”r at all. 2. Tosofos that it is included. 3. The Rosh the
partitions are not including on top of the bridge at all. There are several possibilities. It could be a rsh”r. Or not, but still need a door that closes. Or a door at each end. (As I recall,
there is another Rosh involved here.) Possibly even locked doors at each end. Rav Moshe works out some kulos even in this opinion. But he ends of that it would need doors to the bridges and they
should be locked. This is about half of the teshuva. Any refutation of Rav Moshe’s shitta would take place on this discussion.
Rav Moshe explains that it would further depend if the bridges are included in the city than they would all be rsh”r because of the city according to all shittos. But Rav Moshe reiterates that
perhaps the bridges are outside the city and a tzuras hapesach would be enough even according to the Rosh.”
Correct, but Rav Moshe allows that if the tzuras hapesach is erected in the reshus hayachid it would be sufficient. His issue with the bridges in Manhattan is that they run outside of the
mechitzos, and so the integral tzuras hapesach on the bridges is not included in the mechitzos encompassing the island. Hence, tzuras hapesachim which enclose an area, which is encompassed by
mechitzos, are sufficient. That is exactly what was done in Brooklyn. As to your comment, “Any refutation of Rav Moshe’s shitta would take place on this discussion,” in fact this is another
chiddush of Rav Moshe. Very few poskim would agree with him. On the contrary, most poskim maintain that tzuras hapesachim would suffice for a karmelis.
August 22, 2023 8:19 am at 8:19 am #2218298
“There is also the opinion that even full mechitzos are not enough for an open area where the public gathers. Rav Moshe expands this opinion and then says his shtikkel about Yerushalayim.
Yerushalayim was fully enclosed but yet had a rsh”r inside it. This seems to uphold this opinion. And it is clear that there were times that putting up an eruv in a metropolis is not a given. And
this that the minhag is to be zealous about putting up eruvin that is because they understood that there needs to be 600,000 for a rsh”r. All eruvin that were put up were in cities without
600,000. And we have no minhag to be lenient about this. This is clearly Rav Moshe’s psak not to put up an eruv without precedence.”
Major conflation of the issues. Rav Moshe maintains (O.C. 1:139) that mechitzos would not be sufficient for sratyas. Rav Moshe also states that we do not have a minhag regarding sratyas in cities
containing shishim ribo. Rav Moshe argues that they did no establish an eruv for Yerushalyim even though it is a chiyuv to establish one, was because they did not want that those who come to
Yerushlayim should establish eruvin in cities that are prohibited. Manhattan he compares to Yerushalyim, and Brooklyn he says is possibly like Yerushalyim regarding this issue.
Later Rav Moshe writes (O.C. 4:87-88, and Hapardus) that the minhag was to establish eruvin in cities containing shishim ribo, and hence, he sets forth his chiddush of three million over a 12 mil
by 12 mil area. He also admits that the population of a 12 mil by 12 mil area is reckoned for a srtaya, and not the entire city. Rav Moshe also writes regarding Manhattan that once the rabbanim
establish an eruv he would not make use of his argument regarding Yerushalyim.
There is no three million people over an area of 12 mil by 12 mil in Brooklyn. There is no sratya that services a 12 mil by 12 mil area that contains shishim ribo. If Manhattan after the
establishment of the eruv, Yerushalyim was not a concern, how much more so regarding Brooklyn where he questioned if it can even be compared to Yerushalyim.
[Rav Moshe’s argument regarding Yerushalyim is puzzling. In fact there is no ciyuv to make an eruv for an entire city, only maavaos and chatzeiros. Some Rishonim actually state that an eruv was
made for the maavaos and chatzeiros of Yerushlayim.]
August 22, 2023 8:20 am at 8:20 am #2218302
“1. All the Local Rabbonim agreed. (This has happened almost nowhere else.)”
Right, and all the rabbanim agreed in KGH. Actually the Minchas Asher did not agree. If Rav Moshe was so concerned with a consensus, why did he not tell the rabbanim in Flatbush that how can they
make an eruv there is no consensus (see the beginning of 4:87). How could Rav Moshe have agreed that the rabbanim of Manhattan have a right to establish an eruv, when there was no consensus, and
he himself did not agree (4:89).
“3. It is difficult for it to become ruined.”
Rav Moshe said the same regarding Seagate, and in fact it did get ruined.
Keep on learning the inyan maybe you will figure it out
August 29, 2023 12:55 pm at 12:55 pm #2220904
@Hoo Hoo
A reshus harabim deOraisa cannot have an eiruv, even with 3 mechitzos around it.
August 29, 2023 3:44 pm at 3:44 pm #2220997
Rav Moshe addresses the newly built Flatbush eruv that he hadn’t wanted to get involved in the project because there is many opinions on what is reshus harrabim and what is dalsos neulos and they
can always consult the seforim instead of him. But once it was publicized that Rav Moshe was the one who permitted the eruv because of the previous sentence, he felt compelled to respond with his
personal opinion as it was already laid out in his first teshuva.
Rav Moshe is trying to make five public points:
1) There is rational for an eruv in Flatbush.
2) The reasoning is evident in the seforim.
3) It is not Rav Moshe’s opinion to put an eruv in Flatbush.
4) Rav Moshe’s own opinion is clear from his first teshuvah.
5) People have difficulty reconciling 1 and 2 with 3 and 4.
This really is the whole story. Rav Moshe saw that people are misinterpreting what he he clearly wrote and said, and responded just to refute what they were stating in his name.
The debates that follow to our day are not really about halachah. The center of the debate since the late Seventies was, is Rav Moshe entitled to his opinion. Rav Moshe himself held he was, but
others would say that the majority disagreed with him. And even Rav Moshe himself may have agreed that he wasn’t entitled to his own opinion.
Rav Shmuel Birnbaum was very bothered by this attitude. But not everybody should be a masmid like Rav Shmuel Birnbaum. He was enough for the whole Flatbush.
All the more so, to imagine what Rav Moshe was up against forty years ago, when rabbis still spoke about business acumen as a qualification for being considered an educated Jew.
August 29, 2023 4:15 pm at 4:15 pm #2221011
Rav Moshe explains the 600,000 opinion as a daily fixed number for a stand alone road, like a highway. And for a city it would be the amount of people that are around and about in the streets.
Meaning, that Rav Moshe’s opinion is that the number is only about reshus harrabim itself and nothing else. So for a city, those that are inside wouldn’t contribute to 600,000.
Rav Moshe says that the population would be estimated to produce the amount on the streets, and it is likely that different cities vary on this calculation. Also, the area would be 12 mil square
[slightly over 7 miles] (less than 9 miles) like it was in the dessert encampment. So then if one would measure in such an area enough people to have 600,000 in the streets then it would be a
reshus harrabim for sure. [And Rav Moshe would have told the Rabbonim that there is no way to make an eruv without dalsos and so on.] But if there is a street anywhere in this vicinity that
carries 600,000 by itself, than that street alone would be a reshus harrabim for sure.
Rav Moshe has no intention of counting the city as was evident in his first teshuva. But if one would count and have 600,000 in the streets than Rav Moshe would have forbade the eruv. And it
would be null according to his opinion. Some people use this paragraph as way to calculate, but that is not the point. Such would only achieve that it is not a reshus harrabim beyond any doubt.
It wouldn’t rule out a safek doraissa and it ignores all the nuance of the first teshuva. Additionally, there is no statistical rule here. Rav Moshe lists five different activities to count. In
two different places. And three estimates. Since it has no practical application, there is no reason to be clear about it. The point is to get the idea.
August 29, 2023 4:43 pm at 4:43 pm #2221014
Rav Moshe continues that counting is a chumrah that there would be a definite reshus harrabim, and a kulah that in a place where there isn’t enough people to have 600,000 on the streets we would
be able to build an eruv. But then he goes on to say how none of this applies to Manhattan and Brooklyn. There we would always assume there is enough to get 600,000 on the streets. Then he goes
on to be lenient if there is a larger area. But doesn’t mention the maximum size like he did in the first teshuva.
There is two ways to understand Rashi in Meseches Eruvin. 1) That any city where there resides 600,000 people is always a rsh”r. Or 2) that if people enter the city daily they contribute to this
number. Rav Moshe rejects the first reason because the minhag is to build eruvin even in cities with this number. So he goes with the second explanation. And it follows that commuters count
toward this number.
August 29, 2023 5:12 pm at 5:12 pm #2221016
Rav Moshe makes some additional side points. These may have come from the in person discussions. And then reiterates why he didn’t stop them even though he disagrees with putting up the eruv. His
wording is, ‘that it is against what he holds is the law’.
Rav Moshe Feinstein emphatically giving his opinion was not a game changer for eruvin. A year later, he wrote another teshuvah. It is just his reasons to not build an eiruv in Flatbush.
1) The assumption is that there is 600,000. One would have to prove that there isn’t. And even if they do, people won’t know about their proofs. And similar considerations.
2) The Rashba about public open places (platya). This could be even if the total population is 600,000. There isn’t a clear precedent for this complication.
3) The area can not be measured randomly. It would start at an edge. In Brooklyn’s case this would be the beach and the river. (Then he discusses if the old Coney Island would be a problem since
it was only in the summer.) But it would still be the problem of people aren’t aware of the area calculations either.
Then Rav Moshe writes that all three reasons are valid even according to what they told him that there are not 600,000 in Brooklyn. Meaning, that there is no way to build an eruv in Brooklyn even
if there is not for sure a rsh”r. It simply runs into too many problems that we don’t have a precedent for. So Rav Moshe held not to put up an eruv in Flatbush and avoid the problems. Which is
comparable to Yerushalayim.
Then Rav Moshe mentions the city map that would make Brooklyn a rsh”r with Manhattan. But the river should separate Brooklyn from Manhattan. But still it would be a rsh”r doraisa according to
that map.
For anybody who knows what happened next, Rav Moshe did not protest the eruv based on this map.
August 29, 2023 10:06 pm at 10:06 pm #2221051
Then you have to know what is “Tzidei Reshus Harabim KeRH”R.” The Chazon Ish has a drawing/shitah on this in his sefer, that would extend Ocean Parkway quite some distance of side streets to side
It is also well known that R’ Ahron Kotler z”l would not allow an eiruv in Lakewood because of the amount of traffic on Rt. 9.
August 29, 2023 10:07 pm at 10:07 pm #2221067
“@Hoo Hoo
A reshus harabim deOraisa cannot have an eiruv, even with 3 mechitzos around it.”
So sorry you are simply incorrect. Most poskim maintain that an area encompassed by mechitzos, would be classified as a reshus hayachid, notwithstanding a reshus harabbim contained therein. Most
poskim maintain that even a tzuras hapesach would reclassify me’dOraysa a reshus harabbim as a reshus hayachid.
August 29, 2023 10:09 pm at 10:09 pm #2221069
“Rav Moshe addresses the newly built Flatbush eruv that he hadn’t wanted to get involved in the project because there is many opinions on what is reshus harrabim and what is dalsos neulos and
they can always consult the seforim instead of him. But once it was publicized that Rav Moshe was the one who permitted the eruv because of the previous sentence, he felt compelled to respond
with his personal opinion as it was already laid out in his first teshuva.”
After trying to give a running commentary of Rav Moshe’s teshuvos, the fact that you leave out pertinent points demonstrates that you either don’t get it, or that you learn lekanter. E.g., you
omitted that Rav Moshe did not want issue a p’sak din barrur, because he knew that he was mechudash, and was going against the poskim.
“4) Rav Moshe’s own opinion is clear from his first teshuvah.”
You are discombobulated. The teshuvah prior is regarding Manhattan (you cannot be referring to 1:138, since that has nothing to do with reshus harabbim). Rav Moshe needed to formulate, at this
time, his opinion, since Manhattan’s metzious was unlike Brooklyn.
August 29, 2023 10:14 pm at 10:14 pm #2221070
“The debates that follow to our day are not really about halachah. The center of the debate since the late Seventies was, is Rav Moshe entitled to his opinion. Rav Moshe himself held he was, but
others would say that the majority disagreed with him. And even Rav Moshe himself may have agreed that he wasn’t entitled to his own opinion.
Rav Shmuel Birnbaum was very bothered by this attitude. But not everybody should be a masmid like Rav Shmuel Birnbaum. He was enough for the whole Flatbush.
All the more so, to imagine what Rav Moshe was up against forty years ago, when rabbis still spoke about business acumen as a qualification for being considered an educated Jew.”
What a bunch of gibberish. No one upheld that Rav Moshe was not entitled to his opinion.
August 29, 2023 10:14 pm at 10:14 pm #2221073
“Rav Moshe says that the population would be estimated to produce the amount on the streets, and it is likely that different cities vary on this calculation.”
No, Rav Moshe at the end of the day did give a number. You do not realize that you are not making any sense. Rav Moshe’s final number is three million people. If the population of a large city is
any less, then one may think that it’s a reshus harabbim, and that is why they should not establish an eruv. According to you, Rav Moshe should have argued, when he was made aware that Brooklyn
and Detroit has less than three million (and the reason not to make an eruv was because one may think that it was a reshus harabbim), that it is irrelevant. Since, “different cities vary on this
“Also, the area would be 12 mil square [slightly over 7 miles] (less than 9 miles) like it was in the dessert encampment. So then if one would measure in such an area enough people to have
600,000 in the streets then it would be a reshus harrabim for sure. [And Rav Moshe would have told the Rabbonim that there is no way to make an eruv without dalsos and so on.] But if there is a
street anywhere in this vicinity that carries 600,000 by itself, than that street alone would be a reshus harrabim for sure.”
No. It is slightly over eight miles. Go learn Rav Moshe’s shiur amah, in regard to hilchos Shabbos. Since it is impossible to know how many people are actually in the streets at one time, Rav
Moshe resorted to giving a number, three million, the amount of people in the midbar. You are avoiding this fact because it makes you uneasy, and it would allow an eruv in Brooklyn, if the
metziuos is otherwise.
August 29, 2023 10:14 pm at 10:14 pm #2221074
“Rav Moshe has no intention of counting the city as was evident in his first teshuva. But if one would count and have 600,000 in the streets than Rav Moshe would have forbade the eruv. And it
would be null according to his opinion.”
No. Rav Moshe did not need to count the numbers, because his first teshuvah was regarding Manhattan, where he realized that it was encompassed by mechitzos, and so the numbers are irrelevant.
Only regarding Brooklyn did he need to calculate.
“Some people use this paragraph as way to calculate, but that is not the point. Such would only achieve that it is not a reshus harrabim beyond any doubt. It wouldn’t rule out a safek doraissa
and it ignores all the nuance of the first teshuva. Additionally, there is no statistical rule here. Rav Moshe lists five different activities to count. In two different places. And three
estimates. Since it has no practical application, there is no reason to be clear about it. The point is to get the idea.”
Absolute gibberish. No. Rav Moshe’s teshuvos are only about reshus harrabim, and that some may perceive that it is a reshus harabbim, and not about safek d’Oraysa. It has everything to do with
statistics. If an area of 12 mil by 12 mil has a population of three million, then Rav Moshe maintained that it is a reshus harabbim. If a large city contained less than three million than he
would not recommend an eruv be established (Rav Moshe never argued that it is a safek d’Oraysa, only that one may think that it is a d’Oraysa, hence it is more like a gezeira not to make one). A
sratya would require 600,000 traversing therein to be classified as a reshus harabbim. You do not, “get the idea.”
August 29, 2023 10:14 pm at 10:14 pm #2221081
“Rav Moshe continues that counting is a chumrah that there would be a definite reshus harrabim, and a kulah that in a place where there isn’t enough people to have 600,000 on the streets we would
be able to build an eruv. But then he goes on to say how none of this applies to Manhattan and Brooklyn. There we would always assume there is enough to get 600,000 on the streets. Then he goes
on to be lenient if there is a larger area. But doesn’t mention the maximum size like he did in the first teshuva.”
Nonsensical. Manhattan, according to Rav Moshe is not a reshus harrabim because of 600,000 on the streets, since it is encompassed by mechitzos (only regarding bridges, which are not included in
the mechitzos, does he discuss numbers). Brooklyn, according to Rav Moshe had three million over 12 mil by 12 mil, and hence is a reshus harabbim. Even if Brooklyn did not contain such a number,
since one may think that it does, an eruv should not be established. However, this objection would only be a gezeirah.
August 29, 2023 10:16 pm at 10:16 pm #2221089
“Rav Moshe makes some additional side points. These may have come from the in person discussions. And then reiterates why he didn’t stop them even though he disagrees with putting up the eruv.
His wording is, ‘that it is against what he holds is the law’.”
You again omit that Rav Moshe states that he can’t issue a p’sak din barrur, because he is mechudash.
“1) The assumption is that there is 600,000. One would have to prove that there isn’t. And even if they do, people won’t know about their proofs. And similar considerations.”
You keep on omitting Rav Moshe’s clear words, three million. It is obvious that Rav Moshe’s words are difficult to stomach. Since these numbers would allow an eruv in Brooklyn (and regarding his
gezeirah he would allow a section to be demarcated with a tzuras hapesach).
“2) The Rashba about public open places (platya). This could be even if the total population is 600,000. There isn’t a clear precedent for this complication.”
No. As I mentioned previously, Rav Moshe, in the end allows that a platya is reckoned as part a 12 mil by 12 mil area, which would require a population of three million.
“Then Rav Moshe writes that all three reasons are valid even according to what they told him that there are not 600,000 in Brooklyn. Meaning, that there is no way to build an eruv in Brooklyn
even if there is not for sure a rsh”r. It simply runs into too many problems that we don’t have a precedent for. So Rav Moshe held not to put up an eruv in Flatbush and avoid the problems. Which
is comparable to Yerushalayim.”
No. The only reason not to make an eruv in Brooklyn, according to Rav Moshe, is because he thought there was a population of over three million over 12 mil by 12 mil in Brooklyn. If the numbers
are any less than it is a gezeirah, that some may think that it is a reshus harabbim. However, regarding this issue if a tzuras hapesach would encompass a much smaller area in Brooklyn, there is
no doubt that he would allow an eruv. Regarding Yerushalayim, as I mentioned, if an eruv was already constructed for Manhattan, Rav Moshe allowed, how much more so regarding Brooklyn, where he
was not even sure if it can be compared to Yerushalayim. Moreover, Brooklyn is encompassed by mechitzos, so there is no doubt that Rav Moshe would allow an eruv.
August 29, 2023 10:30 pm at 10:30 pm #2221106
“Then you have to know what is “Tzidei Reshus Harabim KeRH”R.” The Chazon Ish has a drawing/shitah on this in his sefer, that would extend Ocean Parkway quite some distance of side streets to
side streets.”
Ocean Parkway according to most poskim would not be a reshus harabbim because of its numbers. Moreover, it is not mefulash, and it is encompassed by mechitzos.
What tzidei reshus harrabim are you talking about. Furthermore, a tzuras hapesach would be sufficient.
“It is also well known that R’ Ahron Kotler z”l would not allow an eiruv in Lakewood because of the amount of traffic on Rt. 9.”
Fiction. Rav Aharon did not accept shitas Rashi at all.
It is also well known that R’ Ahron Kotler z”l would not allow an eiruv in Lakewood because of the amount of traffic on Rt. 9.
August 30, 2023 9:19 am at 9:19 am #2221153
When I said 3 mechitzos, I was referring to the 3 redundant tzuros hapesech put up by 3 independent batei dininim.
The Teshuvos HaRashba’s psak is accepted, not to use a tzuras hapesach for a RH”R deOirsissa. If shishim riboi is required to constitute a RH”R deOiraissa according to roiv Roishoinim (which I
believe is what you were touching on,) depends on how you count the shitois. This is the famous machloikes of the Mishkonois Yaakov and the Bais Ephraim. See SH”A OR”CH soman 364 se’if 2, Bais
Ephraim OR”CH siman 26 and Chazon Ish OR”CH siman 107 ois 4. [The Mishkanois Yaakov became famous with this teshuvah of his in disagreement with the renowned gadol hadol R’ Ephraim Zalman
Margoliois z”l.]
(I think?)
August 30, 2023 1:02 pm at 1:02 pm #2221333
Dear Youdont,
““4) Rav Moshe’s own opinion is clear from his first teshuvah.”
You are discombobulated. The teshuvah prior is regarding Manhattan (you cannot be referring to 1:138, since that has nothing to do with reshus harabbim). Rav Moshe needed to formulate, at this
time, his opinion, since Manhattan’s metzious was unlike Brooklyn.”
““Rav Moshe has no intention of counting the city as was evident in his first teshuva. But if one would count and have 600,000 in the streets than Rav Moshe would have forbade the eruv. And it
would be null according to his opinion.”
No. Rav Moshe did not need to count the numbers, because his first teshuvah was regarding Manhattan, where he realized that it was encompassed by mechitzos, and so the numbers are irrelevant.
Only regarding Brooklyn did he need to calculate.”
I don’t understand how you reconcile Rav Moshe writing to the Rabbanim in Flatbush (1978) that it is all printed at length in every detail in the first volume of Iggres Moshe, with your opinion
that Rav Moshe only came up with the three million number in this teshuva and then reworked a year later?
And all those who spoke in learning with Rav Moshe, testify that even to his last days he said his shita is like it is published in chelek aleph.
August 30, 2023 8:16 pm at 8:16 pm #2221451
“I don’t understand how you reconcile Rav Moshe writing to the Rabbanim in Flatbush (1978) that it is all printed at length in every detail in the first volume of Iggres Moshe, with your opinion
that Rav Moshe only came up with the three million number in this teshuva and then reworked a year later?”
Show me where Rav Moshe wrote in chelek aleph anything about the numbers which he required. They do not exist. Rav Moshe is referring to that he formulated his shita regarding shishim ribo being
conditional of 12 mil by 12 mil. However, the actual number of people residing in the area that it would require to have 600,000 people traversing the streets at one time, he did not express in
this teshuvah (since it was not nogeia for Manhattan). Rav Moshe originally wanted to argue that 600,000 people residing would be enough. In 4:87 he clearly makes this suggestion, and then argues
that the minhag was not as such. Hence, how can one say that Rav Moshe did not add to his shitos later on, regarding Brooklyn? I do not have to explain these words of Rav Moshe when in fact his
shita evolved over time. To deny that Rav Moshe’s number is three million is denying the entire Igros Moshe O.C. 4:87-88, 5:28-29.
“And all those who spoke in learning with Rav Moshe, testify that even to his last days he said his shita is like it is published in chelek aleph.”
I don’t care what people say they never learnt his teshuvos.
August 30, 2023 8:16 pm at 8:16 pm #2221444
“When I said 3 mechitzos, I was referring to the 3 redundant tzuros hapesech put up by 3 independent batei dininim.”
Huh. Actually, most poskim maintain that a tzuras hapesach is a mechitzah d”Oraysa. In any case, huh
“The Teshuvos HaRashba’s psak is accepted, not to use a tzuras hapesach for a RH”R deOirsissa. If shishim riboi is required to constitute a RH”R deOiraissa according to roiv Roishoinim (which I
believe is what you were touching on,) depends on how you count the shitois.”
Since you are not clear in what Rashba you are referring to, I will include all the possibilities.
1) You are referring to accepting the criterion of shishim ribo (and that is why you mentioned his teshuvos). You are absolutely incorrect. We do not follow the Rashba, we accept shishim ribo
lechatchilah. The Rema and all the Reshonim from Tzarfas and Ashkenaz accept shishim ribo lechatchilah. While the Rishonim argue that the Rashba did not accept the criterion, and therefore we
accept it as a given that he opposes the fundament, it is not at all clear from where we see this in his actual words (I have a lot to say about this matter but alas it is irrelevant because the
Rishonim take it as a given).
2) You are referring to a tzuras hapesach reclassifying a reshus harabbim to a reshus hayachid me’d’Oraysa. The Rashba maintains that the Chachamim maintain that a tzuras hapesach would do so.
[However, the Rashba paskens like Rav Yehudah.] In fact many poskim maintain as such, notably the Shulchan Aruch Harav.
3) You are referring to shitas HaRashba regarding platyas. In fact most Reshonim do not accept the Rashba’s shita at all. Most poskim do not follow the Rashba regarding platyas.
“This is the famous machloikes of the Mishkonois Yaakov and the Bais Ephraim. See SH”A OR”CH soman 364 se’if 2, Bais Ephraim OR”CH siman 26 and Chazon Ish OR”CH siman 107 ois 4. [The Mishkanois
Yaakov became famous with this teshuvah of his in disagreement with the renowned gadol hadol R’ Ephraim Zalman Margoliois z”l.]
(I think?)”
Their machlokas was mainly regarding four issues:
1) Do we accept shishim ribo lchatchilah?
2) Do we pasken like the Chachamim or Rav Yehudah?
3) Are pirtzos esser me’d’Oraysa or me’d’rabbanan?
4) Are delasos required for a karmelis?
The world followed the Bais Ephraim regarding all these issues. We accept shishim ribo lchatchilah. We pasken like the Chachamim. Pirtzos esser is only me’d’rabbanan A karmelis does not require
August 31, 2023 8:11 pm at 8:11 pm #2221899
Dear Youdont,
“Hence, how can one say that Rav Moshe did not add to his shitos later on, regarding Brooklyn?”
One may be inclined to say so based on Raqv Moshe writing “that it is all printed at length in every detail in the first volume of Iggres Moshe. ” Rav Moshe himself clearly writes it, but some
say (At least regarding eruvin in Brooklyn) that Rav Moshe is not entitled to his own opinion.
All this exact numbers stuff is what you say. It is nothing to do with Rav Moshe’s shitta. And is a very messy way to read the teshuvos.
I’m really sorry for you that you miss the essence of this thread and the reasons why the YV was so incensed twenty years ago. But you absolutely twisted Rav Moshe’s teshuva because you miss all
the nuance of this sugya.
Anybody who actually knows how to learn, could see that I was tipping you off to the nuance for months. You were oblivious. You still are. Or you are pretending. I don’t mind that people use the
eruv. I mind very much that people who can’t learn at all, refuse to give their attention to learn from those who do.
September 2, 2023 9:44 pm at 9:44 pm #2222108
It is amazing how obstinate you are. You clearly did not know Rav Moshe’s teshuvos, but fancy yourself as all knowing. Now you are trying to play catch up, but you have the gumption to write that
my learning of the teshuvos are messy. You have the chuzpah to argue that I do not understand the sugya, when you barely know Rav Moshe’s teshuvos, never mind the Rishonim and Achronim on the
sugya. You think that you are making strawman arguments, but in fact you are not making arguments at all. You simply have no inkling regarding the matter.
No one ever claimed that Rav Moshe is not entitled to his opinion. You can repeat this lie as much as you want it will not make it true. You cannot show me one person of repute who made this
statement. It is a figment of your imagination, because you would like it to be true.
As to your exposition of Rav Moshe’s teshuvos, let us see who is omitting pertinent parts or not. (Never mind, the numerus mistakes that I illuminated – all along- that you conveniently do not
answer. So much for understanding sugyos.)
שו”ת אגרות משה אורח חיים חלק ה סימן כח
ומווארשא שעשו שם עירובין, אף שהיה כרך גדול ידוע לפי חשיבות כרכים גדולים שברוסלאנד ופולין, לא היו בה ס’ ריבוא ברחובות שהוא רק בעיר שדרים בה ערך קרוב לשלשת מיליאן נפשות. ובווארשא לא היו שם כל כך
אינשי ואף המחצה מזה לא היו שם. ועוברים ושבים ג”כ לא היו כל כך שישלימו המספר. ובשאר הכרכים לא היו עירובין, אלא בחלק קטן בהעיר מקום שדרו שם רוב היהודים ושם לא היו ס’ ריבוא
שו”ת אגרות משה אורח חיים חלק ה סימן כט
שיהיה שייך לאמוד שימצא בחוצותיה ששים ריבוא, דהוא שייך רק כשיהיו תושבי העיר עם העוברים ושבים ממקומות אחרים לכה”פ חמשה פעמים ס’ ריבוא דהוא ערך שלשה מיליאן
יראה העם וישפוט
I reiterate, regarding Brooklyn Rav Moshe only referred to his Manhattan teshuvah, in regards to his chiddush that shishim ribo is conditional on 12 mil by 12 mil. In light of the above to argue
otherwise, demonstrates dishonesty.
Alas, I understand your issue. You do not accept that all teshuos in Igros Moshe where penned by Rav Moshe, but are afraid to utter this out of your mouth.
September 3, 2023 12:58 pm at 12:58 pm #2222344
Dear Youdont,
Cute post.
I’ll get to it all eventually.
But you really only have to answer one line in Rav Moshe’s teshuva from 1978 he writes that ” “that it is all printed at length in every detail in the first volume of Iggres Moshe.” In the first
teshuva, there is no mention of any number for a city besides 600,000. It would be here:
Clearly, Rav Moshe’s opinion is that counting the city would not change his psak. Is Rav Moshe entitled to that opinion or not?
September 3, 2023 9:30 pm at 9:30 pm #2222537
Oh, your still learning the teshuvos. But you argued in the name of Rav Moshe from the get go. If you never learnt the teshuvos, how could you have claimed to know what you are talking about?
Clearly you are not capable of being modeh al haemes.
While I answered that one line twice, you have not answered the many lines that I cited.
From Rav Moshe’s words in this teshuvah (4:87) there is proof that Rav Moshe’s shitah was not static. You do not have a better answer for the fact that Rav Moshe himself wrote that he initially
(in 1:139) only wanted to reckon those living in the 12 mil by 12 mil towards 600,000. The fact that only in this teshuvah (4:87) did Rav Moshe admit that eruvin were established in cities
containing shishim ribo, is proof that his shita evolved. Clearly, one would need to learn 4:87-88, and 5:28-29, to comprehend Rav Moshe’s final thoughts on the matter, and the Manhattan teshuvah
was not his last words on the matter.
“Anybody who actually knows how to learn,” would realize that you are in over your head. Stay out of matters that you know nothing about.
September 3, 2023 10:07 pm at 10:07 pm #2222562
Dear Youdont,
You still refused to explain how Rav Moshe could write that every detail is explained in the first teshuva and then write about a whole new system.
What is the point of arguing? What Rav Moshe held according to himself isn’t good enough for you.
September 4, 2023 2:16 am at 2:16 am #2222587
Dear Youdont,
Maybe you don’t see it, but 600,000 is the only number that is real according to Rav Moshe. For a city, that is the amount that is out and about daily on the streets. That is what he wrote in the
first teshuva. He reiterates it in the second teshuva. And he points out that it is the foundational point in the third teshuva.
All the other number are ways to come to 600,000.
And Rav Moshe writes that the other numbers are not to be relied on. Not because of a gezeira. Because those are not real numbers.
September 4, 2023 4:23 pm at 4:23 pm #2222755
This is comedy. I continuously write what Rav Moshe meant, but you avoid his clear words that I cite, and I am the one that “refuses to explain”? You hang on to a few words of Rav Moshe, and
refuse to admit that his shita evolved. What about the teshuvos that I cite, are they not Rav Moshe’s words? How utterly ridiculous is your argument. Forget knowing sugyos, how about alef bais.
September 4, 2023 4:23 pm at 4:23 pm #2222759
Correct, 600,000 people traversing the streets at one time, which Rav Moshe states happens when there is three million people over a 12 mil by 12 mil area. However, if the population is less than
three million, then if it is a large city it is only a gezeira, where some may think that it contains such a population. Clearly, Rav Moshe maintained that the number is three million [unless one
can prove that there is 600,000 people traversing the streets of such an area; however, the flipside would possibly be considered a reshu harabbim even without 600,000 in the streets, as it is
similar to the degalei hamidbar]. This number is based on the degalei hamidbar, and so Rav Moshe considered it a “real number.” Nowhere, does Rav Moshe argue that the other numbers are not real,
you simply are making things up.
I realize that many of your ilk, have an issue with the three million number, you would rather that Rav Moshe stuck to 600,000 in the city. Unfortunately for you and your friends Rav Moshe did
not want to argue with precedent. [While your ilk (would have) just lied and claimed that there was no such precedent.] Moreover, it is very difficult to accept this chiddush of Rav Mohse, which
is not mentioned anywhere else.
September 4, 2023 7:27 pm at 7:27 pm #2222822
Dear Youdont,
Do you really mean that Rav Moshe’s opinion evolved immediately after he wrote that every detail is in the first teshuva?
Or that he sat down to write a teshuva about a chiddush that he negates his first teshuva but lied about it?
September 4, 2023 11:03 pm at 11:03 pm #2222855
Your just not able to understand simple matters. No, he was not referring to the numbers required in his first teshuvah, since he never mentioned numbers there. Your being silly for hanging onto
words that are irrelevant to the conversation. Lets try explain Rav Moshe’s shita regarding shishim ribo again:
Like most poskim, Rav Moshe originally maintained (Igros Moshe, O.C. 1:109) that the criterion of shishim ribo was dependent on the street having shishim ribo traversing it. However, later
(ibid., 1:139:5) he formulated his chiddush in which shishim ribo when applied to a city was not dependent on a street but over a 12 mil by 12 mil area. Rav Moshe added that the criterion of
shishim ribo ovrim bo would require a sizable population living and commuting into the 12 mil by 12 mil area so that it could physically satisfy the condition of 600,000 people collectively
traversing its streets. However, at this time Rav Moshe did not quantifying how many people would be required to live in this 12 mil by 12 mil area.
In the first teshuvah quantifying how many people would be required to live in this 12 mil by 12 mil area, Rav Moshe stated (ibid., 4:87) that since in the past eruvin had been erected in cities
with populations exceeding shishim ribo, one could not classify a city as a reshus harabbim solely on the basis of the existence of a population of 600,000. He then added that although the actual
number of inhabitants could possibly vary according to the city, in Brooklyn it would most likely require four to five times shishim ribo. In the final two teshuvos which followed regarding
Brooklyn we see that Rav Moshe codified his chiddush that the requirement is, “just about three million people,” (ibid., 5:28:5) or, “at least five times shishim ribo,” (ibid., 5:29) which could
amount to even more than 3 million people. Consequently, in the Chicago eruv pamphlet (West Rogers Park Eruv, 1993 p. 23) it is stated that Rav Dovid Feinstein shlita was in agreement that
according to his father’s shitah there must be a minimum of 3 million people in order for the city to be defined as a reshus harabbim.
September 5, 2023 12:33 pm at 12:33 pm #2223082
Dear Youdont,
“In the first teshuvah quantifying how many people would be required to live in this 12 mil by 12 mil area, Rav Moshe stated ” (I’ll continue for you.) ” he had said in person that he doesn’t
want to get involved because of all the shittos and sefarim are available, but once there are rumors that Rav Moshe is from the mattirim because of he said he is not getting involved, now he is
forced to write this letter ito directly explain his own view of the matter. And it is explained at length in his first teshuva at length in every single detail…..”
And in your own understanding, he concludes the paragraph with a completely new number, with no indication that this any new thinking. And then in the first lines of the next paragraph, he calls
it pashut and vadei reshus harrabim.
So, he didn’t have this number at the meeting. He wouldn’t have gotten involved. But now he had to explain his opinion…. But it is not the same opinion as he gave in person. It is something that
is unsourced and never heard before.
Is this how you read this?
September 5, 2023 7:39 pm at 7:39 pm #2223191
“that he doesn’t want to get involved… But it is not the same opinion as he gave in person. It is something that is unsourced and never heard before.”
No. He did not give an opinion in person. He did not want to get involved period. Now that he had to write a teshuvah regarding Brooklyn, because he was ‘told’ that there where those who said
that he allowed, he needed to clarify the matter, and write a teshuvah. The teshuvah needed to explore why Brooklyn was osser as well. Rav Moshe did not need to give numbers for Manhattan
(although he may have had it worked out at that time, and may be that’s is what he was referring to), because they were irrelevant in light of the mechitzos encompassing the island.
I am not making anything up, the fact is Rav Moshe is the one who made up these numbers. Where do you get the chutzpah to cite Rav Moshe’s opinion, and leave out his own words. Stop this
nariskeit. Furthermore, you cite the entire beginning of 4:87, and conveniently leave out that Rav Moshe did not issue a psak din barrur for Brooklyn. How deceitful of you.
September 6, 2023 1:42 pm at 1:42 pm #2223450
Dear Youdont,
1) You are still not answering how Rav Moshe can pass it off as his old opinion, if it is being publicized here for the first time.
2) Do you think that the Brooklyn and Manhattan Teshuvos in chelek aleph are not one concept, applying to both?
3) I am not leaving anything out. It doesn’t matter all the other points, if one of us has his opinion completely wrong.
4) You have been on this diyuk from the beginning so I’ll address it. Are you reading ‘psak din barrur’, to mean that Rav Moshe wasn’t clear about this? From the context, it is clear that he is
not telling them they are not allowed to follow anyone else. But if he was giving them a psak din barrur, then he is saying they would be wrong to follow other Rabbonim.
September 7, 2023 5:38 pm at 5:38 pm #2223766
1) You are either thickheaded or a liar. You are not answering Rav Moshe’s own words. I am not the one making up the numbers. So if you believe that these numbers should have been included in
chelek aleph, ask it on Rav Moshe, not me. Anyone who skips the numbers because of the few words that you are harping on, is dishonest. And yes I did offer possible answers to your irrelevant
fake argument.
2) You never learnt the Brooklyn Eruv teshuvah in chelek aleph. It simply has nothing to do with the issue at hand.
3) You again are trying to pass yourself off as one who is all knowing. You do not know the inyan and are trying to play catch up. You are not the one to arbitrate if I am wrong or not.
4) No you have continuously left out numerous issues, but this is very glaring. It is obvious that you do not learn much halacahah. It is simple. Rav Moshe maintained that his chiddush was
correct but since other poskim disagreed with him he did want to issue a clear p’sak. Hence, those following other rabbanim are not really contradicting a clear p’sak of Rav Moshe. [In any case,
even if Rav Moshe would be issuing a clear p’sak one should follow his rav notwithstanding Rav Moshe’s opinion.]
September 7, 2023 10:15 pm at 10:15 pm #2223904
I spent a Shabbat in Mexico City last year. It’s population is multiples of that of Brooklyn. The Ashkenazi community maintains an eruv and I didn’t hear anyone questioning it. Not sure whether
the Syrian community there uses it or not.
September 24, 2023 10:37 am at 10:37 am #2227531
Dear You dont,
What is the point of being evasive? Whatever your shittah is, you should own it. And say ‘I don’t know’ or ‘I can’t explain that’.
I will take the liberty of responding directly on your behalf.
1) Rav Moshe wrote something completely new, and we are left wondering if it is a new opinion or an omission from his first letter. Even though he writes that it is all explained at length in his
teshuva, we see a new idea in the Flatbush letter of three million.
2) There can not be any teshuva that is about Brooklyn as then Rav Moshe could be accused of issuing a psak din barrur. Rav Moshe in 4:87 writes that he is compelled to respond with his opinion
because he writes stuff about his own thinking. These lines are to be ignored.
3) The understanding of the topic is not important. It’s all about what could we get away with in spite of Rav Moshe’s well publicized, thoroughly printed, and until recently, testified by many
great chachamim. Anybody who points to Rav Moshe’s opinion, and doesn’t admit that it can be manipulated, must be ignorant.
4) Rav Moshe doesn’t pasken against other rabbanim. (Except for thousands of instances that are convenient to ignore.) And if one follows those rabbanim he can still not be in contradiction to
Rav Moshe’s opinion. Even though Rav Moshe told them not to build an eruv. Because what he actually said and thought should not be considered when reading his writings.
Okay, that is what I think you are trying to argue. You don’t get that I am trying to give you a chance to talk. I could crush every one of your mistakes with a dozen responses. But then you come
with a bunch more incoherence. Admittedly, this is not a topic I know well. But you are so tripped up, I can’t help myself. You fell into every net I put out. You are completely oblivious to how
the eruv battle aligned with the attack on Orthodoxy. You don’t even know ten percent of the eruv story. And whatever you do know that I don’t you do not want to share. This isn’t a machlokes
leshem shamayim on your part.
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4-digit Hotel Room Safe Code :: Transum Newsletter
4-digit Hotel Room Safe Code
Tuesday 1st January 2019
Happy New Year! This is the January 2019 Newsletter and it begins, just as all new years should begin, with a puzzle.
I spent part of the holidays driving around the northern hills of Thailand. Each night I stayed in a different hotel and each of these hotels had very similar safes in the rooms. A four digit code
number was required to use the safe. I decided to choose a number such that the first digit was the number of noughts in the code, the second the number of ones, the third the number of twos and the
fourth the number of threes. When I realised there was more than one four-digit number that satisfied this requirement I chose the largest one.
What was the code needed to unlock my safe? The answer will be at the end of this newsletter.
In addition to getting ready for Christmas I spent a great deal of time last month adding to and updating the Transum website. Here are some of the highlights:
I realised that the site did not have a really basic activity requiring pupils to plot given coordinates without any other distractions. That gap has now been filled with
Coordinate Plotter
, a drag-and-drop, randomly generated, self-checking learning experience.
For those who have long passed the coordinate plotting learning phase there is now a new addition to the Graphs collection.
Parallel Graphs
is a quick activity requiring pupils to arrange equations of straight line graphs into a form that will help them spot lines with the same gradients. The equations can then be dragged together into
groups representing sets of parallel graphs.
Another new activity is the
Latin Square puzzles
. This set of challenges involves arranging single digit numbers on a four by four grid so that no number appears more than once in any vertical or horizontal line. While doing this you have to also
ensure that the calculations in the lines also produce the given totals.
This puzzle is similar to those appearing in newspapers and puzzle books but, the big difference is the operations are positioned so that the standard order of operations that pupils should know
(BIDMAS or PEMDAS) are not contradicted when working from left to right or top to bottom.
Common Trig Ratios
has been extended to include a level on finding the lengths if the indicated sides on right-angled triangles by solving trigonometric questions with exact solutions.
The activity called
Overdraft Charges
is based on an email I received from Lloyds Bank explaining their new system for calculating the fees charged for being ‘in the red’ in 2019. I thought this information should be available to pupils
who are just starting to learn how to manage personal finances as part of an ongoing financial education, skills-for-life programme.
My driving tour of Northern Thailand took me to the village called Pie. I arrived excitedly expecting to experience many mathematical attractions but sadly there were none. That disappointment did
not stop me writing
In Terms of Pi
, an exercise in which the answers should be given exactly and contain the symbol for pi.
The old activity called
Stamp Sticking
has been split into levels so that it is easier to earn a trophy. I you have pupils in need of some more unusual times tables practice then this might fit the bill. The postage amounts written on the
envelopes are randomly generated so a level can be attempted many different times without repetition if that is the practice required.
I would think that right now you are starting to think about a new Term or even a new School Year (if you are in the southern hemisphere). Don’t forget that there is a
Back To School
collection of activities available to you on the Transum website. Relevant to this time of year I’d like to mention that the
Dodecahedral Calendar for 2019
is waiting for subscribers to print out and construct producing a useful item of desk furniture for the coming year.
I can’t mention my adventure in northern Thailand without letting you know about my most exciting moment. I had just got out of bed one morning when to my surprise I opened the curtains in my chalet
and what was outside but an elephant. It provided entertainment by blowing dust into the air while eating its breakfast by the side of the river. It reminded me of the old joke: “I got up early one
morning and saw an elephant in my pyjamas --- What the elephant was doing in my pyjamas I’ll never know!”.
Finally the answer to the puzzle about the hotel room safe code digits. I found that 1210 and 2020 both satisfy the criteria so the largest one is 2020.
That’s all for now,
PS. I send you my very best wishes for 2019, a year in which you will be older than you have ever been before. Right now you are younger than you will ever be again so seize the day!
Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics
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Lesson 7
Measure Length with Different Length Units
Warm-up: Notice and Wonder: Large Cubes and Small Cubes (10 minutes)
This warm-up prompts students to consider the same object being measured with different length units to set students up to measure with different units in the next activity.
• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.
Student Facing
What do you notice?
What do you wonder?
Activity Synthesis
• “Can anyone restate _____’s idea?”
Activity 1: Measure With Different Objects (20 minutes)
The purpose of this activity is for students to measure the same length using three different length units. Students choose which units they would like to use. When groups compare their measurements,
they notice that although they measured the same length, they got different measurements. They consider why this could be and determine that when they use a different size length unit, they get a
different number to describe the length. Some students may notice that the number of units needed depends on the size of each individual length unit, but this is not a discussion point until grade 2.
Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Social-Emotional Functioning, Organization
Required Preparation
• Create sets of 30 connecting cubes, 50 base-ten cubes (centimeter cubes), twenty 2-inch paper clips, and twenty 1-inch paper clips for each group.
• Put 18-inch strips of tape on the floor for each group.
• Groups of 2–4
• Give each group access to the different length units.
• “Who wears the biggest shoes in your family?”
• Share responses.
• “How long do you think your shoes will be when you are grown up?”
• Share responses.
• “A man named Jeison Orlando Rodriguez Hernandez holds the record for having the longest feet in the world. He has to have his shoes specially made because stores don't sell shoes big enough.”
• “There are strips of tape on the floor that show the length of Jeison’s foot. You will measure the length of his foot using different objects.”
• Read the task statement.
• 12 minutes: partner work time
• “Compare your measurements with another group. What do you notice?”
• 2 minutes: small group discussion
Student Facing
Circle the 3 objects you will use:
Measure the length of Jeison's foot with each object you chose and fill in the table.
Activity Synthesis
• “What did you notice when you measured the length of Jeison's foot using different objects?”
• If needed, ask, “Why did we get different numbers when we used different objects?” (The objects we used had different lengths. Some were shorter and some were longer. We used more small cubes to
measure the length than any other object.)
Activity 2: Measure the Teacher’s Shoe (15 minutes)
The purpose of this activity is for students to use what they have learned about measuring length to determine whether measurements are accurate. Students understand that they need to use same-size
units to be precise in their measurements (MP6).
MLR8 Discussion Supports. Activity: Display sentence frames to support partner discussion: “I think the measurement is accurate because…” and “I think the measurement is not accurate because…”
Advances: Speaking, Conversing
• Read the task statement.
• 8–10 minutes: partner work time
• Monitor for students who can explain whose measurements are accurate and whose are not accurate and why.
Student Facing
1. Andre measured his teacher’s shoe and said it was 15 connecting cubes long.
Is his measurement accurate?
Why or why not?
2. Jada measured her teacher’s shoe and said it was 12 connecting cubes long.
Is her measurement accurate?
Why or why not?
3. Clare measured her teacher’s shoe and said it was 30 small cubes long.
Is her measurement accurate?
Why or why not?
4. Kiran measured his teacher’s shoe and said it was 19 cubes long.
Is his measurement accurate?
Why or why not?
Advancing Student Thinking
If students say that Clare's measurement is inaccurate because she did not use connecting cubes or because the measurement is not 15, encourage students to tell you more about what they mean.
Consider asking:
• “How is the way Clare measured like how we measured objects in the last activity?”
• “Did Clare measure the entire length of the shoe with the small blocks?”
• “How are the small blocks the same? How are they different?”
Activity Synthesis
• Display the image for Jada’s measurement.
• “Is Jada’s measurement accurate? Why or why not?” (It is not accurate because there are gaps between some of the cubes.)
• Display the image for Kiran’s measurement.
• “Is Kiran’s measurement accurate? Why or why not?” (Kiran’s measurement is not accurate because the cubes are different sizes.)
• “What could Kiran change to make his measurement accurate?” (He could use all connecting cubes or all the little cubes, instead of using some of each.)
• “When we measure a length with objects it is important that each object has the same length.”
Lesson Synthesis
Display the images that show Andre’s measurement of his teacher’s shoe from the previous activity.
“Today we looked at how other students measured their teacher’s shoe. We decided that Kiran’s measurement of 19 cubes was not accurate because he used some connecting cubes and some small cubes.”
“Kiran says that Andre’s measurement must not be accurate either because he used two different cubes.”
“What do you think Kiran means? Do you agree with his argument about Andre’s measurement?” (I think he means there are 2 different colors. I disagree. The different colors do not matter. Each
connecting cube has the same length. Each cube is the same size and Andre did not have any gaps or overlaps.)
As time permits, display Clare’s measurement and ask, “What about Clare’s measurement? She used many different colors of small cubes. Is her measurement of 30 small cubes accurate?” (Yes, it’s
accurate. The color isn’t what matters. What matters is that each cube has the same length and that there are no gaps or overlaps.)
Cool-down: The Length of a Shoe (5 minutes) | {"url":"https://im.kendallhunt.com/K5/teachers/grade-1/unit-6/lesson-7/lesson.html","timestamp":"2024-11-13T09:22:11Z","content_type":"text/html","content_length":"95522","record_id":"<urn:uuid:8ccc8df2-3c05-4b4c-bc5c-30d75281be02>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00107.warc.gz"} |
Sturm-Liouville Operators and Applications: Revised Editionsearch
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Sturm-Liouville Operators and Applications: Revised Edition
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Hardcover ISBN: 978-0-8218-5316-0
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Sturm-Liouville Operators and Applications: Revised Edition
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Hardcover ISBN: 978-0-8218-5316-0
Product Code: CHEL/373.H
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN: 978-1-4704-1580-8
Product Code: CHEL/373.H.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $52.00
Hardcover ISBN: 978-0-8218-5316-0
eBook ISBN: 978-1-4704-1580-8
Product Code: CHEL/373.H.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $114.10 $91.35
• AMS Chelsea Publishing
Volume: 373; 2011; 393 pp
MSC: Primary 34; 35; 47
The spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems. Besides the basic results on the structure of the spectrum and the
eigenfunction expansion of regular and singular Sturm-Liouville problems, it is in this domain that one-dimensional quantum scattering theory, inverse spectral problems, and the surprising
connections of the theory with nonlinear evolution equations first become related. The main goal of this book is to show what can be achieved with the aid of transformation operators in spectral
theory as well as in their applications. The main methods and results in this area (many of which are credited to the author) are for the first time examined from a unified point of view.
The direct and inverse problems of spectral analysis and the inverse scattering problem are solved with the help of the transformation operators in both self-adjoint and nonself-adjoint cases.
The asymptotic formulae for spectral functions, trace formulae, and the exact relation (in both directions) between the smoothness of potential and the asymptotics of eigenvalues (or the lengths
of gaps in the spectrum) are obtained. Also, the applications of transformation operators and their generalizations to soliton theory (i.e., solving nonlinear equations of Korteweg-de Vries type)
are considered.
The new Chapter 5 is devoted to the stability of the inverse problem solutions. The estimation of the accuracy with which the potential of the Sturm-Liouville operator can be restored from the
scattering data or the spectral function, if they are only known on a finite interval of a spectral parameter (i.e., on a finite interval of energy), is obtained.
Graduate students and research mathematicians interested in operator theory.
□ Chapters
□ Chapter 1. The Sturm-Liouville equation and transformation operators
□ Chapter 2. The Sturm-Liouville boundary value problem on the half line
□ Chapter 3. The boundary value problem of scattering theory
□ Chapter 4. Nonlinear equations
□ Chapter 5. Stability of inverse problems
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Requests
Volume: 373; 2011; 393 pp
MSC: Primary 34; 35; 47
The spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems. Besides the basic results on the structure of the spectrum and the
eigenfunction expansion of regular and singular Sturm-Liouville problems, it is in this domain that one-dimensional quantum scattering theory, inverse spectral problems, and the surprising
connections of the theory with nonlinear evolution equations first become related. The main goal of this book is to show what can be achieved with the aid of transformation operators in spectral
theory as well as in their applications. The main methods and results in this area (many of which are credited to the author) are for the first time examined from a unified point of view.
The direct and inverse problems of spectral analysis and the inverse scattering problem are solved with the help of the transformation operators in both self-adjoint and nonself-adjoint cases. The
asymptotic formulae for spectral functions, trace formulae, and the exact relation (in both directions) between the smoothness of potential and the asymptotics of eigenvalues (or the lengths of gaps
in the spectrum) are obtained. Also, the applications of transformation operators and their generalizations to soliton theory (i.e., solving nonlinear equations of Korteweg-de Vries type) are
The new Chapter 5 is devoted to the stability of the inverse problem solutions. The estimation of the accuracy with which the potential of the Sturm-Liouville operator can be restored from the
scattering data or the spectral function, if they are only known on a finite interval of a spectral parameter (i.e., on a finite interval of energy), is obtained.
Graduate students and research mathematicians interested in operator theory.
• Chapters
• Chapter 1. The Sturm-Liouville equation and transformation operators
• Chapter 2. The Sturm-Liouville boundary value problem on the half line
• Chapter 3. The boundary value problem of scattering theory
• Chapter 4. Nonlinear equations
• Chapter 5. Stability of inverse problems
Permission – for use of book, eBook, or Journal content
You may be interested in...
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Coq eats expression parentheses
While I was writing a prove for the theorem below, I noticed that parentheses on the right magically disappeared. I would like to keep them and avoid auto simplification in the exercise.
Theorem add_assoc : forall n m p : nat,
n + (m + p) = (n + m) + p.
intros n m p.
induction n as [|n' IHn].
- reflexivity.
goals state:
n', m, p : nat
IHn : n' + (m + p) = n' + m + p
S (n' + (m + p)) = S (n' + m + p)
Set Printing Parentheses.
1 Like
Thanks for the option, but it affects only formatting imho, because following prove is accepted:
Set Printing Parentheses.
Lemma l : forall n m p : nat, n + m + p = (n + m) + p.
Well, Coq can only control what it prints. If you want parentheses in your own theorem statement, you need to put the parentheses in there yourself.
reflexivity tactic is too smart. Is there strcmp tactic in the Coq toolbox?
Such literal reflexivity tactic would be faster.
I’m not sure what you expected here…
Coq does not work with “stream of symbol” expressions. Expressions are naturally tree-shaped, and Coq internally uses that tree everywhere, except when printing and parsing human-readable input.
As such, parentheses are “consumed” during parsing. They affect the tree produced by parsing but there is no “parenthesized expression” internally. Coq did not eat your parentheses, it
accounted for them when it created the syntax tree.
When printing an expression, the printer inserts parentheses wherever necessary. Since + is left-associative, they are not necessary, so none are inserted.
So even if reflexivity is too smart (it runs unification), a dumber version of it (not running unification) would also be able to prove that (a+b)+c = a+b+c because these are the same expression,
syntactically, in Coq. They have the exact same syntax tree, there is no way of telling them apart.
As such, strcmp seems the wrong function for comparing trees. They are not strings.
TL;DR: Parentheses are not real, there are only trees.
1 Like
This is folklore in almost any programming language, but it can be surprising:
Coq reads x+y+z as a parenthesized expression (because + is a binary operator). In this case it applies a “left-associativity parsing rule” so it reads (x+y)+z. The exact same way it would read
a+b*c as a+(b*c).
Moreover the printer by default omits parenthesis when they are not “needed for the expression to be re-read into an identical one”, this is called the “re-entering pretty printing”.
Set Printing Parentheses. will ensure all parantheses are printed
Thanks for detailed explanation. Now I see why my abstract representation about Coq leaked - expression type doesn’t have data constructor for parentheses like Haskell does:
module Language.Haskell.TH.Syntax
data Exp
= VarE Name -- ^ @{ x }@
| ConE Name -- ^ @data T1 = C1 t1 t2; p = {C1} e1 e2 @
| LitE Lit -- ^ @{ 5 or \'c\'}@
| UInfixE Exp Exp Exp -- ^ @{x + y}@
| ParensE Exp -- ^ @{ (e) }@
Indeed, it does not. I also don’t know why you would want to have such a thing, since it has no functionality, and also it suggests the wrong mental model about abstract syntax trees. It makes you
think the parentheses are necessary or affect the meaning, when they don’t.
Edit: Also, you can create a tree that does not have the ParensE constructor, but still requires parentheses to be faithfully “linearized” into a line of text.
I knew that Coq doesn’t have a built-in type for booleans and assumed similar approach for notation’s semantic.
Absence of parentheses complicates formatting preservation during automatic code refactoring (e.g. theorem / variable rename).
coq-lsp must have a forked parser to cover this then. | {"url":"https://coq.discourse.group/t/coq-eats-expression-parentheses/2442","timestamp":"2024-11-06T04:04:51Z","content_type":"text/html","content_length":"43005","record_id":"<urn:uuid:5ea92894-a783-4969-a102-034a895a2f13>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00673.warc.gz"} |
Henderson Hasselbalch Equation Worksheet - Equations Worksheets
Henderson Hasselbalch Equation Worksheet
Henderson Hasselbalch Equation Worksheet – The objective of Expressions and Equations Worksheets is for your child to be able to learn more efficiently and effectively. These worksheets are
interactive as well as problems based on the sequence of operations. With these worksheets, children are able to grasp simple and more complex concepts in a quick amount of period of time. You can
download these free resources in PDF format to aid your child’s learning and practice maths equations. These are helpful for students from 5th through 8th grades.
Get Free Henderson Hasselbalch Equation Worksheet
These worksheets can be used by students in the 5th-8th grades. These two-step word problem are constructed using fractions and decimals. Each worksheet contains ten problems. The worksheets are
available both online and in printed. These worksheets are a great way to exercise rearranging equations. These worksheets can be used to practice rearranging equations and aid students in
understanding the concepts of equality and inverse operations.
The worksheets are intended for students in the fifth and eighth grades. They are great for students who struggle to compute percentages. There are three types of problems to choose from. You can
choose to solve one-step challenges that contain whole or decimal numbers, or utilize word-based strategies to do fractions or decimals. Each page contains ten equations. These Equations Worksheets
can be used by students from the 5th-8th grades.
These worksheets can be used to practice fraction calculations and other concepts in algebra. A lot of these worksheets allow users to select from three types of challenges. You can select a
word-based problem or a numerical. It is important to choose the problem type, because each problem will be different. Each page contains ten problems and is a wonderful resource for students from
5th to 8th grade.
These worksheets help students understand the connection between numbers and variables. These worksheets provide students with practice in solving polynomial equations in addition to solving
equations and getting familiar with how to use these in their daily lives. If you’re in search of a great educational tool to master the art of expressions and equations it is possible to begin by
exploring these worksheets. They will teach you about different types of mathematical issues and the different types of symbols that are used to communicate them.
These worksheets are extremely beneficial for students in the first grade. The worksheets will assist them to develop the ability to graph and solve equations. They are great to practice polynomial
variables. These worksheets can help you simplify and factor them. There are many worksheets to teach children about equations. The best way to learn about equations is by doing the work yourself.
You will find a lot of worksheets on quadratic equations. There are several levels of equations worksheets for each degree. The worksheets are designed to allow you to practice solving problems in
the fourth degree. Once you have completed an amount of work then you are able to work on other types of equations. Then, you can tackle the same problems. For instance, you could discover a problem
that uses the same axis and an extended number.
Gallery of Henderson Hasselbalch Equation Worksheet
Henderson Hasselbalch Equation Derivation And Problems
Buffer And Henderson Hasselbalch Equation Worksheet With Answers
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New benchmark helps solve the hardest quantum problems
Predicting the behavior of many interacting quantum particles is a complicated process but is key to harness quantum computing for real-world applications. A collaboration of researchers led by EPFL
has developed a method for comparing quantum algorithms and identifying which quantum problems are the hardest to solve.
From subatomic particles to complex molecules, quantum systems hold the key to understanding how the universe works. But there’s a catch: when you try to model these systems, that complexity quickly
spirals out of control - just imagine trying to predict the behavior of a massive crowd of people where everyone is constantly influencing everyone else. Turn those people into quantum particles, and
you are now facing a "quantum many-body problem".
Quantum many-body problems are efforts to predict the behavior of a large number of interacting quantum particles. Solving them can unlock huge advances in fields like chemistry and materials
science, and even push the development of new tech like quantum computers.
But the more particles you throw into the mix, the harder it gets to model their behavior, especially when you’re looking for the ground state, or lowest energy state, of the system. This matters
because the ground state tells scientists which materials will be stable and could even reveal exotic phases like superconductivity.
For every problem, a solution: but which one?
For years, scientists have relied on a mix of methods like quantum Monte Carlo simulations and tensor networks (variational wave functions) to approximate solutions to these problems. Each method has
its strengths and weaknesses, but it’s hard to know which one works best for which problem. And until now, there hasn’t been a universal way to compare their accuracy.
A large collaboration of scientists, led by Giuseppe Carleo at EPFL has now developed a new benchmark called the "V-score" to tackle this issue. The V-score ("V" for "Variational Accuracy") offers a
consistent way to compare how well different quantum methods perform on the same problem. The V-score can be used to identify the hardest-to-solve quantum systems, where current computational methods
struggle, and where future methods --such as quantum computing - might offer an advantage.
The breakthrough method is published in
How the V-score works
The V-score is calculated using two key pieces of information: the energy of a quantum system and how much that energy fluctuates. Ideally, the lower the energy and the smaller the fluctuations, the
more accurate the solution. The V-score combines these two factors into a single number, making it easier to rank different methods based on how close they come to the exact solution.
To create the V-score, the team compiled the most extensive dataset of quantum many-body problems to date. They ran simulations on a range of quantum systems, from simple chains of particles to
complex, frustrated systems, which are notorious for their difficulty. The benchmark not only showed which methods worked best for specific problems, but also highlighted areas where quantum
computing might make the biggest impact.
Solving the hardest quantum problems
Testing the V-score, the scientists found that some quantum systems are much easier to solve than others. For example, one-dimensional systems, such as chains of particles, can be tackled relatively
easily using existing methods like tensor networks. But more complex, high-dimensional systems like frustrated quantum lattices, have significantly higher V-scores, suggesting that these problems are
much harder to solve with today’s classical computing methods.
The researchers also found that methods relying on neural networks and quantum circuits - two promising techniques for the future - performed quite well even when compared to established techniques.
What this means is that, as quantum computing technology improves, we may be able to solve some of the hardest quantum problems out there.
The V-score gives researchers a powerful tool to measure progress in solving quantum problems, especially as quantum computing continues to develop. By pinpointing the hardest problems and the
limitations of classical methods, the V-score could help direct future research efforts. For instance, industries that rely on quantum simulations, such as pharmaceuticals or energy, could use these
insights to focus on problems where quantum computing could give them a competitive edge.
List of contributors
• EPFL Computational Quantum Science Lab
• Sorbonne Université
• University of Zurich
• Università di Trieste
• Flatiron Institute
• Vector Institute
• Goethe-Universität
• Collège de France
• CNRS École Polytechnique
• Université de Genève
• University of Waterloo
• Toyota Physical and Chemical Research Institute
• Waseda University
• Sophia University
• Paul Scherrer Institute (PSI)
• University of Zurich
• IBM Quantum
• Columbia University
• New York University
• Keio University
• Université Paris-Saclay
• University of Tokyo
• University of California Irvine
• International School for Advanced Studies (SISSA)
• Politecnico di Torino
• Max Planck Institute
• University of Chinese Academy of Sciences
• College of William and Mary
Dian Wu, Riccardo Rossi, Filippo Vicentini, Nikita Astrakhantsev, Federico Becca, Xiaodong Cao, Juan Carrasquilla, Francesco Ferrari, Antoine Georges, Mohamed Hibat-Allah, Masatoshi Imada, Andreas M.
Läuchli, Guglielmo Mazzola, Antonio Mezzacapo, Andrew Millis, Javier Robledo Moreno, Titus Neupert, Yusuke Nomura, Jannes Nys, Olivier Parcollet, Rico Pohle, Imelda Romero, Michael Schmid, J. Maxwell
Silvester, Sandro Sorella, Luca F. Tocchio, Lei Wang, Steven R. White, Alexander Wietek, Qi Yang, Yiqi Yang, Shiwei Zhang, and Giuseppe Carleo. Variational Benchmarks for Quantum Many-Body Problems.
Science 17 October 2024. DOI: 10.1126/science.adg9774 | {"url":"https://www.myscience.ch/news/2024/new_benchmark_helps_solve_the_hardest_quantum_problems-2024-epfl","timestamp":"2024-11-13T21:55:46Z","content_type":"text/html","content_length":"53982","record_id":"<urn:uuid:c41025a4-1de5-4192-a25d-070ffc0e18c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00145.warc.gz"} |
Moving on a strange diagonal
Here is a puzzle I enjoy giving to students. Given a 4×4 grid of sixteen dots, draw six straight lines that form a continuous path passing through all of the dots. Here, continuous means you must be
able to draw over your six lines in one go without taking your pen off the paper.
This task is easy to complete with seven lines and impossible with five. Six is where the interesting puzzle lies.
I have given this to undergraduate students up and down the UK, and I have noticed a common response that makes me enjoy this puzzle and the lesson it teaches.
Many students start playing with the puzzle, drawing lines on grids right away. This is good because playing is how you start to understand puzzles and by showing their thought processes I can look
out for when they have misunderstood the problem. In my experience, this is a problem often misunderstood.
One approach I see quite often is to draw lines within the grid always moving to the next adjacent dot – either up, down, left, right or diagonally. I ask the student, why aren’t you drawing outside
the grid? “What,” the student typically replies, “you’re allowed to go outside the grid?”
Then I ask, why are you only moving to adjacent dots, why not move at strange diagonals and jump around the grid in a more irregular fashion? Again, the response: “is that allowed?”
The statement of the problem was: “draw six straight lines that form a continuous path passing through all of the dots”. This says nothing to limit how you can move through the dots or where on the
page you can draw your lines.
In fact, I know of fourteen solutions to this problem and each of them involves a strange diagonal and leaving the confines of the grid. By imagining these additional conditions the student creates a
new problem – one that cannot be solved.
Good puzzles lie on a thin line between easy and impossible, and the particular wording in the statement of the problem is likely to be very carefully chosen to give you enough to solve it without
being too easy. A would-be puzzle solver must check the wording to be sure they are attempting the puzzle that has actually been set and not some imagined, impossible version.
16 Comments | {"url":"http://minds.acmescience.com/?p=29","timestamp":"2024-11-13T08:23:55Z","content_type":"application/xhtml+xml","content_length":"25825","record_id":"<urn:uuid:786b409e-48a7-4bbc-bab8-73f132c75be9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00063.warc.gz"} |
The ACT is important to high school students in Missouri and across the U.S. for several reasons. It is an important factor that is considered in college admissions and how scholarships are awarded.
It can help students become eligible for the A+ Scholarship and in some parts of the state it is a requirement to graduate. While the level of importance can differ among students, based on their
plans, it is a tough challenge for everyone their first time. Students take timed tests in multiple subjects, answering questions that can confuse them or make them second guess themselves.
This is the second article of a four-part series on the ACT test, from Journey to College, describing each of the subject tests and how to prepare for them. In this article, the math section of the
test is the subject of discussion.
The basics of the math test
Of all the subject tests, the math portion is the most straightforward. In an hour, you answer 60 problems. The questions get more difficult as you go, meaning the first question is the easiest and
the last question is the most difficult. It mainly covers concepts from algebra and geometry. The last 20 questions pull from more advanced topics, such as trigonometry, pre-calculus, and calculus.
The ACT publishes a test breakdown, which describes how much of the test is focused on one topic (modeling, preparing for higher math, etc.). In each topic, it is further broken down into certain
types of questioning. For example, about 8-12 percent of the test will cover statistics and probability questions.
To help students prepare for the test, the ACT also provides old versions of the test online and in print. Ask your counselor if they have copies of old booklets so you can take practice tests and
time yourself.
How is the test graded?
scoring rubric so you can know ahead of time what to expect. This is very helpful for students who are aiming for a specific score, such as those aiming to earn the A+ Scholarship. Students who
didn’t score proficient or advanced on their Algebra 1 end-of-course exam can substitute an ACT math score to qualify. Depending on your GPA, this score can change. The same is true for students
trying to earn scholarships from a university, especially with the superscore option now available.
Final thoughts
The math section of the ACT is meant to demonstrate the depth of your knowledge in the subject. You either know the material or you don’t. And that is OK. Every student will bring a different level
of expertise, as well as a different desire, whether you are aiming for top marks and the Bright Flight scholarship, or just trying to make a certain threshold for A+. Remember, preparation is key.
Read the other parts of the ACT Series | {"url":"https://journeytocollege.mo.gov/tag/act-math/","timestamp":"2024-11-12T20:21:00Z","content_type":"text/html","content_length":"67431","record_id":"<urn:uuid:01e08a8e-f35b-4a83-b00d-0d6106850013>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00872.warc.gz"} |
Applied Engineering Electromagnetics
Applied Engineering Electromagnetics. Instructor: Dr. Pradeep Kumar K, Department of Electrical Engineering, IIT Kanpur. Applied electromagnetics for engineers is designed to be an application
oriented course while covering all the theoretical concepts of modern electromagnetics. It begins by an in-depth study of transmission lines which play an important role in high-speed digital design
and signal integrity of PCBs. After a brief review of necessary mathematics (coordinate systems, vector analysis, and vector fields), the course covers analytical and numerical solution of Laplace's
and Poisson's equations, quasi-static analysis of capacitors and skin effect, inductance calculations, and Maxwell equations. Wave propagation in free-space, ferrites, and periodic media are covered
along with waveguides (rectangular, planar dielectric, and optical fibers) and antennas. The course includes a balance between theory, programming, and applications. Several case studies will be
discussed. (from nptel.ac.in)
Introduction to Applied Electromagnetics
Applied Engineering Electromagnetics
Instructor: Dr. Pradeep Kumar K, Department of Electrical Engineering, IIT Kanpur. Applied electromagnetics for engineers is designed to be an application oriented course while covering all the
theoretical concepts of modern electromagnetics. | {"url":"http://www.infocobuild.com/education/audio-video-courses/electronics/applied-engineering-electromagnetics-iit-kanpur.html","timestamp":"2024-11-08T15:55:07Z","content_type":"text/html","content_length":"17737","record_id":"<urn:uuid:44d57700-f697-497c-afa3-2955d273aa8c>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00418.warc.gz"} |
Investigations in Projective Geometry
Investigations in Projective Geometry
Dwayne Alvis and Wanda Nabors
Try to imagine yourself as an artist. You are driving out in the country side and you see beautiful landscape scenes of trees, houses, and streams of water. Being an artist you want to capture all
this beauty onto paper. You are brought face to face with the problem of trying to reproduce three-dimensional scenery onto a two-dimensional piece of paper or canvas. What do you do?
Suppose there is some rectangular-shaped builder in the scenario given above and your first attempt is to draw a rectangle on the canvas. You realize that your drawing seems distorted even though you
drew the shape that you saw. This was the problem that was addressed by artists like Leonardo de Vinci in the fifteenth century. At that time in history the artists developed a theory of perspective
in order to realistically paint two-dimensional representations of three-dimensional scenes. By the nineteenth century mathematicians began studding the properties involved in transferring a
three-dimensional object to a two-dimensional canvas. Projective geometry involves the study of these properties.
To begin this investigation, consider the following illustration: Let the point O correspoind to the eye of the viewer, the green plane, P', corresponds to what the artist actually sees; the yellow
plane, P, corresponds to the canvas. Therefore, from the point O, the lines l and l' are the same.
Exercise one involves one of the earliest discoveries pertaining to projective geometry:
Topic One.
1. Construct three lines l, m, & k intersecting in a point P. You should have something like the following:
2. Select 3 points A, B, and C such that A lies on k, B lies on m, and C lies on l. Then select 3 other points A', B', and C' such that A' lies on k, B' lies on m and C' lies on l.
3. Construct triangles ABC and A'B'C'.
4. Construct lines AC and A'C' and mark the point of intersection, X.
5. Construct lines AB and A'B'; mark their point of intersection, Y.
6. Construct lines BC and B'C'; mark their point of intersection, Z.
If you don't want to do this construction, here is GSP file:(Click here.)
7. Is there anything of interest about the points of intersection, X, Y,
8. Investigate what happens to the line when the triangles are
placed in different positions?
9. In particular, consider what happens when 2 corresponding side are parallel?
10. In general, what can be concluded from this exercise?
"If the vertices of two triangles correspond in such a way that the lines joining corresponding vertices are concurrent, then the intersections of corresponding sides are collinear."
The following are exercises pertaining to the above result. These can be assigned as homework/classwork.
1. Suppose you are given a piece of paper on which someone has drawn a point A and two lines l' and l'', neither passing through A and whose point of intersection lies off the paper. You are asked to
draw the line through A which goes through that point of intersection. How do you do it?
(Hint: Pick a on l', b on l'', and let A=c. Let a' be on l' and b' on l'' be chosen so that lin ab is parallel to a'b', and choose c' so that line ac is parallel to line a'c' and line bc is parallel
to line b'c'.)
2. You are given a short straightedge and two points P & Q a great distance apart. Use your short straightedge to draw the line joining P and Q. (Hint: Use 1 above)
Topic two.
1. Construct 2 non-parallel .lines l and m
2. Pick 3 points, P, Q, and R on l and 3 points P', Q', and R' on m.
3. Construct lines PQ', PR', QP', QR', RQ', and RP'.
4. Consider PQ' and QP'; PR' and RP'; and RQ' and QR'. Your figure
should look similar to the following:
5. What is interesting about these points of intersection?
(They are collinear.)
6. Move all points around even to the extent of changing their
relative positions. Make a general statement you believe to be
true about the relationship(s) you have noted.
The general statement that can be conjectured from the above result is known as Pappus's Theorem; i.e., Let P, Q, R and P', Q', R' be collinear triples of points. Denote the intersection of line
PQ' and P'Q by X; the intersection of line PR' and RP' by Y; the intersection of line QR' and line Q"R by Z. Then the points X, Y, & Z lie on a line also, i.e., X, Y, Z are collinear points.
The following can be used as homework/classwork.
1. If the point A is the intersection of lines PQ and P'Q' ; point B the intersection of lines QR and Q'R'; and point C the intersection of lines PR and P'R', then A, B, & C lie on the same line.
Topic 3.
Extension of topic 2: (Pascal's Theorem)
Let A, B, C, A', B', C' be six distinct points on a nondegenerate conic. Then the three points of intersection, X, Y, & Z , where X is the intersection of lines AB' and BA', Y is the intersection of
AC' and CA', and Z is the intersection of lines BC' and CB', lie on the same line.
Topic 4.
Cross Ratio.
Consider the following situation: If you have 3 points, A, B, and C on a line, when you project to another line as in the picture below, the ratio of the lengths AB and BC is not the same as the
corresponding ratio of A'B' and B'C'.
Now observe what happens when we add a fourth point, D. (Click here.) | {"url":"https://jwilson.coe.uga.edu/emt669/Student.Folders/Alvis.Dwayne/instruct/iu.html","timestamp":"2024-11-13T08:27:20Z","content_type":"text/html","content_length":"7529","record_id":"<urn:uuid:a843b938-fb1b-4b20-b5d0-2955421aa466>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00110.warc.gz"} |
The Genius' Secret: Boost Your Task Management with Fibonacci Prioritisation Funneling
The Genius' Secret: Boost Your Task Management with Fibonacci Prioritisation Funneling
Unlock the Genius' Secret to mastering task management with the Fibonacci Prioritisation Funnel. Boost productivity and maintain focus by prioritizing tasks efficiently while reducing mental
overwhelm. Adapt this versatile system to various tools and easily conquer your digital life.
1️⃣ Introduction
There are many ways to prioritise tasks, and dozens of books have been published on each method. Five popular methods include:
1. MoSCoW Method: This method helps teams categorise tasks or features based on their importance and urgency. It is particularly useful for managing project requirements and addressing critical
elements first.
2. Time Management Matrix: Also known as the Eisenhower Matrix, this method classifies tasks based on urgency and importance. The matrix is divided into four quadrants: important and urgent,
important but not urgent, not important but urgent, and not important and not urgent. This approach helps individuals and teams allocate time and resources more effectively.
3. Weighted Shortest Job First (WSJF): This approach is used in Agile and Lean methodologies to prioritise tasks based on their cost of delay and value.
4. Kano Model: This technique focuses on customer satisfaction by categorising product features into three main categories: basic, performance, and excitement.
5. Value vs Effort Matrix: This method plots tasks or features on a two-dimensional grid, with value on one axis and effort on the other.
In addition, classic methods include prioritising tasks in priority order (Ivy Lee method) and prioritising with labels A, B or C.
I have used methods 1, 2, and 5 for several years and found them effective. In particular, the MoSCoW rule has done an excellent job.
However, all of these methods overlook an essential point.
Humans have limited mental capacity for priorities, and no matter which method is used, huge lists accumulate over time, with low-priority buckets becoming so full of ideas and tasks that they
overwhelm us.
One solution is to review the list rigorously weekly or monthly and discard anything that has no chance of implementation, no matter how nice, urgent or important it may seem. Unfortunately, this is
a mentally exhausting task (as a decision must be made for each task), and few of us go about it this way.
I have another solution for you.
I've found a unique prioritisation approach for myself, which I call Fibonacci Prioritization Funnel, that focuses on avoiding overwhelm.
Today, I'll share this concept with you.
2️⃣ Logarithmic Thinking is Part of Human Nature
The human brain is naturally inclined to think logarithmically, which can be traced back to our evolutionary roots. Logarithmic thinking is a cognitive bias where people tend to perceive changes in
stimuli on a logarithmic scale rather than a linear one.
This means we are more sensitive to differences in smaller numbers, while larger numbers are perceived as less distinguishable.
This thinking pattern is not limited to numbers but extends to various aspects of our lives, including time, space, and priority management.
Logarithmic thinking is thought to have developed as a survival mechanism for our ancestors. In the early days of human evolution, reacting quickly to environmental changes or potential threats was
crucial. The ability to rapidly distinguish between varying levels of danger or scarcity helped our ancestors make quick decisions, ensuring their survival.
In his book "Sapiens", author Harari cites the example that encountering one vs three tigers can make a significant difference, but encountering 1000 or 1001 doesn't matter much.
3️⃣ Understanding Logarithmic Scales
If you've ever seen a graph or chart with a logarithmic scale, you may have wondered what that means. A logarithmic scale is used in mathematical representations based on a value's logarithm rather
than its actual value. This cannot be very clear initially, but once you understand how it works, you'll see that logarithmic scales can be handy.
To understand logarithmic scales, it's essential first to understand what a logarithm is. A logarithm is simply the power to which a given number, called the base, must be raised to produce a
particular value. For example, the logarithm of 100 to the base 10 is 2 because 10 to the power of 2 is 100.
Visual Comparison: Logarithmic vs Linear Scale.
In a logarithmic scale, the distance between two values is not equal but based on their logarithmic ratio. This means that each increment on the scale represents a multiplication by a constant factor
rather than a consistent increase. For example, on a logarithmic scale with a base of 10, each increment represents a multiplication by 10.
Logarithmic scales are commonly used to represent large ranges of values, such as in earthquakes, sound intensity, or the brightness of stars. In these cases, the values can span many orders of
magnitude, making it difficult to represent them accurately on a linear scale. For example, if you were to plot the brightness of stars on a linear scale, the difference between the brightest and
dimmest stars would be so vast that the faintest stars would be practically invisible.
Using a logarithmic scale, however, we can compress this range of values into a more manageable size. This allows us to see more detail in the data and make meaningful comparisons between different
values. For example, on a logarithmic scale, we can easily see the difference between a star with a brightness of 1 and a star with a brightness of 10,000.
As you can see, the log scale has a better resolution in the lower parts.
4️⃣ Logarithmic Thinking for Task Management
Humans think and perceive logarithmically; in my observation, this also affects our focus on tasks. Just as it is crucial whether there are 2 or 4 tigers, but it doesn't matter whether there are 1000
or 1001, we need higher resolution and concentration on the next three tasks than on the 20th or 23rd task is somewhere farther ahead on the horizon.
How many problems can we juggle mind-term in our minds? According to Richard Feynman, it would be a dozen. Feynman was fond of giving the following advice on how to be a genius:
You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state.
Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps.
Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”
I consider 12 an ideal number as it closely corresponds to the number of deliverables that, from my experience, can cause overwhelming feelings when exceeded.
This means we need to prioritise in a way that allows us to quickly identify the 12 most important tasks with clarity amidst the many unessential ones.
It's pretty alarming that many individuals have many dozens, or even hundreds, of tasks with varying degrees of granularity on their to-do lists. Accomplishing all of these tasks is simply an
illusion, and the size of the backlog often leads to mental blocks.
5️⃣ The logarithmic scale leads to intuitive priority levels.
How can we make use of this phenomenon? From my perspective, it's about building a priority funnel that ensures we look at the next tasks with a very high resolution while only seeing distant tasks
roughly on the horizon not to be overwhelmed by them.
We must sequence the tasks accordingly.
Logarithmic thinking allows us to form an order that does not necessarily require us to weigh ALL tasks against each other but will enable us to categorise them into resolution buckets.
If we follow the logarithmic scale to base two, for example, the following levels could be used:
1, 2, 4, 8, 16, 32, 64
This would imply that we can assign precisely one task as the top priority, two as a second priority, four as a third priority, eight as a fourth priority, and so on.
This is quite good, but the resolution between 8 and 32 is too insensitive in my experience.
Photo by Alex Eckermann / Unsplash
Therefore, in practice, I use a similar scale, which does not have a logarithmic background but can depict many natural phenomena well similarly: the Fibonacci scale.
1, 2, 3, 5, 8, 13, 21, 34, 55
Like the logarithmic scale, Fibonacci numbers have interesting applications in mathematics and science. They are ideal for modelling growth and patterns in nature, such as the branching structures of
trees, the arrangement of leaves on a stem, and the spiral patterns of shells and other natural structures.
And now, we'll take them to create our Fibonacci Prioritisation Funnel.
6️⃣ Introducing the Fibonacci Prioritisation Funnel
This is how you can imagine the prioritisation funnel.
Now, after all that preamble, we come to the actual application. And in comparison to the logarithmic and Fibonacci number background knowledge, you only need to know that you need to sort your tasks
into the following buckets:
1, 2, 3, 5, 8, 13, 21, 34, 55
The big and crucial difference from other prioritisation systems such as ABC or MoScoW rule is the limited space! In the ABC system, you can give dozens or even hundreds of tasks priority A - the
system allows for it. However, this defeats the idea of priority and ignores the limits of your cognitive capacity.
You can only have one important priority in the present moment. And 2-5 that you may already be juggling and preparing for as the following tasks.
And that's precisely what the Fibonacci Prioritization Funnel perfectly represents:
• Bucket 1 has space for precisely one task - your Most Important Task (MIT).
• Bucket 2 has space for precisely two tasks - these are the following two tasks that will come once you have cleared Bucket 1
• Bucket 3 contains the following three tasks accordingly.
• … And so on.
• Bucket 55 is a placeholder for tasks that may arise in the future. These tasks are collected in a very rough manner.
7️⃣ Bringing Fibonacci Prioritisation Funnel to Practice
I have used the Fibonacci prioritisation funnel in different tools, and it is versatile to implement. Here are a few suggestions on how you can do this.
You have to remember that with all three tools, you have to pay attention to how many items are in your buckets and that you do not overfill any of the buckets.
You do that in all three tools by dragging and dropping the items from bucket to bucket.
🗂️ Database Note-taking tools (e.g. Notion, ...)
In Notion, you build a matrix from two selects:
• Select 1: The Fibonacci numbers
• Select 2: Your areas of life or responsibilities
Here's a template to make it more tangible for you.
A simple example of how to implement the Fibonacci Prioritisation Funnel in Notion.
📔 Folder-based Note-taking tools (e.g. Evernote, Obsidian, ...)
In Evernote, you can create a task stack and have a notebook for each bucket.
Be sure to turn on the option that Evernote displays the number of notes per notebook in the sidebar. This way, you can easily see in which bucket there is still space and which buckets are
Each task has its own note; you drag and drop it through the different funnel levels.
A simple example of how to implement the Fibonacci Prioritisation Funnel in Evernote.
🧾 Outlining and Graph-based Note-taking Tools (e.g. Roam, Tana, Workflowy, Logseq, ...)
In graph-based tools, you have two options:
• You can implement the Fibonacci Prioritisation Funnel using [[concepts]] or #tags (no drag and drop, though)
• Or you can use hierarchical bullet point lists
A simple example of how to implement the Fibonacci Prioritisation Funnel in Workflowy.
8️⃣ Conclusion
The Fibonacci Prioritisation Funnel offers an innovative and practical approach to task management that aligns with our inherent logarithmic thinking. This method enables us to focus better on the
most important tasks, reducing the mental overwhelm that can arise from extensive to-do lists. By organising tasks into limited capacity buckets, the Fibonacci Prioritisation Funnel ensures that we
allocate our attention and resources to the tasks that truly matter.
This versatile system can be adapted to various tools, such as Notion, Evernote, or graph-based note-taking applications. By implementing the Fibonacci Prioritisation Funnel, we can increase our
productivity and accomplish more meaningful work while maintaining a healthy balance.
Embracing this genius' secret to task management will empower you to dominate your digital life and achieve your personal and professional goals more efficiently.
Feel free to add tips and thoughts to this page's comment section, Twitter or LinkedIn!
Best regards,
-- Martin from Deliberate-Diligence.com | {"url":"https://www.deliberate-diligence.com/the-genius-secret-boost-your-task-management-with-fibonacci-prioritisation-funneling/","timestamp":"2024-11-14T00:15:30Z","content_type":"text/html","content_length":"106977","record_id":"<urn:uuid:0991a7d6-e8fd-425d-8a28-c927f11f4478>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00106.warc.gz"} |
Some Demonstration Programs for Use in Teaching Elementary Probability and Statistics: Parts 3 and 4
Bruce E. Trumbo
California State University, Hayward
Journal of Statistics Education v.3, n.2 (1995)
Copyright (c) 1995 by Bruce E. Trumbo, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the
author and advance notification of the editor.
Restrictions. The programs accompanying this article are also copyrighted 1995 by Bruce E. Trumbo, all rights reserved. Permission to use the programs and any portion of this article for any
nonprofit educational purpose is hereby granted, except that any use that involves gambling for money or for items or services of value is not permitted.
Although extensively tested, the computer programs should be regarded as developmental. They may contain errors, some of which may cause unpredictable results, including computer "crashes." Source
code is not available. Please report errors to the author. NO WARRANTY OR REPRESENTATION OF FITNESS FOR ANY PURPOSE IS MADE OR IMPLIED. The user assumes all risks of any kind and waives the right to
claim damages.
Requirements. Programs in this series are intended for use with IBM-PC compatible machines equipped with EGA graphics or better, although some of them may run successfully on other machines.
Key Words: Bivariate normal distribution; Correlation; Expectation; Hypergeometric distribution; Gambling; Keno; Pedagogy; Simulation.
In this second paper of a series, two programs for EGA-equipped IBM-PC compatible machines are included with indications of their pedagogical uses in the teaching of elementary probability and
statistics. Concepts illustrated include the coefficient of correlation, the expectation of a discrete distribution, the concept of a fair game, and the hypergeometric distribution. Three datasets
useful for illustrating correlation are also documented and appended.
1. Introduction
1 As in the first paper in this series (Trumbo 1994), the emphasis here is on building student intuition and understanding of probability concepts through the use of simple computer programs in
class. Both of the programs made available with this article are based on simulation. Of course, the results of simulations can be presented without having a computer, but the advantage of
interactive computer use is that simulations can be repeated often enough that general principles can be discerned above the inevitable eccentricities of each individual simulation run.
2 The package of probability demonstration programs included with Trumbo (1994) consisted of the following: BRUN40.EXE (Microsoft utility), PROBDEMO.EXE (Entry program), PDMEN.EXE (Menu program),
PDLLN.EXE (Part 1), and PDPPR.EXE (Part 2). The present article adds PDBNS.EXE (Part 3) and PDKEN.EXE (Part 4) to the list of programs released.
3 Each of the four demonstration programs released to date is called from a Main Menu activated by the command PROBDEMO. The menu program is written to detect the presence of these four programs (as
well as others that may be released later), displaying only programs currently loaded into the same directory. The title page of each program tells how to start running it. For all except the program
in Part 4, the F1 and F2 keys give technical details and additional options for more advanced users.
4 These programs were written for student use in computer labs, and most of the material provided here is intended to help instructors and students use them to the best advantage in that setting.
However, if suitable projection equipment is available, the programs could also be used in large lecture sessions, perhaps with commentary partially based on the laboratory notes provided.
5 Readers are referred to Trumbo (1994) for further comments on the rationale, design, development, testing, advantages, limitations, and classroom use of the programs in this series.
6 Part 3 is intended to give students an intuitive grasp of the correlation coefficient, including an understanding of the meaning of various numerical values of correlation between -1 and +1.
(Advanced students can benefit from establishing some of the distribution theory involved in the workings of the program.)
7 Part 4 allows students to play a simulation of the casino game Keno. It provides insight into the lure of a gambling game in which highly advertised large winnings are possible, but at which the
player will, on average, lose more than a quarter of the money bet. The hypergeometric probability distribution is used to compute probabilities and expected winnings for this game---two crucial
kinds of information not generally provided by casinos.
2. Part 3: Simulated Samples from a Bivariate Normal Distribution to Illustrate Various Correlations
2.1. Operation of the Program and Discussion of Its Purpose
8 This is a simulated drawing of 500 observations from a bivariate normal distribution. Both means are fixed at zero and both standard deviations are fixed at unity. The user may select any value of
the population correlation \rho between -1 and 1.
9 A bivariate normal density function can be viewed as a mound-shaped surface above a plane. The shape of the mound varies as \rho varies, but it is always possible to find an ellipse in the plane
above which exactly 95% of the probability (volume) of the mound lies. The shape of this ellipse can vary from circular (for \rho = 0) to "football" shaped (for \rho around +.7 or -.7) to "cigar"
shaped (for \rho nearer to +1 or -1). In the present graphical display, the inclination of the principal axis of the ellipse changes from +45 degrees to -45 degrees depending on whether \rho is
positive or negative. As the simulation begins, this ellipse appears in deep blue background on the plotting axes. Its shape provides a valuable visual link to the numerical value of \rho. When the
simulation is complete this ellipse should contain about 95% of the points. Thus, about 5% of the 500 points (about 25) will lie outside the ellipse.
10 As the simulated points are sampled from the bivariate normal distribution they are plotted on the axes, and the updated sample correlation r is printed after every 5 observations. After all 500
points are plotted, the user is asked if he or she wants to do another simulation. The possible responses are K for "Keep the same value of \rho and run again," Y for "Yes, with the opportunity to
select a different value of \rho," and N for "No, return to the title page."
11 Not surprisingly, most students who encounter the idea of correlation for the first time have no feel for what the various numbers in the allowable range from -1 to +1 mean. In particular, many
expect that any correlation above .9 or below -.9 will correspond to almost perfect fit to a line. Exploring with this program helps to build intuition about sample correlations in several ways:
1. Students can experiment with several runs each of various values of the population correlation. The sample correlation in each case will be fairly near the selected value of \rho. Thus, they
can form an impression of what a correlation of .5 or .9 or -.99 looks like.
2. The distinction between population and sample correlation is illustrated. The population correlation \rho is a parameter, which is chosen when one specifies a particular bivariate probability
model. The sample correlation is a random quantity, slightly different with each run, resulting from the sampling process. The near agreement of the sample correlation with the population
correlation shows that the sample correlation can be used as an estimate of the population correlation.
3. The ellipse in the background ("football" or "cigar" shape) gives students a vivid graphical image to associate with the population correlation.
12 The F1 key shows technical details about the generation of the bivariate normal observations. The Box-Muller method is used to convert pairs of independent uniform random variables, generated
easily by the computer, into pairs of independent standard normal random variables, which are then transformed to yield a bivariate observation with the appropriate correlation. (See Zelen and Severo
in Sec. 26.8 of Abramowitz and Stegun (1972) or Sec. 6.5.1 of Kennedy and Gentle (1980).) Plotting peculiarities for the special cases where \rho is equal to +1 or -1 are also discussed briefly. The
F2 key shows additional options (monochrome operation, if needed for projection, and speed controls). Return to the title page from either the F1 or F2 screen by pressing ESC.
2.2. Before Going to the Lab
13 Ideally, the following should be done just after the concept of correlation or formula for the coefficient of correlation has been introduced in lecture:
14 Ask students what their reaction would be if you were to run your eye around the classroom and guess that the average height of the students present is about 75 inches (6'3"). Ask what a better
guess would be.
15 Recall the Empirical Rule, which says that (for roughly normal data) 95% of the observations will fall within two standard deviations of the mean. Armed with this information, ask them what their
reaction would be if you guessed the standard deviation of the weights in the class to be about two pounds. Ask what a better guess would be.
16 Ask for 10-15 student volunteers willing to disclose (truthfully) their heights and their weights. As each student provides this information, plot the point on an overhead projector or blackboard
for all to see. Also input these numbers into a computer with projection display running a statistical package such as Minitab, or enter the numbers into a hand calculator. Prepare to reveal the
correlation, but do not do so yet.
17 Alternative scenario: Gather the height and weight information at the end of the previous class and have an overhead projection slide and sample correlation figure ready. (I like to appear to live
dangerously, but always have a back-up dataset from some previous class just in case not enough volunteers emerge or the computer crashes.)
18 Typically, of course, there is a positive association between heights and weights. Ask what the class thinks would be a good guess for the correlation represented by the scatterplot of heights and
weights. If this discussion is taking place when the concept of correlation is very new to the class, you will mainly get blank stares and bad guesses. Reveal the correct answer.
19 Tell students that a major purpose of the computer demonstration to follow in lab is for them to be able to gain enough intuition that they will be able to make rough, but intelligent guesses as
to the correlation represented by a scatterplot.
20 Optional additional demonstration: Show overhead projection slides with scatterplots of several bivariate datasets. (These should be scaled so that the height and width of each scatterplot are
approximately equal, and so that each scatterplot uses most, but not all, of the height of the slide.) Ask students which slides represent a clear positive association, which represent clear negative
association, and which represent no significant association at all. Comparing two datasets that exhibit positive association, point out the one with the larger association and stress that it will
have a correlation nearer to +1 than the other.
21 Several datasets are documented in Appendix B and provided with this article. Other datasets might be obtained from the text you are using. The height and weight data collected in class might also
be made available. For the larger datasets you will probably want to use Minitab or some other statistical package to produce the scatterplots.
22 In designing a scatterplot, attention needs to be given to the selection of the scales. The amount of unused space surrounding the data cloud can influence perception of the correlation (Cleveland
et al. 1982). I think it is a mistake to introduce this complication too early while students are learning to use scatterplots to understand correlation. Thus the program has been designed so that
the only changeable parameter is \rho. (Both population means are fixed at 0 and both population standard deviations are fixed at 1.) The parenthetical suggestion just above on scaling scatterplots
for classroom demonstration is in this same spirit.
23 After students have had experience using scatterplots and begin to feel comfortable using them to interpret correlation, you may want to show them examples in which the same data are presented on
several scatterplots with different scales. (See, for example, Moore 1995, page 112.) My preference for an elementary course is to do just enough of this so that students will be aware that proper
scaling is important. Many computer packages for statistical analysis (including Minitab) give you the ability to manipulate scatterplot scales at will. At the suggestion of a referee I have added to
my program the capability to double the scale of either or both axes; the blue ellipse (which, of course, is not a feature of ordinary scatterplots where scaling may be an issue) may also be
suppressed. (From the Title Page of the program, press F2 for instructions on the use of the scaling and ellipse suppression options.)
24 There is considerable literature on the making and interpretation of scatterplots. Additional references, some dealing with issues beyond correlation, are Strahan and Hansen (1978), Shipp and
Margolin (1982), Cleveland and McGill (1984), Raveh (1985), Huber (1987), Lewandowsky and Spence (1989), Meyer and Shinar (1992), and Spence and Garrison (1993).
2.3. Lab Instructions for Students
25 Type PROBDEMO at the DOS prompt and press ENTER at the "copyright" page to get to the "Main Menu," where you should select item 3. To adjust your monitor for the most effective use of this
program, press ENTER to start the program and type in the value 0 for "rho". On this adjustment run, ignore the points and concentrate on the blue figure in the background.
(a) Adjust the brightness of your monitor so that this figure is clearly visible, but not really bright, and
(b) if your monitor has a vertical size adjustment, try to adjust it so that the figure is a perfect circle.
26 Notes: (1) The symbol \rho is not conveniently available on the PC screen, so the name of this symbol "rho" is spelled out. We use the symbol \rho in the rest of these lab notes. (2) In your work
with this program, you may wish to change the speed at which it runs. Press \ (slow) or / (fast) at the title page or continuation prompt to do so; return to normal speed with |.
27 Type N to return to the title page. On the title page it is suggested that you try the following values of \rho: 0, -.2, .4, -.8, .9, -.99, .999, and 1. Do this. (Press Y after each run to do
another run with a different value of \rho.) Then let a neighbor pick several of these values (perhaps changing the sign of some of them), covering the bottom line of the screen so that the values of
\rho and r are not visible to you. Can you figure out which values of \rho your neighbor chose for each run? Pick several values of \rho for your neighbor to guess from scatterplots.
28 Some special features of the program may help in such guessing games: If the person guessing does not watch the simulation run, the person running the computer can press L when it is complete to
hide the legend on the bottom line. If you are working on your own, return to the Title Page and press G. This will put the program into a mode in which it selects values of \rho for you, and hides
the resulting numbers until the run is completed and you have had a chance to make your guess. Also, you can press X, at the Title Page or when asking for the next simulation run, to keep the ellipse
from being plotted in the background; you will probably find that this makes guessing more challenging.
29 How accurate are your guesses expected to be? You are not trying to substitute your eyeball for a computer. You are only trying to get a rough idea what various values of r look like in practice.
Correlations range from - 1 to +1. For values near +1 and -1 you should be able to guess r correct to the nearest 0.1 or even better; for values nearer to 0 you may be off by as much as 0.2 or even
0.3. If r is far enough from 0 you should have no trouble seeing whether it is positive or negative.
2.4. Questions
1. Do five runs of the program using any values of \rho you wish (for easiest viewing, it may be best to keep \rho between -.9 and +.9). The blue ellipse shown for each run is supposed to contain
95% of the 500 points. Thus, it is expected that roughly (.05)(500) = 25 points in each run will fall outside the ellipse. For each of the five runs do your best to count the points that fall
outside. (For some of the ones near the boundary you may have to guess.) Do your results seem to confirm that about 5% of the points fall outside of the ellipse?
2. The value of the POPULATION correlation \rho that you choose is a constant for any one probability model; it is a fixed "population parameter." Notice, however, that the SAMPLE correlation r
is a random variable. It will be different for each sample of 500 observations you select from the population. Why is r different from \rho?
(a) Choose \rho = .9, then do five runs with this same value of \rho (press K after all but the last run), writing down the five values of r that result.
(b) Repeat the steps in (a), but use \rho = .3.
(c) Are your values of r more variable in (a) or in (b)?
(Answer: Of course we cannot say for sure what happened in your particular simulation runs, but the theoretical variance of r [i.e., V(r)] is smaller when \rho = .9 than when \rho = .3. It would
be very surprising if your sampling did not produce results in the same direction.)
3. For this exploration put the program into slow mode by pressing \ at the title page or the continuation prompt before starting each run. (You will see the \-symbol in the lower right-hand
corner of the screen when you do.) In slow mode the first 20 points are plotted very slowly. For five runs with \rho = .9, try to note the value of r when n is about 10. (Values print only at n =
5, 10, 15, etc. Do not worry if you have to settle for 15 instead of 10 or if you botch a run completely and must try again.) For each of the five runs, note the values of r very early in the run
(n = 10 or 15) and at the end of the run (n = 500). Anytime after you have written down the value of r early in each run, you can press | for medium or / for fast speed.
The sample correlation r is an estimator of the population correlation \rho. As with other widely used estimators in statistics, r tends to be a better estimator when it is based on a larger
sample. Do your records from the five runs confirm this principle? (Answer: During a run, the values of r tend to fluctuate at first before beginning to settle down to something near \rho. Of
course, it is possible for a run to begin with nearly the correct value of \rho and then to drift somewhat afield, but this is much less common.)
30 Note: Questions 4-6 use datasets provided in Appendix B. The instructions below are written for use with Minitab (and the Minitab worksheet versions of the data files), but they can be adapted for
use with other statistical packages (and the ASCII versions of the data files). Especially if you are using Minitab, you may want to provide your students with a copy of Appendix C which shows
explicitly how to proceed with Question 4.
4. In a study of the concentration of red blood cells in blood samples from newborn babies, two measures of red blood cells were used: The "hematocrit" is a measure of the percent by volume of
blood that consists of red blood cells. The "hemoglobin" is a quantitative chemical analysis for the protein hemoglobin, the substance that gives these cells their characteristic red color. These
are two ways to try to attach quantitative measures to the concept "concentration of red blood cells." They should be highly correlated. Anemic babies (ones with not enough red blood cells)
should be low on both scales; polycythemic babies (ones with too many red cells) should be high on both scales.
In Minitab, retrieve the worksheet REDCELL.MTW and make a scatterplot of hematocrit (HCrit) against hemoglobin (Hgb). Try to find the center of gravity of the data cloud by estimating the sample
means of each variable and finding the corresponding point on the plot. Try to imagine an ellipse centered there, of the right shape to match the data cloud, and of the right size to contain 95%
of the data points---all but perhaps a couple of them. Try to make a rough intuitive guess as to the value of r. (Write down your guesses for the means and the correlation before you continue. It
would be very surprising if your guesses were exactly correct, but you won't develop your ability to make educated guesses without practicing.)
Finally, use Minitab to find the means of the two variables and the correlation between them. How well did you locate the center of gravity? How good was your estimate of r?
5. The worksheet EUROPREC.MTW contains annual precipitation (in mm) for three European cities, Manchester, Paris, and Madrid, for the 100-year period 1870-1969. "Precipitation" is rainfall plus
snowfall converted to equivalent amounts of rain. Weather patterns in Europe would lead to the supposition that rainfall for Manchester and Paris will show a significant correlation, whereas
precipitation for Manchester and Madrid (distant cities in different climates) would not. Follow the same procedures (scatterplot, guessing, correlation, etc.) as in Question 4 twice: once for
Manchester-Paris and once for Manchester-Madrid. What values of r did you guess? What are the actual computed values? Which pair of cities shows the highest correlation?
6. The worksheet RAINGRAD.MTW shows high school graduation rates (in percents) and typical annual precipitation (in inches) for the 50 United States plus the District of Columbia. Follow the same
procedures as in Question 4 for plotting and guessing r. Can you think of a mechanism or rationale to explain this correlation? If so, write down your speculation. If not, write an explanation of
how you think it was possible to find data with a value of r so far from 0. (Do this before you look at the answer.)
(Answer: Here is how the data were actually obtained: For such a small sample size (here n = 51) it does not take much looking around through an almanac to find two variables that happen to show
a correlation quite different from 0. It would be very difficult to imagine a causative link between rainfall and high school graduation rates.)
2.5. Theoretical Problems for Advanced Students
31 Students near the end of a first calculus-prerequisite course in probability theory or students in a second probability course should be able to demonstrate the validity of the Box-Muller
transformation and of the transformation used to obtain bivariate normal observations (as presented briefly on the page that shows when F1 is pressed from the title page of Part 3). These exercises
are phrased formally below as Advanced Problems 1 and 2. Students can also be expected to derive the equation for the ellipse that contains 95% of the probability---Advanced Problems 3 and 4 below.
Advanced Problem 1. Let U and V be independent random variables, each distributed uniformly on the interval [0, 1). Show that the random variables W and X defined below are independent random
variables, each distributed standard normal:
W = \sqrt{(-2 ln U)} \sin{(2 \pi V)},
X = \sqrt{(-2 ln U)} \cos{(2 \pi V)}.
Advanced Problem 2. If W and X are independent random variables, each distributed standard normal, and if Y is defined as below, then show that (X,Y) has a bivariate normal distribution with E(X)
= E(Y) = 0, V(W) = V(Y) = 1, and correlation \rho:
Y = \rho X + \sqrt{(1 - {\rho}^2)} W.
Advanced Problem 3. With W and X defined as in Advanced Problem 2, show that the random vector (W,X) falls inside the circle
W^2 + X^2 = 5.99
with probability .95.
Advanced Problem 4. Use the result of Advanced Problem 3 and the definitions of X and Y in Advanced Problem 2 to find the equation of the ellipse that contains the point (X,Y) with probability
2.6. Concluding Comments on Part 3
32 I usually use this program in class as soon as the idea of correlation has been introduced. Some elementary books introduce the sample correlation r without mentioning the population correlation \
rho. In this case, I explain briefly in lecture that just as \mu is the population parameter corresponding to \bar{X} (quantifying centrality) and \sigma is the population parameter corresponding to
s (dispersion), so---for bivariate data---\rho is the population parameter corresponding to r (association). Later, when the population correlation has been formally introduced, we take another look
at the program, emphasizing the distinction between the population parameter and the statistic, which can be used as its estimate.
33 Finally, towards the end of a first calculus-based probability course or in a second course, I have found that students are motivated to look at the mathematics behind the program, as outlined in
the advanced problems given above.
34 A demonstration somewhat parallel to this program, but not quite as graphically elegant or easy to use, can be done using Minitab (see Appendix A). This demonstration uses Minitab's procedure for
generating standard normal random variables, so the procedure for obtaining standard normal variates from uniform ones is not explicitly displayed.
35 Note: Instructors interested in a comprehensive collection of simulation experiments in statistics using Minitab may wish to consider Keller (1994). Although this book has no experiments
specifically dealing with correlation, and its primary emphasis is computational rather than graphical, many of the author's purposes are similar to the ones that prompted my programs. Spurrier et
al. (1995) also contains laboratory material appropriate for elementary statistics and probability courses.
3. Part 4: The Casino Game "Keno"
3.1. Operation of the Game and Discussion of Its Purpose
36 Keno is a lottery game played in many gambling casinos. Typically, 80 balls numbered from 1 through 80 are agitated in an air stream by a machine so that 20 of them can be selected at random
during play. Before play begins the gambler marks a ticket printed with the numbers from 1 through 80 in an effort to predict some of the numbers that will be among the 20 drawn. Many variations of
the game are possible, but we concentrate here on the simplest, in which the gambler decides to "mark" a certain number of "spots" (i.e., predict a certain number of balls that will be selected); for
us the number of predictions must be between one and nine.
37 After the 20 balls are selected by the machine, the casino notes how many "hits" (successful predictions) the gambler has accomplished. Each casino publishes lists of payoffs that depend on the
amount bet, the number of spots marked, and the number of hits achieved. Relatively small proportions of hits receive no payoff. Relatively large proportions of hits (although extremely unlikely) can
yield very large payoffs. The payoff schedules used in this program are ones recently advertised by Harrah's casinos in Reno and South Lake Tahoe, Nevada. The maximum possible payoff, for perfect
tickets with large numbers of spots marked, is $50,000. Here is the payoff schedule for $2 bet on a game with only five spots marked, in which the maximum payoff is $1640.
Number Corresponding
of Hits Payoff
38 One goal of the casino, evident from the way Keno games are promoted, is to get the gambler to focus on the largest possible payoff. The minuscule probabilities of such bonanzas are not
advertised. Another goal seems to be to foster the illusion among gamblers that Keno is a game of strategy and skill in which there is some advantage in making wise guesses. It would be illegal to
make this claim forthrightly. (But it is not illegal to encourage players to give very careful consideration to the numbers they pick, nor to act as if it is worthwhile to go through this thought
process afresh for each new game.) Honestly played, Keno is a game of pure chance owing to the randomness of the selection of balls.
39 The probabilities of each number of hits can be computed using the hypergeometric probability distribution. The total number of outcomes from the drawing is the combinations of 80 things taken 20
at a time: C(80, 20) = 3.54 x 10^18. The number of ways to achieve exactly four hits is the number of ways to pick four hits out of five spots marked, C(5, 4) = 5, times the number of ways to pick 16
non-hits out of the 75 spots not marked, C(75, 16) = 8.55 x 10^15. The product is 4.28 x 10^16. Thus, the probability of getting exactly four hits is 4.28/354 = .01209.
40 Similar computations are used to fill in the rest of the probabilities in the table below for our example of a five-spot game, on which $2 is bet.
Hits Payoff Probability Product
0 0 .22718 0.00
1 0 .40569 0.00
2 0 .27046 0.00
3 2 .08394 0.17
4 18 .01209 0.22
5 1640 .00064 1.06
1.00000 1.44
41 The expected amount won E(W) is computed as the sum of the products shown in this table: $1.44. Thus, in our example, the expected return on a bet of $2.00 is about $1.44. It is typical of most
Keno games that the gambler loses on average a little more than 25% of the amount bet. Based on the criterion of the expected percentage of the bet lost on each play, Keno is among the least
favorable among the common "honest" games of pure chance legally available anywhere in the United States. Only the state lotteries with their payoffs of approximately 50% are worse, but with these
there is the hope that the proceeds will be used more or less efficiently for some public good.
42 One can only speculate on the popularity of a game with such miserable odds. One reason is surely the possibility (if not the probability) of winning big for only a small bet. Many players will
have a vivid image in their minds of how they would feel if they "win big" and what they might do with the money. They may feel a thrill as each number is drawn and posted. Another reason might be
that the game requires a minimum of knowledge or concentration---or even sobriety---to play. In fact, "Keno runners" are available throughout the casinos, even in restaurants and cocktail lounges, to
submit marked Keno tickets for play, and the results of each game appear on ubiquitous screens. A third reason might be that it is possible to imagine that one's failure on the game just finished
was, nevertheless, "nearly" a success. "If only I had picked 15 instead of 25" (15 instead of 14, 15 instead of 16, 15 instead of 5, Aunt Sue's birthday which is the 15th instead of Uncle Dan's
birthday which is the 27th, etc.) There are so many ways to imagine that one "almost" won that the probability of "almost" winning (if quantifiable at all) may be quite large indeed.
43 The program in Part 4 gives students the opportunity to try their luck at a simulated game of Keno. They begin with a "grant" of $20 in play money and can continue playing until it is lost. Then
it is easy enough to start the program again with a fresh $20 stake. The illusion that it makes a difference what numbers are chosen is maintained to an extent by the ability to make changes until
the ticket is marked just the way the player desires. However, unlike in the casino atmosphere, the probabilities of winning and the expected loss are clearly posted for each game.
44 From the title page the user has the option to start play at once or to read an introduction in which the game is explained and an indication (much less thorough than provided above in this paper)
is given as to how probabilities are computed. Since several pages of explanation are provided in the introduction, there is no F1-page to provide technical details.
45 Since the look and feel of this program depends on the display of color text and since different computers with monochrome displays treat text so diversely, I could see no feasible way to write a
program that would use a monochrome display with appealing and predictable results. Also the game seems to lose something if the speed is changed much either way. For these reasons there is no
F2-page offering monochrome or speed options. For similar reasons, no Minitab analogue is offered.
3.2. Before Going to the Lab
46 Some preliminary words of caution are necessary. Somehow computers and recreational games seem to go together in the minds of students at all levels. Before introducing ANY demonstration into a
statistics or probability lab, it is important to make sure that there is something tangible to be gained and that adequate thought is given to preventing confusion or abuse. In the case of a game,
however, an extra degree of foresight may be necessary.
47 Before using this game in class it is important to understand exactly what you expect to accomplish by its presentation and to make sure students understand what this is. Written instructions on
how to proceed and a requirement for a written report on what was accomplished are especially important here. It is also wise to make Keno the last item presented at a particular lab session so that
there is a natural ending point to playing it.
48 One approach for introducing the Keno program into an elementary or intermediate probability course is to explain in lecture how Keno is played and to present the payoff table for some specific
instance, such as the one given above for a $2 bet on a five-spot game. (Any relevant payoff table can be obtained by running the program.) Ask if anyone has ever played the game in a casino. Ask
students if it looks like a game they think they could win. Ask how they would judge whether the game is "fair" and what they mean by fair.
49 If the idea of using expectation as a criterion for fairness does not emerge, propose a simpler lottery in which 1000 tickets have been sold and there is only one prize---$1500 for the single
winner. Does $1.50 emerge as the "fair" price for a ticket in such a lottery?
50 Depending on how the discussion goes and on the interests of the instructor, there might be some room at this point for a brief and intuitive mention of the idea commonly called "utility" by game
theorists. (A presentation of the formal ideas would not be difficult, but would take the discussion away from its main purpose.) Here are some points that might be profitably mentioned: Some people
may consider $1.50 to be too small to be of any value, except for the momentary entertainment value it might have. After all, people put such sums of money into video games with NO chance of a
monetary return. On the other hand, $1500 might seem to be a large enough sum to buy them some truly valued item. For such a person $1500 might be worth more than a thousand times $1.50---formally, a
"non-linear utility function." If so, $1.50 might seem like a bargain price for the lottery ticket. What about such a person who gets carried away and participates in 1000 lotteries within some short
span of time---the beginnings of gambling fever?
51 Next, whether or not the hypergeometric distribution is covered in the text for the course, one might show from combinatorial principles how to compute several of the probabilities that go with
the payoff table. The additional probabilities can be supplied without computation and left as an exercise. Then the expected winnings can be computed, and the profitability of the game for the
casino can be discussed.
52 It might also be worth showing an overhead projection slide of a Keno ticket, using the layout on the computer screen as a model. Show a game with five spots marked. Then show an overlay with 20
balls selected that produce only three hits but lots of "near" hits in the immediate vicinity of spots marked. Would students feel that they had "almost" won $18---or maybe even $1640---and be
encouraged to try again?
53 Finally, one might discuss the public perception of how often gamblers win compared with the known, computable probabilities of winning. How likely is someone who does win big at Keno to let
everyone know? How likely is someone who gambles all weekend and loses heavily to advertise his or her lack of success? Certainly, within the confines of the casino, games with no winners are quietly
ignored and the next game is started at once. A game with a big winner is the subject of hoopla for hours if not days or weeks in every medium of exposure available to the casino.
54 (It might also be added here that hoopla governs the public perception of events other than gambling. The news media cover the most interesting events, and do not emphasize more common but
uninteresting occurrences. What are the real chances of being devastated by an earthquake in California? Is it more dangerous to fly from Chicago to Atlanta than it is to drive? Statistical methods
based on random sampling are necessary in the pursuit of truth precisely because our intuitive data gathering about the relative frequencies of even simple events is so easily biased.)
3.3. Lab Instructions For Students
55 Type PROBDEMO at the DOS prompt and then press ENTER at the "copyright" page to get to the "Main Menu," where you should select Item 4. Begin by reading the introduction. Then select a $2 bet on a
five-spot game. Before you play the game notice the number of hits that is most likely. In the long run, over 40% of five-spot games will result in one hit (which yields no payoff).
56 Continue by playing several more five-spot games with $2 bets. Keep track of the number of hits you get on each game. (Also keep track of the number of games with no payoff in which you feel you
"almost won" either an $18 or a $1640-payoff.) Compare notes with other students nearby. Does the 40% figure seem reasonable? (According to your definition of "almost winning," what percentage of the
games fall into that category?)
3.4. Questions
57 Pedagogical Note: The Keno game works well at a variety of academic levels. Letters in parentheses indicate the level of difficulty: E for elementary, M for intermediate, and A for advanced.
58 Questions 1-4 refer to $2 five-spot games.
1. (E) If you play one game, what are the chances of losing your $2? (Answer: .22718 + .40569 + .27046 = .90333.)
2. (E) If you started with $20,000 and played 10,000 games, about how much money would you expect to have left? (Answer: $14,400.)
3. (M) In Problem 2, what is the probability that you will lose all of your money? (Answer: (.90333)^10,000 or very nearly zero.)
4. (M) Find the variance of the amount won in playing one game. (Answer: Using rounded numbers from the table in the program, we have E(W^2) = 4(.08394) + 324(.01209) + 2,689,600(.00064) = .33576
+ 3.91716 + 1721.344 = 1725.59692, V(W) = 1725.60 - 1.44^2 = 1723.)
5. (M) (Continuation of 2 and 4) Find the standard deviation of the amount won in playing 10,000 independent games. Suppose that the Empirical Rule holds so that there is about a 95% chance that
you will wind up with an amount of money that is within two standard deviations of the expected winnings. What interval of likely winnings does this give? (Answer: For the 10,000 games the
variance is 10,000 times the answer to Question 4. The square root of this is $4151. The interval is $14,400 plus or minus twice $4151 or about $6100 to $22,700. This answer cannot be exact
because we are using some rounded numbers from the computer screen for input.)
6. (E) Suppose you play a two-spot game and mark the spots 13 and 66. What is the probability that you will get exactly one hit? (Answer: From the program: .37975. Computed: C(2,1) C(78,19) / C
7. (M) If you bet $2 on the game in Question 6, what payoff for two hits would make this a fair game? (Answer: The probability of two hits is .06013. Since fewer hits will not pay, we need the
product of this probability and the payoff to be equal to the $2 bet; $33.26 comes very close.)
8. (A) (Continuation of 6) Further suppose in the two-spot game in Question 6 that you consider the numbers 3, 12, 14, and 23 to be "near" to 13, and the numbers 56, 65, 67, and 76 to be "near"
to 66. What is the probability that you will get exactly one hit, but feel that one or more other numbers drawn "nearly" gave you a second hit? (Answer: First, we find the probability of exactly
one hit and EXACTLY ONE near hit. In the numerator select which of the two marked numbers is the hit, select one of four allowable numbers for the near hit, and then select the 18 other numbers:
thus, the probability of exactly one hit and exactly one near hit is:
C(2,1) C(4,1) C(74,18) / C(80,20) = .1644.
Probabilities of one hit along with exactly two, three, and four near hits are computed similarly. The total probability of one hit and one or more near hits is .2585. Over a quarter of the time
you will think you "nearly" won.)
3.5. Concluding Comments on Part 4
59 I have used this program most often in elementary probability courses as soon as the concept of expectation for discrete distributions has been introduced. I have used it successfully even when
the text does not include an explanation of the hypergeometric distribution. I either offer my own brief treatment keyed specifically to Keno, or just say that it is possible to compute the
probabilities, provide them without proof, and then focus on how they are used to find the expected winnings. (In the latter case, several students usually insist on private outside-of-class
explanations of how to find the probabilities.)
60 Keno can be used at a much lower academic level than the other programs in this package. At California State University at Hayward we have a variety of periodic events in which high school
students and junior high school students come to campus (usually on a Saturday) for a day of classes, demonstrations, and exhibits. Some of these events are targeted at disadvantaged students or
gifted students, and sometimes any students in the area and their parents are invited. The Keno program has become a standard attraction for these events. I admit that attempts to accompany it with
some appreciation of the probability principles involved meet with varying degrees of success depending on the audience. The sessions always have waiting lines, even with 20 or more available
computer stations, and they are never boring. I see no harm in this sort of carnival atmosphere if one goes into it with eyes open and worthy ulterior motives in mind.
4. Conclusion
61 The programs presented here can be used in an introductory probability course to build student interest and understanding of the concepts of randomness, expected value, and correlation. The
programs' most effective use is in an interactive laboratory setting, with (1) adequate introduction and guidance to focus attention on the concepts to be learned, (2) an opportunity for students to
compare and discuss results, and (3) a structure for students to provide written answers to relevant questions and to summarize what they have learned.
Preparation of this article and development of the programs provided with it were partially supported by NSF Grant USE 91-50433 and by California State University, Hayward. Newton Wai, a statistics
graduate student at Cal State Hayward, has read drafts of this paper and made helpful suggestions. The author also wishes to thank the editor and the referees for their careful and helpful comments.
Computer programs were compiled using Microsoft QuickBasic (Version 4.0). The utility BRUN40.EXE, which must be present to run the programs, is property of Microsoft Corporation and is used with
Appendix A
MINITAB Alternative to Part 3
The attached programs should run on any IBM-PC compatible machine with a display that is EGA or better. In a Windows environment it is usually possible to run DOS programs; many recent Macintosh
machines are also able to emulate or run DOS. However, for those users who need or prefer to use Minitab, macros are included that capture most of the spirit of Part 3. (Values of r based on only
part of a simulation, such as those used in Question 3 of Part 3, are not available in the Minitab macros.) The stored programs BIVCOR.MTB, BIVCOR1.MTB, and BIVCOR2.MTB are intended to be used
together and should work in most pre-Windows releases of Minitab. The stored programs WBIVCOR.MTB, WBIVCOR1.MTB, and WBIVCOR2.MTB will run on Minitab for Windows.
Technical notes:
(1) The three non-Windows Minitab stored programs were written using Release 7 for DOS. BIVCOR2.MTB uses the GPLOT command with a LINE subcommand, neither of which is supported by Windows
versions. WBIVCOR2.MTB uses the new Windows PLOT * command and the new LINE subcommand that goes with it. All of the Minitab stored programs are annotated with comments (following #-symbols) that
explain the key steps.
(2) To operate these programs make sure Minitab addresses the path that contains them (use the CD command, "change drive/disk," if necessary), and at the MTB > prompt type the appropriate one of
the following commands:
MTB > exec 'bivcor'
MTB > exec 'wbivcor'
The appropriate command must be repeated for each run.
Appendix B
Documentation of Data Sets in Part 3
Three datasets are presented; each is provided in three formats. The format with the extension .MTW is a worksheet in the format produced by Minitab Release 7 for DOS and readable by many other DOS
and Windows versions of Minitab. The extension .MTP indicates a Minitab worksheet saved using the PORTABLE subcommand and retrievable by all versions of Minitab. The extension .dat.txt indicates DOS
ASCII format with unlabeled variables, which appear in columns specified below for each dataset. These can be used with almost any statistical computer software.
Data Set 1. Data are taken from Herzog and Felton (1994) based on blood samples from 43 newborn babies at a Northern California community hospital. Variables are labeled HCrit (hematocrit in percent)
and Hgb (hemoglobin in grams per deciliter). Data files are named REDCELL.MTB, etc. In the ASCII file, hematocrit appears in columns 1-4, hemoglobin in columns 8-11. Both variables carry one digit
after the decimal point.
A printout of a Minitab session in response to Question 4 of Part 3 is shown in Appendix C.
Data Set 2. Data are taken from Mitchell (1975). Variables are Year (1870-1969), Manchester (Manchstr), Paris, Madrid. The columns for cities give annual precipitation in millimeters. Data files are
named EUROPREC.MTW, etc. In the ASCII file, Year is in columns 1-4, Manchester in columns 6-9, Paris in columns 11-15, and Madrid in columns 17-20. All variables are integer-valued. I thank Jason
Stover, a graduate student in statistics at California State University, Hayward, for calling these data to my attention.
Data Set 3. Data are taken from The World Almanac and Book of Facts (1994). Variables are State (two letter abbreviation), typical annual rainfall (data in inches, credited to National Climatic Data
Center, U.S. Department of Commerce), and percentage graduation rates from public high schools (credited to National Center for Educational Statistics, U.S. Department of Education). Files are named
RAINGRAD.MTB, etc. In the ASCII file, State appears in columns 1-2, Rainfall in columns 8-9, Graduation Rate in columns 16-17. State is an alphanumeric variable and the other two are integer
All three of these datasets are taken from Trumbo (forthcoming), a compendium of classroom materials based the exploration of real datasets.
Appendix C
Sample MINITAB Run for a Data Set Used in Part 3
This Appendix presents a printout for Question 4 of Part 3 made using Minitab Release 7 for DOS. A character graphics scatterplot was used here so that results could be printed using ASCII
characters. A higher quality plot could be obtained using the Minitab command GPLOT (or the new PLOT command or menu selection in Minitab for Windows).
Technical notes: (1) To be readable, this output should be printed in a format with line widths of at least 76 characters and in a mono-spaced font (such as Courier). (2) It is assumed that the
Minitab worksheet is located in the path C:\JSE.
MTB > retr 'c:\jse\redcell'
[output confirming successful retrieval suppressed]
MTB > plot c1 c2
HCrit -
- * ** *
60+ 3 * *
- * *
- * 2 *
- * *
- 2222
50+ 2* 3 *
- * * 2
- *
- *
- 2
40+ * * *
14.0 16.0 18.0 20.0 22.0
MTB > desc c1 c2
N MEAN MEDIAN TRMEAN STDEV SEMEAN
HCrit 43 52.43 52.10 52.44 6.62 1.01
Hgb 43 17.826 17.500 17.810 2.292 0.349
MIN MAX Q1 Q3
HCrit 39.40 64.70 48.80 57.00
Hgb 13.400 22.700 16.600 19.800
MTB > corr c1 c2
Correlation of HCrit and Hgb = 0.990
Analogous Minitab sessions should be performed in response to Questions 5 and 6 in Part 3.
Cleveland, W. S., Diaconis, P., and McGill, R. (1982), "Variables on Scatterplots Look More Highly Correlated When the Scales Are Increased," Science, 216, 1138-1141.
Cleveland, W. S., and McGill, R. (1984), "The Many Faces of a Scatterplot," Journal of the American Statistical Association, 79, 807-822.
Herzog, B., and Felton, B. (1994), "Hemoglobin Screening for Normal Newborns," Journal of Perinatology, 14(4), 285-289.
Huber, P. J. (1987), "Experiences With Three-Dimensional Scatterplots," Journal of the American Statistical Association, 82, 448-453.
Keller, G. (1994), Statistics Laboratory Manual: Experiments Using Minitab, Belmont, CA: Duxbury Press.
Kennedy, W. J., Jr., and Gentle, J. E. (1980), Statistical Computing, New York and Basel: Marcel Dekker, Inc.
Lewandowsky, S., and Spence, I. (1989), "Discriminating Strata in Scatterplots," Journal of the American Statistical Association, 84, 682-688.
Meyer, H., and Shinar, D. (1992), "Estimating Correlations From Scatterplots," Human Factors, 34, 335-349.
Mitchell, B. R. (1975), European Historical Statistics, New York: Columbia University Press.
Moore, D. S. (1995), The Basic Practice of Statistics, New York: W. H. Freeman and Company.
Raveh, A. (1985), "On Quick Estimates of Pearson's r From Scatter Diagrams [Letter]," The American Statistician, 39, 239-240.
Shipp, C. E., and Margolin, C. G. (1982), "Graphical Display of Scatter Data Using the Standard Deviation Ellipse," in Proceedings of SAS Users Group International Conference, 7, pp. 171-175.
Spence, I., and Garrison, R. F. (1993), "A Remarkable Scatterplot," The American Statistician, 47, 12-19.
Spurrier, J. D., Edwards, D., and Thombs, L. A. (1995), Elementary Statistics Laboratory Manual, Belmont, CA: Duxbury Press.
Strahan, R. F., and Hansen, C. J. (1978), "Underestimating Correlation From Scatterplots," Applied Psychological Measurement, 2, 543-594.
Trumbo, B. E. (1994), "Some Demonstration Programs for Use in Teaching Elementary Probability: Parts 1 and 2," Journal of Statistics Education, v.2, n.2.
Trumbo, B. E. (forthcoming), Exploring Real Data With Minitab (provisional title), Belmont, CA: Duxbury Press.
The World Almanac and Book of Facts (1994), Mahwah, NJ: World Almanac Books.
Zelen, M., and Severo, N. C. (1964), "Probability Functions," in Handbook of Mathematical Functions (1974 ed.), eds. M. Abramowitz and I. A. Stegun, U.S. Department of Commerce, National Bureau of
Standards, Applied Mathematics Series #55 (Tenth Printing with corrections), Washington: U.S. Government Printing Office, pp. 927-995.
Bruce E. Trumbo
Department of Statistics
California State University, Hayward
Hayward, CA 94542
To unpack the files BRUN40.EXE, PROBDEMO.EXE, PDMEN.EXE, PDLLN.EXE, PDPPR.EXE, PDBNS.EXE, and PDKEN.EXE, type prob02 at the DOS prompt.
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Elevators - Force and Power
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Work done by Lifting the Elevator
The work done by lifting an elevator from one level to an other can be expressed as
W = m a[g ] (h[1] - h[0] ) (1)
W = work done (J, ft lb[f] )
m = mass of elevator and passengers (kg, lb[m] )
a[g ] = acceleration of gravity (9.81 m/s^2, 32.17 ft/s^2)
h[1] = final elevation (m, ft)
h[0] = initial elevation (m, ft)
The generic equation for work done by a force can be expressed as
W = F[c ] s (2)
W = work done by force (J, ft lb[f] )
F[c ] = force acting on the elevator at constant speed (N, lb[f] )
s = distance moved by elevator (m, ft)
Forces acting on the Elevator
Since works done in (1) and (2) are equal - the equations can be combined to
F[c ] s = m a[g ] (h[1] - h[0] ) (3)
Force at constant Speed
Since the difference in elevation and the distance moved by the force are equal - (3) can be modified to express the force required to move the elevator at constant speed to
F[c ] = m a[g ] (4)
Force at start/stop
When the elevator starts or stops - the acceleration or deceleration force in addition to the constant speed force can be expressed as
F[a ] = m (v[1] - v[0] ) / t[a ] (5)
F[a ] = acceleration force (N, lb[f] )
v[1] = final velocity (m/s, ft/s)
v[0] = initial velocity (m/s, ft/s)
t[a ] = start or stop (acceleration) time (s)
Power required to move the Elevator
The power required to move the elevator can be calculated as
P = W / t
= m a[g ] (h[1] - h[0] ) / t (6)
P = power (W, ft lb[f] )
t = time to move the elevator between levels (s)
Example - Force and Power to Lift an Elevator
An elevator with mass 2000 kg including passengers are moved from level 0 m to level 15 m . The force required to move the elevator at constant speed can be calculated as
F[c ] = (2000 kg) (9.81 m/s^2)
= 19820 N
The power required to move the elevator between the levels in 20 s can be calculated as
P = (2000 kg) (9.81 m/s^2) ((15 m) - (0 m)) / (20 s)
= 14865 W
= 14.9 kW
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The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.
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Power is the rate at which work is done or energy converted.
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28.75 An Hour Is How Much A Year
If you are earning $28,75 per hour, you will be making $5,606,250 a year. This is a very generous amount, and most companies also offer time off to work over holidays. An eight hour workday is equal
to around 2,080 hours in 2022. Therefore, a full-time employee will need to work a total of about 2800 hours a month to earn the same amount.
If your hourly earnings total about $28.75 per year, it is important to remember that you will have to pay taxes on the entire amount. Since you will earn approximately $6,500, you will pay tax on
the sum of these two figures. In order to calculate your tax burden, multiply the annual salary by 6.25 percent. Then, multiply the result by 25. You will need to add up your yearly expenses and make
sure that you don’t have any other pending bills.
The effective tax rate for individuals is usually around 25%. Therefore, you would multiply your hourly earnings by six times and subtract the tax rate. Your effective tax rate is approximately 25
percent. The difference in your earnings between the two figures will be zero. You should also note that your tax bracket is about the same as that of a full-time employee. If you want to calculate
your tax liabilities, you can use the following formula.
If you are paying yourself, you would expect to pay a quarter-hour’s worth of tax. Since your effective tax rate is around 25%, this means that your hourly earnings are multiplied by six times. Thus,
your effective tax rate is approximately 25. That means that you should subtract your hourly salary from your yearly wage. Then, you would need to divide the amount by six, and the difference would
be one-quarter of your annual income.
If you work twenty-eight hours a week, you would earn approximately $8,750 a year. Then, if you pay a quarter of that, you would need to deduct a quarter’s worth of tax. Hence, the total income you
would earn in the month would be about $675. This will be your taxable earnings. The amount of money you earned is about two and a half.
The effective tax rate in the United States is about 25%. The hourly earnings would be multiplied by zero and six times by the effective tax rate. Assuming that you work for the same employer every
day, you would make $365 per year. If you work forty-eight hours a week, the income from your overtime will be $5,250. For the same wage, you will earn around $2,800 in a year.
The effective tax rate is about 25%. Then, multiply the hourly earnings by zero and by six. In this case, you would have paid out $1,875.75 for the year. A quarter of that would have been taxable,
but you will still be paying out the rest. This is a very reasonable amount. If you are self-employed, your tax rate would be lower than if you are self-employed.
The effective tax rate is around 25%. Therefore, if you earn $28,75 per hour, you will make $6,875 a year. If you are self-employed, you will pay approximately half of this amount in taxes. You
should also consider the effect of inflation on your income. You must ensure that you are comfortable with the taxes before you can make any big decisions. If you are unsure about whether or not you
should get a job, consult with a lawyer before you do so.
The effective tax rate is around 25%. If you are self-employed, you will pay half of your earnings to the taxman. Then, your hourly earnings will be multiplied by zero and divided by six to get your
tax bill. You will pay back the rest of your taxes if you are self-employed. The effective tax rate depends on your state and the amount of income you are making. | {"url":"https://sonichours.com/28-75-an-hour-is-how-much-a-year/","timestamp":"2024-11-11T03:07:19Z","content_type":"text/html","content_length":"69878","record_id":"<urn:uuid:a1cfc265-ae61-47b6-8ed6-824ae723e3c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00292.warc.gz"} |
Recurrence relations of the Hermite polynomials - Mono Mole
Recurrence relations of the Hermite polynomials
The recurrence relations of the Hermite polynomials describe how each polynomial in the sequence can be obtained from its predecessors.
Expanding eq18 and assuming $n$ is an even number,
Taking the derivative of eq26, we have
Substituting eq27 in eq28,
If we repeat the steps from eq26 through eq29 on the assumption that $n$ is an odd number, we too end up with eq29. Therefore, $n$ in eq29 represents any number. Taking the derivative of eq29 again,
Substitute 29 and 30 in eq23, we have
Replacing the dummy index $n$ in eq31 with $n+1$ gives
Eq29 and eq32 are the recurrence relations of the Hermite polynomials.
Given $H_0(y)=1$, show that eq32 can be used to generate the Hermite polynomials.
Substituting $n=0$ in eq32, we have $H_1(y)=2yH_0(y)=2y$. Substituting $n=1$ in eq32, we have $H_2(y)=4y^2-2$. Repeating the logic, we can generate the rest of the polynomials.
Using eq32, $\small{y=\sqrt{\frac{m\omega}{\hbar}}x}$ and eq46, show that $\small{\langle\psi_k\vert x\vert\psi_n\rangle=\sqrt{\frac{\hbar(n+1)}{2m\omega}}\delta_{k,n+1}+\sqrt{\frac{n\hbar}{2m\
Substituting eq32 in eq46, we have
Multiplying the above equation by $\psi^*_k(x)$ and integrating over all space,
$\langle\psi_k\vert x\vert\psi_n\rangle=\sqrt{\frac{\hbar(n+1)}{2m\omega}}\delta_{k,n+1}+\sqrt{\frac{n\hbar}{2m\omega}}\delta_{k,n-1}\;\;\;\;\;\;\;\;32a$ | {"url":"https://monomole.com/recurrence-relations-of-the-hermite-polynomials/","timestamp":"2024-11-03T09:38:26Z","content_type":"text/html","content_length":"102191","record_id":"<urn:uuid:5a957a9f-4205-40fb-9210-1d2d208b159b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00552.warc.gz"} |
Mathematics Colloquium
Energy Identity for Stationary Yang Mills
Speaker: Aaron Naber, Northwestern University
Location: Warren Weaver Hall 1302
Date: Monday, November 14, 2016, 3:45 p.m.
Yang Mills connections over a principle bundle are critical points of the energy functional \(\int |F|^2\), the \(L^2\) norm of the curvature, and thus may be viewed as a solution to a nonlinear pde.
In many problems, e.g. compactifications of moduli spaces, one considers sequences \(A_i\) of such connections which converge to a potentially singular limit connection \(A_i \rightarrow A\). The
convergence may not be smooth, and we can understand the blow up region by converging the energy measures \(|F_i|^2 dv_g \rightarrow |F|^2dv_g +\nu\), where \(\nu=e(x)d\lambda^{n-4}\) is the \(n-4\)
rectifiable defect measure (e.g. think of \(\nu\) as being supported on an \(n-4\) submanifold). It is this defect measure which explains the behavior of the blow up, and thus it is a classical
problem to understand it. The main open problem on this front is to compute \(e(x)\) explicitly as the sum of the bubble energies which arise from blow ups at \(x\), a formula known as the energy
identity. This talk will primarily be spent explaining in detail the concepts above, with the last part focused on the recent proof of the energy quantization, which is joint with Daniele Valtorta.
The techniques may also be used to give the first apriori higher derivative estimates on Yang Mills connections, and we will discuss these results as well. | {"url":"https://math.nyu.edu/dynamic/calendars/seminars/mathematics-colloquium/458/","timestamp":"2024-11-06T08:27:45Z","content_type":"text/html","content_length":"49103","record_id":"<urn:uuid:ccd1cfc3-31f0-467b-9108-d585f31aa9dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00578.warc.gz"} |
Abstract : This research came up because of the students’ decreased achievement of class X and X1 in the entrepreneurship subject of the first semester in the academic year of 2015/2016.183 out of
645 students gained under minimum passing grade (KKM) in the entrepreneur subject. This phenomenon happened because of some factors. One of them was an unmotivated learning indicated by the laziness
to engage the teaching learning process, such as the frequent going in and out of the class, absence, and not doing the task given by the teacher. This research aimed at investigating in what extent
the influence of students’ learning motivation towards the learning achievement of the students having under minimum passing grade at the entrepreneurship subject at SMK Muhammadiyah 1 Pekanbaru. The
population of this research was 183 students with 65 students as the samples. Probability sampling, simple random sampling, is used as the technique to gain the sample in this research. The data were
collected using observation, questionnaire, and documentation. Based on the data analysis, it can be concluded that the significance value in the coefficient table 0.000 is less than critical value
0.1. Thus, it is obtained that 0.000 < 0.1, which means that the students’ learning motivation influences the students’ learning achievement in the entrepreneurship subject at SMK Muhammadiyah 1
Pekanbaru. Viewed from the R square 0.207 shows that the independent variable student learning motivation used in this research is influenced by the dependent variable student learning achievement at
the rate of 20.7%. Thus, it can be said that the capability of the independent variable student learning motivation in influencing the dependent variable student learning achievement is high. Based
on the calculation of t test, the t table 4,057 > t quantification 1.998 (Ho) is rejected or sig 0.000 < 0.05 (Ha) is accepted. It means that there is an influence of the students’ learning
motivation towards the students’ learning achievement. Based on the regression coefficient, if the students’ learning motivation happens to increase at rate of 1%, the students’ learning achievement
also will increase at the rate of 0.284 or 28.4%. The positive coefficient shows that there is positive correlation between students’ learning achievement and students’ learning motivation. The more
students are motivated, the more students will reach the achievement.
Keywords: Students’ Learning Motivation, Students’ Learning Achievement2
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Real World Impacts of Learning Math
Mathematics. Once it becomes hard to visualize, caring about learning it can become a real struggle of internal nagging questions like…
• “Why do I need to learn this?”
• “How will this ever be useful in my life?”
• “What is the point of knowing how to solve these problems?”
These thoughts are perfectly understandable. I wondered too, if I would ever “use” math regularly. In high school, I saw it more as puzzle-work and not applicable to most common situations. In
certain careers, sure, but in my day-to-day life? I was not so sure.
Now, I understand the impact of learning math better. Not just because I have seen its use in industry and academia, but because I see how logical reasoning as a skill is remarkably useful and
applicable to ordinary scenarios. And the development of logical thinking skills is a major effect of engaging in the process of learning and doing math.
Viewing math as just formulaic puzzle-work misses the mark of what the actual impact is. Memorizing formulas and knowing how to apply them is not the long term takeaway. Developing the deductive and
logical thinking skills involved in solving the problems is. Because of the field’s strict logical rules, solving mathematical problems requires the application of logical reasoning skills. These
skills can be applied during real-life situations.
Consider being challenged with the task of creating a method to practice applying logical thinking. Certainly you couldn’t plan exercises that work through every scenario that you may face that would
require logical reasoning, so the method must generalize to be applicable to a range of situations. Creating this would be very difficult.
Mathematics solves this challenge. It provides problems that can only be solved in a logical way. The process of learning math requires actively practicing logical reasoning to solve mathematical
“Even if you don’t remember the quadratic formula, the process of learning it made you a clearer thinker. That’s how the entire world teaches math.” - Svetlana Jitomirskaya, Math Professor, U
California Irvine
Concepts and features of mathematical problem solving that are applicable in real-world scenarios:
• Understanding the concept of multiple factors influencing a problem
• Understanding how different initial circumstances lead to different outcomes
• Solving the same problem in different ways / from different viewpoints
• Contextualizing a problem’s scope, limitations, and nuances
• Developing, organizing, and applying a process structure while working towards solutions
• Saving solution strategies and methods internally to apply later on similar problems
• Prioritizing correctness and punctuality based on what is known to be true, not believed
• Checking other’s work by thinking for yourself using logical reasoning
• Analyzing a problem quantitatively and objectively
Furthermore, understanding the concept of “quantitative proof” and being able to do math is empowering in the sense that it gives people the power to recognize when they are being manipulated - by a
lengthy car loan that will cost them much more over the long term - by the statistics fed to them by politicians and PACs - by the “scientists have proven” articles that have zero quantitative
foundation. In a world of clickbait and endless manipulation of the regular person, understanding mathematics and quantitative logic, and applying it appropriately, functions like a strong piece of
armor against such forces.
Short and sweet. I just wanted to put this in writing somewhere. Math is extraordinarily “pure” in its logical foundations, but similar arguments could be made for numerous other fields like
rhetoric, computer science, and more. There are other ways to view the impacts than my descriptions here, it is just my two cents. My undergrad degree is in applied mathematics. | {"url":"https://cornbreadjournals.com/mathimpacts/","timestamp":"2024-11-13T04:58:35Z","content_type":"text/html","content_length":"6239","record_id":"<urn:uuid:d8e0d014-53d5-4efe-bbf0-540417db4cad>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00622.warc.gz"} |
Humus - News
"If there's one thing worse than a program that doesn't work when it should, it's a program that does work when it shouldn't."
- Bob Archer
Wednesday, January 14, 2009 |
When you look at a highly tessellated model it's generally understood that it will be vertex processing heavy. Not quite as widely understood is the fact that increasing polygon count also adds to
the fragment shading cost, even if the number of pixels covered on the screen remains the same. This is because fragments are processed in quads. So whenever a polygon edge cuts through a 2x2 pixel
area, that quad will be processed twice, once for both of the polygons covering it. If several polygons cut through it, it may be processed multiple times. If the fragment shader is complex, it could
easily become the bottleneck instead of the vertex shader. The rasterizer may also not be able to rasterize very thin triangles very efficiently. Since only pixels that have their pixel centers
covered (or any of the sample locations in case of multisampling) are shaded the quads that need processing may not be adjacent. This will in general cause the rasterizer to require additional
cycles. Some rasterizers may also rasterize at fixed patterns, for instance an 4x4 square for a 16 pipe card, which further reduces the performance of thin triangles. In addition you also get
overhead because of less optimal memory accesses than if everything would be fully covered and written to at once. Adding multisampling into the mix further adds to the cost of polygon edges.
The other day I was looking at a particularly problematic scene. I noticed that a rounded object in the scene was triangulated pretty much as a fan, which created many long and thin triangles, which
was hardly optimal for rasterization. While this wasn't the main problem of the scene it made me think of how bad such a topology could be. So I created a small test case to measure the performance
of three different layouts of a circle. I used a non-trivial (but not extreme) fragment shader.
The most intuitive way to triangulate a circle would be to create a fan from the center. It's also a very bad way to do it. Another less intuitive but also very bad way to do it is to create a
triangle strip. A good way to triangulate it is to start off with an equilateral triangle in the center and then recursively add new triangles along the edge. I don't know if this scheme has a
particular name, but I call it "max area" here as it's a greedy algorithm that in every step adds the triangle that would grab the largest possible area out of the remaining parts on the circle.
Intuitively I'd consider this close to optimal in general, but I'm sure there are examples where you could beat such a strategy with another division scheme. In any case, the three contenders look
And their performance look like this. The number along the x-axis is the vertex count around the circle and the y-axis is frames per second.
Adding multisampling into the mix further adds to the burden with the first two methods, while the max area division is still mostly unaffected by the added polygons all the way across the chart.
35 comments
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Oracle SIGN() function - w3resource
Oracle SIGN() function
The SIGN() returns the sign of an argument.
This function is used to return the sign of a given number. This function takes as an argument any numeric data type and any nonnumeric data can also comes in the argument but that can be implicitly
converted to number and returns number.
The function returns:
• 1 when the value of the argument is positive
• -1 when the value of the argument is negative
• 0 when the value of the argument is 0
For binary floating-point numbers (BINARY_FLOAT and BINARY_DOUBLE), this function returns the sign bit of the number. The sign bit is:
• -1 if n<0
• +1 if n>=0 or n=NaN
Uses of Oracle SIGN() Function
• Determine the sign of a number: Identify if a number is positive, negative, or zero.
• Conditional logic: Use in conditional statements to execute different logic based on the sign of a number.
• Mathematical operations: Assist in various mathematical computations where the sign of a number influences the result.
• Data analysis: Facilitate data analysis by categorizing numbers based on their sign.
• Binary floating-point operations: Return the sign bit of binary floating-point numbers.
Name Description
n A number whose sign is to be retrieved.
Pictorial Presentation of SIGN() function
SELECT SIGN(-145), SIGN(0), SIGN(145) FROM dual;
Here is the result.
SIGN(-145) SIGN(0) SIGN(145)
---------- ---------- ----------
-1 0 1
The above statement will return the sign of given numbers -145, 0 and 145.
Previous: ROUND
Next: SIN
It will be nice if you may share this link in any developer community or anywhere else, from where other developers may find this content. Thanks.
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This project for a Calculus I class was adapted from a similar set of problems from the Calculus series by James Stewart.
\documentclass[12pt]{amsart} \addtolength{\hoffset}{-2.25cm} \addtolength{\textwidth}{4.5cm} \addtolength{\voffset}{-2.5cm} \addtolength{\textheight}{5cm} \setlength{\parskip}{0pt} \setlength{\
parindent}{15pt} \usepackage{amsthm} \usepackage{amsmath} \usepackage{amssymb} \usepackage[colorlinks = true, linkcolor = black, citecolor = black, final]{hyperref} \usepackage{graphicx} \usepackage
{multicol} \usepackage{ marvosym } \usepackage{wasysym} \usepackage{tikz} \newcommand{\ds}{\displaystyle} \pagestyle{myheadings} \setlength{\parindent}{0in} \pagestyle{empty} \begin{document} \
thispagestyle{empty} {\scshape Math 2300} \hfill {\scshape \Large Project \#4} \hfill {\scshape Fall 2017} \medskip \hrule \bigskip \bigskip In this project, you will investigate the most economical
shape for a can. We're considering food storage cans here. Think soup, tunafish, beans, corn, soda, etc. These are cylinders. For this entire project, use the variable $V$ for the volume of the can,
$r$ for the radius of the cylinder, and $h$ for the height. You are free to look up formulas for the volume and area of any shapes that are helpful. You may use a computer program for simplifying
algebraic expressions or even to find helpful derivatives, but you should not include information or findings that you do not fully understand, nor skip any steps that you could not justify with
computations by hand. \bigskip The goal here is to look at a variety of scenarios to try to determine the optimal ratio of the height of a can to its radius, $\frac{h}{r}$. It should work out best to
complete these problems in the order they are listed. You do not need to copy the questions word for word in your write-up, nor do you even need to number them. However, a reader should be able to
read your report without knowing anything about the project or having access to this document\dots and you need to answer {\bf all} the questions. This means you need to use your words. \bigskip \
bigskip \begin{enumerate} \item Measure some cylindrical cans from your cupboard, pantry, or the grocery store. Choose a few different sizes and write down the height and diameter of each. Then,
compute the ratio of height to radius. Include pictures of your can examples. \vfill \item Suppose that the cost of the metal is the same for the top, bottom, and side of a cylindrical can. Show that
the cost for materials is minimized when $h = 2r$. What will cans fitting this description look like when viewed from the side? \vfill \item The idea above only considers the cost of the metal that
shows up in the final product, but cutting circular tops and bottoms from rectangular sheets means that some of that metal is paid for, but not used. If the tops and bottoms are cut from squares as
in the picture below, show that the amount of needed metal is minimized when $\frac{h}{r} = \frac{8}{\pi}$. \bigskip \begin{center} \begin{tikzpicture}[scale = 1.732] \draw (0,0) grid (4,3); \foreach
\x in {.5, 1.5, 2.5, 3.5} \foreach \y in {.5, 1.5, 2.5} \draw[very thick, fill = gray!20] (\x,\y) circle (.5cm); \end{tikzpicture} \end{center} \vfill \item How do cans with these parameters compare
to those from Problem 2? Taller? Wider? Skinnier? etc. Why does that make sense? \vfill \newpage \item A more efficient way of cutting out the disks is obtained by dividing the metal sheets into
hexagons rather than squares. If this strategy is chosen, show that the minimal amount of material happens when $\frac{h}{r} = \frac{4\sqrt{3}}{\pi}$. \bigskip \begin{center} \begin{tikzpicture} \
clip (-.5,.5) rectangle (6,4); \foreach \i in {-1,...,2} \foreach \j in {-1,...,2} { \foreach \a in {0,120,-120} \draw (3*\i,2*sin{60}*\j) -- +(\a:1); \foreach \a in {0,120,-120} \draw (3*\i+3*cos
{60},2*sin{60}*\j+sin{60}) -- +(\a:1); \draw[very thick, fill = gray!20] (3*\i+.5,2*sin{60}*\j+.866) circle (.866cm); \draw[very thick, fill = gray!20] (3*\i+3*cos{60}+.5,2*sin{60}*\j+sin{60}+.866)
circle (.866cm);} \end{tikzpicture} \end{center} \vfill \item How do these cans compares with previous ones? Why does that make sense? \vfill \item So far, we've only considered the costs of the
materials, not the cost of assembly of those materials. Let's assume that most of the cost in manufacturing is incurred in joining the pieces of metal and that we cut disks from hexagons as in
Problem 5. Call the cost per square unit of metal $M$ and the cost per linear unit to join the metal $J$. Explain why (or show that) the cost to produce one can is given by $$C = M\left(4\sqrt{3} r^2
+ 2\pi rh\right) + J\left(4\pi r + h\right).$$ \vfill \item Show that this cost is minimized when $$\frac{M\sqrt[3]{V}}{J} = \sqrt[3]{\frac{\pi h}{r}}\cdot\frac{2\pi - h/r}{\pi h/r - 4\sqrt{3}}.$$ \
vfill \item Plot the function $\ds{\frac{M\sqrt[3]{V}}{J}}$ as a function of $x = \frac{h}{r}$. You will want to replace each $\frac{h}{r}$ on the right side with $x$, and use that as the independent
variable along the horizontal axis of the graph. Which portion of the graph has no meaning in this situation? What are some interesting features of this graph? \vfill \item What would a larger value
of $\ds{\frac{M\sqrt[3]{V}}{J}}$ represent in terms of can manufacturing? What does the graph tell us about the optimum value for the ratio $\frac{h}{r}$ in those cases? \vfill \item What would a
smaller value of $\ds{\frac{M\sqrt[3]{V}}{J}}$ represent in terms of can manufacturing? What does the graph tell us about the optimum value for the ratio $\frac{h}{r}$ in those cases? \vfill \item
This analysis should show that larger volume cans should be nearly square (the diameter of the can is very close to its height), while it is more economical to make smaller cans in a tall and thin
shape. Make sure that statement agrees with your findings, and compare that to the shapes of the cans in Problem 1. If there are differences, what may have caused that? \end{enumerate} \end{document} | {"url":"https://cs.overleaf.com/articles/fsu-math2300-project4/jhtthqgwwdcc","timestamp":"2024-11-04T12:17:21Z","content_type":"text/html","content_length":"42479","record_id":"<urn:uuid:6f453a83-8fa4-466f-a328-8f0781d93ea8>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00563.warc.gz"} |
Moment of inertia problem
What is the moment of inertia about an axis through its center of a sphere of radius R, mass M and density varying with radius as Ar.
Can anyone explain this? This is a Conquering physics book problem and I did not get this solution.
Re: Moment of inertia problem
I assume you are referring to Q2 in the 1.4.5 problem set from "Conquering the Physics GRE".
I also found this to be a tricky problem. What I would do is first take the volume integral of the density and set it equal to the mass. Then I solve for A. I believe this is shown pretty well in the
solution, giving us $$M = 4 \pi \int \rho (r^2 dr) = \pi A R^4 \rightarrow A = \dfrac{M}{\pi R^4}$$. (Note that the $$r^2$$ is because it's a volume integral in spherical coordinates and that the $$4
\pi$$ is from integrating over $$\theta$$ and $$\phi$$.
Once you have this you integrate $$I = \int s^2 \pho dV$$, where s is not the same as r. This is the really tricky part of this problem. s, as is explained in the solution, is the perpendicular
distance from the axis of rotation to a point on the sphere. Moments of inertia are always computed using this, which is also known as a Moment Arm. Thus we can define $$s = r \sin(\theta)$$Thus,
your fully integral would be $$\int \left( r \sin(\theta) \right)^2 \rho r^2 \sin(\theta) dr d \theta d \phi$$.
I hope that this was helpful. If you have any questions please feel free to ask.
Re: Moment of inertia problem
So how did find that
s = r sin(theta)
Re: Moment of inertia problem
tazi wrote:So how did find that
s = r sin(theta)
If you draw it you'll see immediately why. | {"url":"https://physicsgre.com/viewtopic.php?f=19&t=6120","timestamp":"2024-11-07T12:53:43Z","content_type":"text/html","content_length":"29397","record_id":"<urn:uuid:f31661cf-d293-4c8f-a438-9b0b061e84b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00653.warc.gz"} |
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Proof of the Law of Conservation of Energy
• Thread starter superfahd
• Start date
In summary, Leonard Susskind discusses the law of conservation of energy in a complicated way. He uses the formula F = - \vec{}\nabla U (x, y). He also mentions that differentiation concepts need a
lot of review.
Hi guys. I've decided to review my physics after a long time through Leonard Susskind's youtube lectures. I'm at lecture 2 and I'm already confused!
in the 1st half hour, he gives a proof of the law of conservation of energy. In the course of this proof he uses the formula: F = - [tex]\vec{}\nabla[/tex] U (x, y). Where U(x,y) is the potential
energy of a particle at position (x,y). I don't remember any such formula from my secondary school classes. Can someone please explain to me how this formula comes out and what it even means?
Also he then writes: d U(x,y) / dt = [tex]\Sigma[/tex][i][tex]\partial[/tex]U/[tex]\partial[/tex]X[i].[tex]\dot{}X[/tex] +[tex]\partial[/tex]U/[tex]\partial[/tex]Y[i].[tex]\dot{}Y[/tex] (I'm not sure
if I rendered the formula correctly. This latex thing is confusing). How does he arrive to this? I realize that differentiation concepts need a lot of review but I'm only doing this as a hobby so can
someone explain it to me as such? Thanks a lot
Staff Emeritus
Science Advisor
Gold Member
To be honest, I don't think that you've seen that in high school (depends maybe on the high school in question).
The inverted triangle stands for a vector differentiation operator. If you apply it to a function of 3 variables x, y and z, it produces a vector where the first component is the partial derivative
of the function wrt x, the second component is the partial derivative of the same function, but wrt y this time, and the third component is the partial derivative wrt z.
For instance, if U(x,y,z) = x^3*y^2 + 5*y*z
then you get as a first component: 3 x^2 * y^2
as a second component: 2*x^3 * y + 5 * z
and as a third component: 5 * y
So this gives you a vector in every point in space. For instance, in the point (1,2,3) this becomes the vector with components (12,19,10)
(if I didn't make any silly mistake...)
Staff Emeritus
Science Advisor
Gold Member
Whilst it may not be crucial for your understanding here, it is important to note that the quantity U is not the potential energy, but simply the potential. The two are very closely related, but not
A simpler version of this, using a single dimension:
I don't see how using math could 'prove' anything in physics though. All we can do is confirm that physics theories closely approximate (as best as we can measure) reality via experiments.
Hootenanny said:
U is not the potential energy, but simply the potential.
U is normally used for potential energy, while Φ or V is used for potential. Wiki link, note the part about gravitational potential:
Last edited:
Staff Emeritus
Science Advisor
Gold Member
Jeff Reid said:
I don't see how using math could 'prove' anything in physics though. All we can do is confirm that physics theories closely approximate (as best as we can measure) reality via experiments.
I disagree. For example, Noether's theorem states that we can obtain a law of conservation if the action of a physical system has a differentiable symmetry. Admittedly, we have to start from
somewhere, say from a physical law. However, if we assume that a physical law holds (and has some symmetry), then we can prove a conservation law, for example.
Proof of law of conservation of energy?...it's an assertion!...well that's what most books say.
Hootenanny said:
I disagree. For example, Noether's theorem states that we can obtain a law of conservation if the action of a physical system has a differentiable symmetry. Admittedly, we have to start from
somewhere, say from a physical law. However, if we assume that a physical law holds (and has some symmetry), then we can prove a conservation law, for example.
But aren't the symmetries confirmed by experimental observations as are any physical laws.
There was the famous observation which showed that gravity bends light, tending to validate one of Einstein's theories. This was celebrated as front page news the world over.
If I recall correctly, Einstein remarked, "A thousand experiments could not prove the theory, and a single experiment could disprove it. The theory, however, is quite correct."
Maybe my recollection is off but the principle of proving things is based upon models that are internally self-consitent, and open ended theories are tested repeatedly by gaining more and more
empirical experience.
Jeff Reid said:
A simpler version of this, using a single dimension:
I don't see how using math could 'prove' anything in physics though. All we can do is confirm that physics theories closely approximate (as best as we can measure) reality via experiments.
U is normally used for potential energy, while Φ or V is used for potential. Wiki link, note the part about gravitational potential:
Thanks a lot jeff. This was precisely the definition i was looking for but was having trouble finding. I mean how the heck do you google a formula anyway!
Also now that my curiosity has peaked to an annoying level, I'll probably be bugging you guys about a lot of such stuff as I go on with the lectures!
Keyword search gets easier when you recognize more patterns. I usually google a term of art with "wiki" or "hyperphysics" or "nasa" and sometimes use the Image rather than text box. However when
you're still learning a new area it can yield confusing information.
For example, in google, try "del operator". You'll get the Wiki and Hyperphysics search links right at the top!
FAQ: Proof of the Law of Conservation of Energy
1. What is the Law of Conservation of Energy?
The Law of Conservation of Energy states that energy cannot be created or destroyed, it can only be transformed from one form to another. This means that the total amount of energy in a closed system
remains constant.
2. How was the Law of Conservation of Energy discovered?
The Law of Conservation of Energy was first proposed by Julius Robert von Mayer in 1842 and later independently by James Prescott Joule in 1847. It was further developed and refined by other
scientists, including Hermann von Helmholtz and William Thomson (Lord Kelvin).
3. What evidence supports the Law of Conservation of Energy?
There is overwhelming evidence that supports the Law of Conservation of Energy. Numerous experiments have been conducted in various fields such as thermodynamics, mechanics, and electromagnetism that
consistently show that the total amount of energy in a closed system remains constant.
4. Are there any exceptions to the Law of Conservation of Energy?
The Law of Conservation of Energy is considered a fundamental principle in physics and has been extensively tested and confirmed. However, there are some rare cases, such as in nuclear reactions,
where mass can be converted into energy and vice versa, as described by Einstein's famous equation E=mc².
5. How is the Law of Conservation of Energy applied in everyday life?
The Law of Conservation of Energy is applied in many everyday phenomena, such as the conversion of chemical energy into heat and light in a candle, or the conversion of electrical energy into motion
in a motor. It is also used in energy conservation efforts, where the goal is to minimize energy waste and maximize energy efficiency. | {"url":"https://www.physicsforums.com/threads/proof-of-the-law-of-conservation-of-energy.359037/","timestamp":"2024-11-06T21:45:30Z","content_type":"text/html","content_length":"116423","record_id":"<urn:uuid:8017c3c6-139e-461e-9fea-1ef40be4fe5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00096.warc.gz"} |
Polynomials whose coefficients are generalized Tribonacci numbers
Let a[n] denote the third order linear recursive sequence defined by the initial values a[0] = a[1] = 0 and a[2] = 1 and the recursion a[n] = pa[n-1] + qa[n-2] + ra [n]-[3] if n ≥ 3, where p, q, and
r are constants. The a[n] are generalized Tribonacci numbers and reduce to the usual Tribonacci numbers when p = q = r = 1 and to the 3-bonacci numbers when p = r = 1 and q = 0. Let Q[n](X) = a[2]x^n
+ a [3]x^n-1 +⋯+ a[n+1]x + a[n+2], which we will refer to as a generalized Tribonacci coefficient polynomial. In this paper, we show that the polynomial Q[n](X) has no real zeros if n is even and
exactly one real zero if n is odd, under the assumption that p and q are non-negative real numbers with p ≥ max{l, q}. This generalizes the known result when p = q = r = 1 and seems to be new in the
case when p = r = 1 and q = 0. Our argument when specialized to the former case provides an alternative proof of that result. We also show, under the same assumptions for p and q, that the sequence
of real zeros of the polynomials Q[n](x) when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence a[n]. In the casep = q = r = 1,
this convergence is monotonie. Finally, we are able to show the convergence in modulus of all the zeros of Q[n](x) when p ≥ 1 ≥ q ≥ 0.
• Linear recurrences
• Tribonacci numbers
• Zeros of polynomials
All Science Journal Classification (ASJC) codes
• Computational Mathematics
• Applied Mathematics
Dive into the research topics of 'Polynomials whose coefficients are generalized Tribonacci numbers'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/polynomials-whose-coefficients-are-generalized-tribonacci-numbers","timestamp":"2024-11-03T00:10:42Z","content_type":"text/html","content_length":"50445","record_id":"<urn:uuid:0df6d6fd-64e9-476f-9ba7-e213005a3d4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00294.warc.gz"} |
Solutions of Right Triangles Class-9 Concise Selina ICSE Maths - ICSEHELP
Solutions of Right Triangles Class-9 Concise Selina ICSE Maths
Solutions of Right Triangles Class-9 Concise Selina ICSE Mathematics Solutions Chapter-24 . We provide step by step Solutions of Exercise / lesson-24 Solutions of Right Triangles for ICSE
Class-9 Concise Selina Mathematics by R K Bansal.
Our Solutions contain all type Questions with Exercise-24, to develop skill and confidence. Visit official Website CISCE for detail information about ICSE Board Class-9 Mathematics .
Solutions of Right Triangles Class-9 Concise Selina ICSE Mathematics Solutions Chapter-24
Exercise 24
Question 1
Fin ‘x’ if:
Question 2
Find angle ‘A’ if:
Question 3
Find angle ‘x’ if:
Question 4
Find AD, if:
Question 5
Find the length of AD.
Given: ABC = 60^o.
DBC = 45^o
And BC = 40 cm
Question 6
Find the lengths of diagonals AC and BD. Given AB = 60 cm and BAD = 60^o.
Thus AB = AC + CD + BD = 54.64 cm.
Question 8
In trapezium ABCD, as shown, AB // DC, AD = DC = BC = 20 cm and A = 60^o. Find:
(i) length of AB
(ii) distance between AB and DC.
First draw two perpendiculars to AB from the point D and C respectively. Since AB|| CD therefore PMCD will be a rectangle.
Consider the figure,
Question 9
Use the information given to find the length of AB.
Question 10
Find the length of AB.
Question 11
In the given figure, AB and EC are parallel to each other. Sides AD and BC are 2 cm each and are perpendicular to AB
Given that AED = 60^o and ACD = 45^o. Calculate:
(i) AB(ii) AC(iii) AE
Question 12
In the given figure, ∠B = 60^° , AB = 16 cm and BC = 23 cm,
(i) BE (ii) AC
Question 13
(i) BC
(ii) AD
(iii) AC
Question 14
In right-angled triangle ABC; B = 90^o. Find the magnitude of angle A, if:
(i) AB is √3 times of BC.
(ii) BC is √3 times of AB.
Question 15
A ladder is placed against a vertical tower. If the ladder makes an angle of 30^o with the ground and reaches upto a height of 15 m of the tower; find length of the ladder.
Given that the ladder makes an angle of 30o with the ground and reaches upto a height of 15 m of the tower which is shown in the figure below:
Question 16
A kite is attached to a 100 m long string. Find the greatest height reached by the kite when its string makes an angles of 60^o with the level ground.
Question 17
Find AB and BC, if:
Question 18
Find PQ, if AB = 150 m, P = 30^o and Q = 45^o.
Question 19
If tan x^o =, 5 /12
tan y^o = 3/4 and AB = 48 m; find the length of CD
Question 20
The perimeter of a rhombus is 96 cm and obtuse angle of it is 120^o. Find the lengths of its diagonals.
We also know that in rhombus diagonals bisect each other perpendicularly and diagonal bisect the angle at vertex.
Hence POR is a right angle triangle and
— End of Solutions of Right Triangles Class-9 Concise Selina Solutions :–
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"Graphs, Morphisms and Statistical Physics"
DIMACS Series in
Discrete Mathematics and Theoretical Computer Science
VOLUME Sixty Three
TITLE: "Graphs, Morphisms and Statistical Physics"
EDITORS: Jaroslav Nesetril & Peter Winkler
Ordering Information
This volume may be obtained from the AMS or through bookstores in your area.
To order through AMS contact the AMS Customer Services Department, P.O. Box 6248, Providence, Rhode Island 02940-6248 USA. For Visa, Mastercard, Discover, and American Express orders call
You may also visit the AMS Bookstore and order directly from there. DIMACS does not distribute or sell these books.
The intersection of combinatorics and statistical physics has been an area of great activity over the past few years, fertilized by an exchange not only of techniques but of objectives. Spurred by
computing theorists interested in approximation algorithms, statistical physicists and discrete mathematicians have overcome language problems and found a wealth of common ground in probabilistic
Close connections between percolation and random graphs, between graph morphisms and hard-constaint models, and between slowmixing and phase transition, have led to new results and new perspectives.
These connections can help in understanding typical, as opposed to extremal, behavior of combinatorial phenomena such as graph coloring and homomorphisms.
Any "nearest neighbor" system of statistical physics can be interpreted as a space of graph morphisms-for example, morphisms to the two-node graph with one edge and one loop correspond to the
"hard-core lattice gas model" and to random independent sets. Given the set of morphisms from a (possibly infinite) graph G to a graph H, when do we see long range order? When does changing the
morphism site by site (heat bath) yield rapod mixing, or even eventual mixing? The special case of proper coloring (corresponding to the anti-ferromagnetic Potts model at 0 temperature) is especially
interesting to graph theorists but more general notions of graph colorings may also yield some intriguing new questions.
Inspired by these issues, and encouraged by Fred Roberts and other colleagues, we organized a "DIMACS/DIMATIA Workshop on Graphs, Morphisms and Statistical Physics" which took place at Rutgers
University, March 19-21, 2001. The workshop was attended by 53 scientists from the U.S. and abroad, and the present volume is an outgrowth of this meeting.
Some of the topics we cover here are: percolation, random colorings, homomorphisms from and to a fixed graph, mixing, combinatorial phase transitions, threshold phenomena, scaling windows, and some
purely combinatorial aspects of graph coloring.
In addition to the participants we asked several other colleagues whose research complemented the scope of the volume. The volume also contains several contributions which were directly influenced by
the fruitful atmosphere of this meeting. We thank all participants for their time and effort which made the success of this meeting possible.
The workshop was a joint activity of DIMACS, New Jersey and DIMATIA, Prague and it was supported by both of these centers. Most of the organization of the volume was done by Robert Samal(DIMATIA-ITI)
and without his effort the volume would hardly be possible.
The final stage of preparation of the volume was supported by Institute of Theoretical Computer Science(ITI) at Charles University, Prague (under grant LNOOAO56)
Forward vii
Preface ix
Photographs xi
List of delivered talks xiii
List of Participants xv
Efficient Local Search Near Phase Transitions in Combinatorial
S. Boettcher 1
On the Sampling Problem for H-Colorings on the Hypercubic Lattice
C. Borgs, J.T. Chayes, M. Dyer, and P. Tetali 13
Graph Homomorphisms and Long Range Action
G.R. Brightwell and P.Winkler 29
Random Walks and Graph Homomorphisms
A. Daneshgar and H. Hajiabolhassan 49
Recent Results on Parameterized H-Colorings
J. Diaz, M. Serna, and D. M. Thilikos 65
Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs
M. Dyer, M. Jerrum, and E. Vigoda 87
On Weighted Graph Homomorphisms
D. Galvin and P. Tetali 97
Counting List Homomorphisms and Graphs with Bounded Degrees
P. Hell and J. Nesetril 105
On the Satisfiability of Random k-Horn Formulae
G. Istrate 113
The Exchange Interaction, Spin Hamiltonians, and the Symmetric
J. Katriel 137
A Discrete Non-Pfaffian Approach to the Ising Problem
M. Loebl 145
Survey-Information Flow on Trees
E. Mossel 155
Chromatic Numbers of Products of Tournaments-Fractional
Aspects of Hedetniemi's Conjecture
C. Tardif 171
Perfect Graphs for Generalized Colouring-Circular Perfect Graphs
X. Zhu 177
Contacting the Center
Document last modified on April 14, 2004. | {"url":"http://dimacs.rutgers.edu/archive/Volumes/Vol63.html","timestamp":"2024-11-02T12:13:41Z","content_type":"text/html","content_length":"6532","record_id":"<urn:uuid:873d043d-4b40-4fde-bed4-4cbbde9f4615>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00143.warc.gz"} |
Acceptable Age Gap Calculator - Online Calculators
Enter the values in fields of basic and advanced acceptable age gap calculator to find out the answer.
Acceptable Age Gap Calculator
Enter any 1 value to calculate the missing variable
Knowing the right age gap in relationships can be difficult, but the “half your age plus 7” rule provides a useful answer. This rule, often used to calculate socially acceptable age differences,
helps people to know whether they can go relationship as per their age. You can use this acceptable age gap calculator to find out the instant answer.
The formula is:
$\text{AG} = \frac{O}{2} + 7$
Variable Meaning
AG Acceptable Age Gap (the minimum acceptable age of a partner)
O Older Person’s Age (the age of the older person in the relationship)
How to Calculate?
First of all start with determining the age of the older person in the relationship (O). Now, divide this age by 2. Finally, add 7 to the result to calculate the Acceptable Age Gap (AG).
Solved Calculations:
Example 1:
• Older Person’s Age (O) = 40 years
Calculation Instructions
Step 1: AG = $\frac{O}{2} +$ Start with the formula.
Step 2: AG = $\frac{40}{2} + 7$ Replace O with 40 years.
Step 3: AG = $20 + 7$ Divide 40 by 2 to get 20.
Step 4: AG = 27 years Add 7 to get the Acceptable Age Gap.
The acceptable age gap is 27 years.
Example 2:
• Older Person’s Age (O) = 30 years
Calculation Instructions
Step 1: AG = $\frac{O}{2} + 7$ Start with the formula.
Step 2: AG = $\frac{30}{2} + 7$ Replace O with 30 years.
Step 3: AG = $15 + 7$ Divide 30 by 2 to get 15.
Step 4: AG = 22 years Add 7 to get the Acceptable Age Gap.
The acceptable age gap is 22 years.
What is Acceptable Age Gap Calculator | Half your age plus 7
The acceptable age gap calculator is often referred by the “half your age plus 7” rule. It is a formula that is normally used to know the acceptable age in society, a minimum for dating. It means
that you divide your age by 2 and then add 7 years, what comes is the acceptable age that you can date. For example if you are of 28 old, then the acceptable age for you to date would be 21 years.
Here we need to mention that this calculator is not the only that only provides you the age, there are others available in the market too. It is therefore, useful not only for your knowledge but also
lets you to decide for date if you are in a relationship.
The Acceptable Age Gap Calculator is a handy tool for anyone curious about the age factor in relationships. Whether you’re simply looking for romance or simply want to know the social norms of a
given society, this calculator would definitely help you. | {"url":"https://areacalculators.com/acceptable-age-gap-calculator/","timestamp":"2024-11-03T03:50:38Z","content_type":"text/html","content_length":"110838","record_id":"<urn:uuid:a9043543-efba-434b-989f-140f0202ae7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00497.warc.gz"} |
A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation
A code C ¶ {0,1}k → {0,1}n is a q-locally decodable code (q-LDC) if one can recover any chosen bit bi of the message b {0,1}k with good confidence by randomly querying the encoding x = C(b) on at
most q coordinates. Existing constructions of 2-LDCs achieve n = exp(O(k)), and lower bounds show that this is in fact tight. However, when q = 3, far less is known: the best constructions achieve n
= exp(ko(1)), while the best known results only show a quadratic lower bound n ≥ ω(k2/log(k)) on the blocklength. In this paper, we prove a near-cubic lower bound of n ≥ ω(k3/log6(k)) on the
blocklength of 3-query LDCs. This improves on the best known prior works by a polynomial factor in k. Our proof relies on a new connection between LDCs and refuting constraint satisfaction problems
with limited randomness. Our quantitative improvement builds on the new techniques for refuting semirandom instances of CSPs and, in particular, relies on bounding the spectral norm of appropriate
Kikuchi matrices.
Original language English (US)
Title of host publication STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
Editors Barna Saha, Rocco A. Servedio
Publisher Association for Computing Machinery
Pages 1438-1448
Number of pages 11
ISBN (Electronic) 9781450399135
State Published - Jun 2 2023
Externally published Yes
Event 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: Jun 20 2023 → Jun 23 2023
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print) 0737-8017
Conference 55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/Territory United States
City Orlando
Period 6/20/23 → 6/23/23
All Science Journal Classification (ASJC) codes
• CSP refutation
• Locally decodable codes
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How To Convert Metric Tons To Cubic Yards
A metric ton, or tonne, is the metric equivalent of a ton and converts approximately to 1.1 U.S. tons, or short tons as they are sometimes called. Mass-to-volume conversions depend on the density,
which is mass or weight per unit of volume. You can convert from metric tons to cubic yards by multiplying the material's mass by its density and then doing the metric conversion.
Step 1
Multiply the amount in metric tons by 1,000 to convert to kilograms. For example, five metric tons converts to 5,000 kilograms.
Step 2
Find the volume in cubic meters. Divide the mass in kilograms by the density of the substance. Different substances have different densities. In the example, if the material is solid ice, which has a
density of 919 kilograms per cubic meter (see the SI Metric link in Resources), then the volume is 5.44 cubic meters (5,000 / 919).
Step 3
Convert to cubic yards. Multiply the amount in cubic meters by 1.3079. To conclude the example, the volume is about 7.12 cubic yards (5.44 cubic meters x 1.3079).
Things Needed
• Calculator
• Material density
Cite This Article
Gartneer, Chance E.. "How To Convert Metric Tons To Cubic Yards" sciencing.com, https://www.sciencing.com/convert-metric-tons-cubic-yards-7835951/. 24 April 2017.
Gartneer, Chance E.. (2017, April 24). How To Convert Metric Tons To Cubic Yards. sciencing.com. Retrieved from https://www.sciencing.com/convert-metric-tons-cubic-yards-7835951/
Gartneer, Chance E.. How To Convert Metric Tons To Cubic Yards last modified March 24, 2022. https://www.sciencing.com/convert-metric-tons-cubic-yards-7835951/ | {"url":"https://www.sciencing.com:443/convert-metric-tons-cubic-yards-7835951/","timestamp":"2024-11-11T00:33:44Z","content_type":"application/xhtml+xml","content_length":"70005","record_id":"<urn:uuid:896be71b-0fa7-445a-9cdd-287dbc45b3e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00579.warc.gz"} |
How to know the linear and angular magnification of an optical system? | Zemax Community
Suppose I’ve designed an optical system in zemax, how can I find the linear and angular magnification of the system.
Hi smax,
I’d suggest using the Merit Function. In the Merit Function, you’ll find two operands PMAG (paraxial magnification) and AMAG (angular magnification). The Help File (F1) for those operands reads:
AMAG Angular magnification. This is the ratio of the image to object space paraxial chief ray angles at the wavelength defined by Wave. Not valid for non-paraxial systems.
PMAG Paraxial magnification. This is the ratio of the paraxial chief ray height on the paraxial image surface to the object height at the wavelength defined by Wave. Only useful for finite conjugate
systems. Note the paraxial image surface is used even if the system is not at paraxial focus.
You can also calculate the real magnification using ray operands. For example, REAY returns a real (non-paraxial) ray Y-coordinate at a particular surface, if you calculate REAY at two surfaces for
an off-axis chief ray and use operand DIVI to calculate the ratio of the two, this will give you the magnification. Similarly you can use RAID, which returns the ray angle of incidence in degrees at
a particular surface, to compute the angular magnification.
For example, in that simple telescope (two paraxial lenses, 100-mm focal length on the left, and 200-mm focal length on the right):
The Merit Function gives:
The first REAY operand is the off-axis chief ray (Hy = 1.0, Hx = Px = Py = 0.0) Y-coordinate (or height) in the image plane (on the right-hand side), and the second operand is the same ray in the
object plane (on the left-hand side). DIVI takes the ratio image height over object height and returns -2X, which makes sense because for such a telescope the magnification is minus the ratio of the
focal lengths (-200/100 = -2).
The same can be done with the angular magnification in the same telescope:
As you increase the field angle, you see the real angular magnification value: 0.5001X departing from the paraxial one: 0.5X.
Does it make sense? If you have an example in mind we help you setup the Merit Function if that is not clear.
Take care,
Hello David,
Thanks for the response . Yeah, I’ve a system. Will you please set up the merit functions for both type of magnifications !
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oil fired vs. electric
Welcome! Here are the website rules, as well as some tips for using this forum.
oil fired vs. electric
Chris, this is the basis of what you seek.
Domestic hot water usage varies so widely. There is no average family. A single dad with five pre-teen sons will use less hot water than an elderly widow.
A single mom with five teenage girls will use more hot water than a Denny's restaurant with an attached car wash.
Let's assume your incoming water is 50 degrees and you heat it to 130 degrees (80 degree rise). Every gallon per minute (8.33 lbs. per minute) you take "from street to temperature" is 666.4 BTU's per
minute. 39,984 call it 40,000 BTU's per hour if continuous flow, and that is net output.
If oil-fired, I would figure 75-80% efficiency or 49,980 to 53,300 BTUH input per GPM input rate. I would figure about 15% of your fuel usage to be standby losses, just a guess. Through-put of cold
water is the obvious biggest loss.
If your shower is 106 degrees (mixing 50 degree cold at the valve for a 3 GPM total head flow) your typical shower net hot water rate will be 2.1 GPM (70% at 130F, 30% at 50F). Ten minute shower? 21
gallons times how many showers?
Laundry? Variable, immensely so. Dishwashing, take your pick. Point being, you should take an inventory of hot water usage. If I had to guess, for a family of five, maybe 50 gallons of HW per person
per day? 21 for the shower and the rest shared? Just a guess. 250 GPD total?
If electric at 100% efficiency assumed, you would need 11.7 kW in that same hour. Most 40 gallon heaters have 9.0 kW in two (2) 4500 Watt elements which explains why the pick-up rate/recovery rate is
so low. Commercial tanks can better this but the elements become huge. I have seen 50 gallon heaters with over 50 kW in them. Yikes.
Sorry to ramble, just wanted to get the numbers out there.
Could someone give me a breakdown or send me in the direction to find this answer. What would be the breakdown of cost per operation between a 40 gallon oil fired water heater compared to a 40
gallon electric water heater for the year.Thanks
• Op Costs
There was a topic similar to this that was posted only a few days ago. In that case the home owner had an extremely low rate on electric, (about 1/2 of what I pay in NE Pennsylvania), and a
fairly high rate on fuel oil.
That is unusual as Fuel Oil is normally much more cost effective in comparison.
First thing you need to do is compare your BTU prices in your area. The link below if a basic way to plug in your per unit costs and start.
If you decide that they are close, then you can ask additional questions regarding types of systems as you wish.
Good Luck
Ed Carey
• PS To pevious post
There is no comparison to the quantity of DHW that can be made by an electric 40 gal vs an oil 40 gal.
The oil DHW heater will make DHW about 4x as fast. (Much more DHW from oil than elect).
They are not even close to equal, and the oil will cost more to install as a result.
I'm trying to come up with a rough estimate on the savings between switching a electric water heater to an oil fired. Based on the link above comparing the cost of each to produce the same btu
output, how would I breakdown each? Does anyone know what the average household (say a family of 5) use for total btus in a given year?
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Maximize Your Experimental Efficiency: The Hidden Power of KKT Conditions
KKT Conditions and Optimal Experiment Design: Beyond Support Vector Machines
While watching a random video on drone motion planning today, I stumbled into some of the deeper theoretical aspects, which unexpectedly triggered a flashback to my early PhD days. Remember those
Karush-Kuhn-Tucker (KKT) conditions you likely encountered in machine learning, perhaps with Support Vector Machines? Well, it turns out they’re not just for that! In fact, they were crucial for my
very first PhD project, where I tackled the challenge of measuring the surface curvature of biomolecules. It suddenly clicked: both the drone motion planning problem* and my old research revolved
around optimizing under constraints, using the same powerful Lagrange dual framework.
The Essence of Optimal Experiment Design
Imagine you’re a scientist with limited resources but a burning desire to learn as much as possible about a phenomenon. You need to design experiments carefully to maximize the information you
gather. This is where optimal experiment design comes in — it’s the mathematical framework that helps you choose the most informative experiments given your constraints.
KKT Conditions: Unlocking the Secrets of Optimality
At the heart of optimal experiment design lies optimization problems. KKT conditions are a set of necessary and sufficient conditions for a solution to be optimal in these problems. Let’s delve into
the two key concepts within KKT conditions:
1.Duality: Most optimization problems have a primal form (the original problem) and a dual form (a transformed version). The magic of duality is that solving either problem also solves the other. In
experiment design, the primal problem might be about choosing experiments, while the dual problem might be about assigning weights to those experiments.
Example: Duality in Portfolio Optimization
Consider the classic portfolio optimization problem:
• Primal: Minimize the risk of a portfolio subject to a return constraint.
Minimize: x^T Σ x
Subject to:
p^T x ≥ r_min (Minimum return constraint)
1^T x = 1 (Budget constraint)
x ≥ 0 (Non-negativity constraint: no shorting)
x is the vector of portfolio weights (fractions of wealth invested in each asset)
Σ is the covariance matrix of asset returns
p is the vector of expected returns for each asset
r_min is the minimum acceptable expected return
• Dual: While the primal problem directly tackles minimizing the portfolio’s risk (variance) while ensuring a minimum return, the dual problem focuses on understanding the trade-offs and
sensitivities associated with these constraints. The optimal dual variables λ*, ν*, and μ* represent the shadow prices of the constraints.
The KKT conditions link the optimal solutions of these two problems, providing valuable insights into how to balance risk and return.
2. Complementary Slackness: This condition connects the primal and dual solutions. It states that for every constraint in the primal problem, either the constraint is active (holds with equality) or
the corresponding dual variable is zero. Mathematically, for a constraint g_i(x) ≤ 0, complementary slackness means:
λ_i * g_i(x) = 0
where λ_i is the dual variable associated with the constraint. This means that if a constraint is not fully utilized (i.e., there’s some “slack”), the corresponding dual variable must be zero. This
helps us pinpoint the most impactful experiments or, in the case of portfolio optimization, the assets that contribute most to risk.
In our portfolio optimisatin problem, complementary slackness conditions provide key insights into which constraints are binding (active) and which are not:
• If the return constraint (𝜇’𝑥 ≥ 𝑟_min) is exactly met (binding), then the corresponding dual variable 𝜆 is positive. This tells us that increasing the required return 𝑟_min will increase the
portfolio’s risk.
• If the return constraint is not tight (the portfolio return exceeds 𝑟_min), then 𝜆=0. This means the return constraint is not influencing the optimal solution.
• If a particular asset weight 𝑥_𝑖=0, the corresponding dual variable 𝛾𝑖 can be positive, indicating that including this asset in the portfolio (allowing it to have a positive weight) would
increase the portfolio’s risk due to the no-shorting constraint.
Minimum Volume Ellipsoid (MVE): A Geometric Intuition
One elegant result in optimal experiment design is the concept of the minimum volume ellipsoid (MVE). Think of it as the smallest possible ellipsoid that encompasses all your potential experiment
outcomes (or, in portfolio optimization, the possible returns of your portfolio). In portfolio optimization, risk is often represented by the variance of returns, which can be visualized as an
ellipsoid in the return space. The covariance matrix Σ defines the shape and orientation of this ellipsoid. The dual problem in portfolio optimization seeks to find the optimal trade-offs between
return and risk. The dual variables (Lagrange multipliers) indicate how much the risk increases for a unit increase in the expected return.
• Dual Problem’s Role: In the dual problem of some experiment design formulations, the solution actually describes this MVE. For instance, the dual of the D-optimal design problem aims to find the
MVE that contains all candidate measurement vectors. When we solve the dual problem, we essentially identify the ellipsoid (defined by the covariance matrix Σ) that touches the constraint
boundary (e.g., the minimum required return 𝑟_min). This ellipsoid is the MVE because it is the smallest one that still satisfies the return constraint. The MVE corresponds to the optimal
portfolio that minimizes risk (variance) for a given set of expected returns.
• Complementary Slackness in Action: The complementary slackness condition tells us that the optimal experiment design will only involve experiments that lie on the surface of the MVE. In portfolio
optimization, this translates to holding only those assets that contribute to the maximum risk boundary.
Why This Matters
• Efficiency: Optimal experiment design helps you get the most out of your limited resources, whether it’s time, budget, or materials.
• Insight: Understanding the MVE gives you an intuitive way to grasp the trade-offs between different experiments or assets.
• Broad Applicability: While we’ve focused on experiment design and portfolio optimization, these concepts extend to many other fields, including economics, engineering, and finance. | {"url":"https://abhijit038.medium.com/maximize-your-experimental-efficiency-the-hidden-power-of-kkt-conditions-38ad311f0a7b","timestamp":"2024-11-10T15:13:01Z","content_type":"text/html","content_length":"109041","record_id":"<urn:uuid:b7582e29-dace-4dcd-9ac9-443d46200411>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00637.warc.gz"} |
Valentina Parigi obtained her PhD at LENS (European Laboratory for Non-Linear Spectroscopy) in Florence under the supervision of Marco Bellini. During her PhD she worked on generation, manipulation
and characterization of non-classical states of light and she realized the first experimental test on quantum commutation rules. She then worked on non-linear effects at the single photon level by
exploiting the long-range interactions in ensemble of Rydberg atoms, atomic quantum memories and near-field characterization of disordered nanostructures.
She is Full Professor since 2023 and had been Associate Professor (2015-2023) in the Quantum Optics group at LKB, working in the Multimode Quantum Optics team.
She is leading the activity on complex quantum networks in a multi-mode continuous-variables scenario, awarded (2018) with an – ERC Consolidator Grant.
Her interests range from the foundations of quantum mechanics to the experimental implementation of basic tools for quantum information technologies.
Memberships and responsibilities
• Member of the international team organizing the Quantum Science Seminar (2020-2022)
• Organizer of Complex Systems: Quantum information and computation, satellite conference of the CCS2022 in Complex Systems conference (Palma de Mallorca; October 17 -22, 2022); Organizer of the
Scientific Meeting of the Royal Society – Foundations of quantum mechanics and their impact on contemporary society, December 11-12, 2017, London. Member of the organizing committee of the ICOQC
2018 conference – International Conference on Quantum Computing November 26-30, 2018, Paris. Member of the organizing committee of the European project Open FET QCUMbER- Quantum Controlled
Ultrafast Multimode Entanglement and Measurement. Organizer of the Cargese School of Quantum Information and Quantum Technology summer school June 21 – 25, 2021, Cargèse
• Member of the scientific committee of the conferences: QTech 2018 Quantum technology International conference, September 5-7, 2018, Paris; CEWQO 2018- Central European Workshop on Quantum Optics;
quantum information and quantum information thematic committee for CLEO- Laser Science for photonic applications- year 2019 and year 2020. Member of the EQEC/CLEO 2023 committee in Quantum
Scientific Track Record and awards
• Médaille de bronze CNRS 2020
• Awarded 2018 – ERC Consolidator Grant. Project: COQCOoN: “Continuos Variables Quantum Complex Networks”
• Emergence CNRS -2016 on “Information quantique optique non-linéaire” – Spectrally broadened optical frequency comb by non-linear photonic crystal fiber for manipulating hundreds of q-modes in
quantum information processes. Role in the project: main investigator.
• Emergence Sorbonne Université 2016 PiCQuNet: Photonic platform for Complex Quantum Networks. Role in the project: main investigator
Recent publications:
• J Henaff, M Ansquer, MC Soriano, R Zambrini, N Treps, V Parigi Optical phase encoding in pulsed approach to reservoir computing arXiv preprint arXiv:2401.14073
• V. Cimini, M. Barbieri, N. Treps, M. Walschaers, and V. Parigi, Neural Networks for Detecting Multimode Wigner Negativity, Phys. Rev. Lett. 125, 160504 (2020)
• D. Barral, M. Walschaers, K. Bencheikh, V. Parigi, J. A. Levenson, N. Treps, and N. Belabas, Versatile Photonic Entanglement Synthesizer in the Spatial Domain, Phys. Rev. Applied 14, 044025
• F. Sansavini and V. Parigi, Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation, Entropy 22, 26 (2020)
• L. La Volpe, S. De, T. Kouadou, D. Horoshko, M. I. Kolobov, C. Fabre, V. Parigi, and N. Trep, Multimode single-pass spatio-temporal squeezing, Optics Express, Vol.28 pp 12385-12394 (2020)
• V. Parigi, E. Perros, G. Binard, C. Bourdillon, A. Maître, R. Carminati, V. Krachmalnicoff, and Y. De Wilde, Near-field to far-field characterization of speckle patterns generated by disordered
nanomaterials, Opt. Express, 24, 7019 (2016)
• V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino and J. Laurat, Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory, Nature Communications
6, 7706 (2015).
• E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold
atom memory, Phys. Rev. Lett. 112, 033601 (2014).
• J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, and P. Grangier, Dispersive optical non-linearities in a Rydberg electromagnetically- induced- transparency medium, Phys. Rev. A 88, 053845
• V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, Observation and Measurement of Interaction-Induced Dispersive Optical
Nonlinearities in an Ensemble of Cold Rydberg Atoms, Phys. Rev. Lett. 109, 233602 (2012).
• J. Stanojevic, V. Parigi, E. Bimbard, A. Ourjoumtsev, P. Pillet, and P. Grangier, Generating non-Gaussian states using collisions between Rydberg polaritons, Phys. Rev. A 86, 021403(R) (2012).
• J. Stanojevic, V. Parigi, E. Bimbard, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, Controlling the quantum state of a single photon emitted from a single polariton, Phys. Rev. A 84, 053830
• A. Zavatta, V. Parigi, M.S. Kim, H. Jeong, and M. Bellini, Experimental Demonstration of the Bosonic Commutation Relation via Superpositions of Quantum Operations on Thermal Light Fields, Phys.
Rev. Lett. 103, 140406, (2009).
• V. Parigi, A. Zavatta, and M. Bellini, Implementation of single-photon creation and annihilation operators: experimental issues in their application to thermal states of light, J. Phys. B: At.
Mol. Opt. Phys. 42, 114005, (2009).
• A. Cere, V. Parigi, M. Abad, F. Wolfgramm, A. Predojevic and M. W. Mitchell, Narrowband tunable filter based on velocity-selective optical pumping in an atomic vapor, Opt. Lett. 34, 1012, (2009).
• M. S. Kim, H. Jeong, A. Zavatta, V. Parigi and M. Bellini, Scheme to prove the bosonic commutation relation using single-photon interference, Phys. Rev. Lett. 101, 260401 (2008).
• A. Zavatta, V. Parigi, M. S. Kim and M. Bellini, Subtracting photons from arbitrary light fields:experimental test of coherent state invariance by single photon annihilation, New Journal of
Physics 10, 123006 (2008).
• A. Zavatta, V. Parigi, and M. Bellini, Toward quantum frequency combs: Boosting the generation of highly nonclassical light states by cavity-enhanced parametric downconversion at high repetition
rates, Phys. Rev. A 78, 033809 (2008).
• T. Kiesel, W. Vogel, V. Parigi, A. Zavatta and M. Bellini, Experimental determination of a nonclassical Glauber-Sudarshan P function, Phys. Rev. A 78, 021804(R) (2008).
• V. Parigi, A. Zavatta, and M. Bellini, Manipulating thermal light states by the controlled addition and subtraction of single photons, Laser Physics Letters 5, 246-251 (2008).
• V. Parigi, A. Zavatta, M.S. Kim and M. Bellini, Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field, Science 317, 1890 (2007).
• A. Zavatta, V. Parigi and M. Bellini, Experimental nonclassicality of single-photonadded- thermal light states, Phys. Rev. A 75, 052106 (2007).
• M. D’Angelo, A. Zavatta, V. Parigi and M. Bellini, Remotely prepared single-photon time-encoded ebits: homodyne tomography characterization,Journal of Modern Optics 53 (16-17), 2259-2270 (2006).
• A. Zavatta, M. D’Angelo, V. Parigi and M. Bellini, Two-mode homodyne tomography of time-encoded single-photon ebits, Laser Physics 16 (11), 1501- 1507 (2006).
• M. D’Angelo, A. Zavatta, V. Parigi and M. Bellini, Tomographic test of Bell inequality for a time-delocalized single photon, Phys. Rev. A 74, 052114 (2006).
• A. Zavatta, M. D’Angelo, V. Parigi and M. Bellini, Remote preparation of arbitrary time encoded single-photon ebits, Phys. Rev. Lett. 96, 020502 (2006). | {"url":"https://www.lkb.upmc.fr/quantumoptics/project/valentina-parigi/","timestamp":"2024-11-11T13:34:09Z","content_type":"text/html","content_length":"168285","record_id":"<urn:uuid:c365b063-681b-4e06-af16-625e449950ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00430.warc.gz"} |
Free worksheet for simplifying exponents
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Potassium Bromide
Bromide compounds, especially potassium bromide, were frequently used as sedatives in the 19th and early 20th century. This gave the word "bromide" its colloquial connotation of a boring cliché, a
bit of conventional wisdom overused as a sedative.
One can test for a bromide ion by adding dilute nitric acid (HNO[3]), then silver nitrate (AgNO[3]). A cream precipitate forms that disappears in concentrated ammonia solution.
Bromide is present in typical seawater (35 PSU) with a concentration of aroud 65 mg/l, which is around 0.2% of all dissolved salts. | {"url":"https://www.trivenichemical.com/bromide1.html","timestamp":"2024-11-04T15:22:32Z","content_type":"text/html","content_length":"19284","record_id":"<urn:uuid:ceb8f001-37e1-43bf-bcf9-c1b53c59ae4e>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00520.warc.gz"} |
Core Course Performance Objectives (Ccpos) s2
Campus Location:
/ Georgetown, Dover, Stanton /
Effective Date:
Course Number and Title:
/ ELC 127 Digital Electronics
Prerequisite: / ENG 090 or concurrent or ENG 091 or concurrent, MAT 020 or concurrent or higher, SSC 100 or concurrent
Course Credits and Hours:
/ 4 credits
3 lecture hours/week
3 lab hours/week
Course Description:
/ This course covers digital concepts, including logic levels, pulse waveforms, number systems, logic gates, Boolean algebra, DeMorgan's theorem, systematic reduction of logical expressions,
universal property of negative-AND (NAND) and NOR gates, pulsed operations, adders, comparators, encoder/decoders, multiplexers/demultiplexers, parity circuits, flip-flops, and synchronous and
asynchronous counters.
Required Text(s):
/ Obtain current information at https://www.dtcc.edu/student-resources/bookstores, or visit the bookstore. (Check your course schedule for the course number and section.)
Additional Materials:
/ Digital Parts Kit, Digital Probe, TI-84+ or TI-89 Calculator.
Method of Instruction:
/ Classroom
Core Course Performance Objectives (CCPOs):
1. Interpret basic digital concepts, number systems, and codes. (CCC 1, 2, 5, 6; PGC 1, 2)
2. Describe and analyze digital logic levels, pulse waveforms, data transmission methods, and digital integrated circuits. (CCC 1, 2, 5, 6; PGC 1, 2)
3. Explain the principles of basic logic gates as used in digital circuitry. (CCC 1, 2, 5, 6; PGC 1, 2, 3, 4)
4. Apply Boolean algebra concepts and techniques to simplify combinational logic circuits. (CCC 1, 2, 5, 6; PGC 1, 2, 3, 4)
5. Apply the principles of binary arithmetic operations to solve digital mathematical problems. (CCC 1, 2, 5, 6; PGC 1, 2)
6. Describe the operating characteristics of binary adders, comparators, encoders, decoders, multiplexers, demultiplexers, parity generators, and detectors as used in digital electronic based
circuits, sub-systems, and systems. (CCC 1, 2, 5, 6; PGC 1, 2, 3, 4)
7. Explain the uses and operating characteristics of digital latches and flip-flops. (CCC 1, 2, 5, 6; PGC 1, 2, 3, 4)
8. Explain the uses and operating characteristics of monostable and astable multivibrators and digital counters. (CCC 1, 2, 5, 6; PGC 1, 2, 3, 4)
See Core Curriculum Competencies and Program Graduate Competencies at the end of the syllabus. CCPOs are linked to every competency they develop.
Measurable Performance Objectives (MPOs):
Upon completion of this course, the student will:
1. Interpret basic digital concepts, number systems, and codes.
1.1 Differentiate between analog and digital signals.
1.2 Discuss how voltage levels are used to represent digital quantities.
1.3 Convert decimal form to and from binary form.
1.4 Convert decimal form to and from hexadecimal form.
1.5 Convert decimal numbers to binary coded decimal (BCD) form.
1.6 Convert between the binary and octal number systems.
1.7 Convert between the binary and hexadecimal number systems.
2. Describe and analyze digital logic levels, pulse waveforms, data transmission methods, and digital integrated circuits.
2.1 Explain and test various parameters of a pulse waveform such as leading edge, trailing edge, rise time, fall time, pulse width, frequency, period, and duty cycle.
2.2 Describe how digital data is transmitted using either serial or parallel communication, and explain their differences.
2.3 Identify pin numbers on integrated circuit packages.
2.4 Interpret and use logic gate datasheets.
2.5 Identify the differences between common types of logic families.
2.6 Define propagation delay time, power dissipation, input and output current, and fan-out in relation to logic gates.
2.7 Recognize digital instruments, and understand how they are used in troubleshooting digital circuits and systems.
3. Explain the principles of basic logic gates as used in digital circuitry.
3.1 Describe the operation and list truth tables for AND, OR, NAND, NOR, NOT, EX OR, and EX NOR logic gates.
3.2 Identify digital devices that are used to implement the logical functions above.
3.3 Construct timing diagrams showing the proper time relationships of inputs and outputs for the various logic gates.
3.4 Construct logic circuitry using the universal capability of NAND and NOR gates using acceptable industry standards and proper tools and equipment.
4. Apply Boolean algebra concepts and techniques to simplify combinational logic circuits.
4.1 Express the operation of AND, OR, NAND, NOR, NOT, EX OR, and EX NOR gates with Boolean algebra expressions.
4.2 Apply the basic laws and rules of Boolean algebra to simplify expressions.
4.3 Apply DeMorgan’s theorems to Boolean expressions.
4.4 Convert Boolean expressions of any form into sum-of-products form.
4.5 Use AND-OR and AND-OR-INVERT circuits to implement sum-of-products (SOP) and product-of-sums (POS) expressions.
4.6 Use a Karnaugh map to simplify Boolean expressions and truth table functions.
5. Apply the principles of binary arithmetic operations to solve digital mathematical problems.
5.1 Solve binary number problems using addition, subtraction, multiplication, and division techniques.
5.2 Compute the 1’s and 2’s complements of a binary number.
5.3 Express signed numbers in binary form.
5.4 Solve arithmetic operations with signed binary numbers.
6. Describe the operating characteristics of binary adders, comparators, encoders, decoders, multiplexers, demultiplexers, parity generators, and detectors as used in digital electronic based
circuits, sub-systems, and systems.
6.1 Explain the difference between half adders and full adders.
6.2 Use full adders to implement multibit parallel binary adders.
6.3 Use the magnitude comparator to determine the relationship between two binary numbers and cascaded comparators to handle the comparison of larger numbers.
6.4 Construct and test a basic binary decoder using acceptable industry standards and the tools and equipment required in your work environment.
6.5 Construct and test a basic binary encoder using acceptable industry standards and the tools and equipment required in your work environment.
6.6 Construct and test basic binary multiplexers using acceptable industry standards and the tools and equipment required in your work environment.
6.7 Construct and test basic binary demultiplexers using acceptable industry standards and the tools and equipment required in your work environment.
7. Explain the uses and operating characteristics of digital latches and flip-flops.
7.1 Explain the differences between an S-R latch and a D latch.
7.2 Recognize the differences between a latch and a flip-flop.
7.3 Explain how S-R, D, and J-K flip-flops differ.
7.4 Explain how master-slave flip-flops differ from the edge-triggered devices.
7.5 Sketch timing diagrams showing the proper time relationships of inputs and outputs for the various flip-flop devices.
7.6 Explain the significance of propagation delays, set-up time, hold time, maximum operating frequency, minimum clock pulse widths, and power dissipation in the application of flip-flops.
7.7 Apply and troubleshoot flip-flops in basic applications using acceptable industry standards and the tools and equipment required in your work environment.
8. Explain the uses and operating characteristics of monostable and astable multivibrators and digital counters.
8.1 Describe the difference between an asynchronous and a synchronous counter.
8.2 Analyze counter-timing diagrams, and create a timing diagram.
8.3 Design a simple controlled synchronous digital counter (or equivalent) circuit employing sequential design techniques.
8.4 Predict and modify the modulus of a counter.
8.5 Describe and predict the sequences of various configured counters, such as four bit types, decade, up/down, and divide by N counters.
8.6 Use integrated circuit (IC) counters in various applications.
8.7 Use cascading to achieve higher modulus counts sequences.
8.8 Explain how retriggerable and nonretriggerable one-shots differ.
8.9 Use a timer to operate as either an astable or monostable multivibrator.
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Students must demonstrate proficiency on all CCPOs at a minimal 75 percent level to successfully complete the course. The grade will be determined using the DTCC grading system:
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2. Use critical thinking to solve problems.
3. Collaborate to achieve a common goal.
4. Demonstrate professional and ethical conduct.
5. Use information literacy for effective vocational and/or academic research.
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1. Perform the duties of an entry-level technician using the skills, modern tools, theory, and techniques of the electronics engineering technology.
2. Apply a knowledge of mathematics, science, engineering, and technology to electronics engineering technology problems that require limited application of principles but extensive practical
3. Conduct, analyze, and interpret experiments using analysis tools and troubleshooting methods.
4. Identify, analyze, and solve narrowly defined electronics engineering technology problems.
5. Explain the importance of engaging in self-directed continuing professional development.
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Reduce multivariate polynomial coefficients to 1
Reduce multivariate polynomial coefficients to 1
Hi all,
I'm new to Python and SAGE and would like to ask a question: I have a multivariate polynomial, of a degree 3, i.e:
g.<x1,x2,x3>=x1^2 + 2x1x2 + x2^2 + 2x1x3 + 2x2x3 + x3^2 + 2x1 + 2x2 + 2*x3 + 1 Its type is <type 'sage.rings.polynomial.multi_polynomial_libsingular.mpolynomial_libsingu\="" lar'="">
The question is how can I reduce all its leading coefficients to 1, i.e. transform ig to:
g.<x1,x2,x3>=x1^2 + x1x2 + x2^2 + x1x3 + x2x3 + x3^2 + *x1 + *x2 + *x3 + 1
I tried to get the coefficients of the polynomial (which is a list), iterate through its items, and set the values of the items to 1, i.e.
for s in g.coefficients(): if s==2: s=1
I think that the problem is that list items can't be cast to integer (TypeError).
How could I solve this? Is there a more efficient way to do this? Thank you for your responses! Regards, Natassa
If speed matters, see compared timings in my answer.
To display blocks of code, separate them by a blank line from the rest of the text, and either indent them with 4 spaces, or select the corresponding lines and click the "code" button (the icon with
'101 010').
Can you edit your question to do that?
4 Answers
Sort by » oldest newest most voted
The shortest way i could think of is to sum its nonzero monomials:
sage: R = PolynomialRing(ZZ,'x',4)
sage: R.inject_variables()
Defining x0, x1, x2, x3
sage: g = x1^2 + 2*x1*x2 + x2^2 + 2*x1*x3 + 2*x2*x3 + x3^2 + 2*x1 + 2*x2 + 2*x3 + 1
sage: g
x1^2 + 2*x1*x2 + x2^2 + 2*x1*x3 + 2*x2*x3 + x3^2 + 2*x1 + 2*x2 + 2*x3 + 1
sage: sum(g.monomials())
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1 + x2 + x3 + 1
The code you provide is not valid, so I don't think this is what you have... The example below show the definition of a polynomial ring, a random polynomial from this ring, and then the polynomial
with all coefficients replaced by $1$. The idea is to get the list of monomials, and then simply sum this list.
sage: R.<x1, x2, x3> = QQ[] # polynomial ring over QQ
sage: g = R.random_element(10) # random polynomial of degree 10
sage: g
3*x1^4*x2^5*x3 - 4*x1^7*x2*x3 + 8*x1*x2^7*x3 + 1/3*x1^4*x3^5 - 5*x1^6*x3
sage: g.monomials() # the monomials of g
[x1^4*x2^5*x3, x1^7*x2*x3, x1*x2^7*x3, x1^4*x3^5, x1^6*x3]
sage: sum(g.monomials()) # their sum
x1^4*x2^5*x3 + x1^7*x2*x3 + x1*x2^7*x3 + x1^4*x3^5 + x1^6*x3
Note that the use of sum(...) is equivalent to the following:
sage: g0 = R.zero() # the polynomial equal to zero
sage: for m in g.monomials():
....: g0 += m
Thank you all,
Indeed, considering sum of all monomials does the trick! As a reference, a slightly tweaked solution below that worked for me:
g # the polynomial
list(g) # convert polynomial to list
g1=sum(mon for coeff, mon in g) # get the sum of all monomials
Reducing coefficients to one for multivariate polynomials
Taking the sum of monomials as suggested in the other answers works fine.
It is however faster to map all coefficients to one using map_coefficients.
sage: R.<x1, x2, x3> = QQbar[]
sage: P = x1^2 + 2*x1*x2 + x2^2 + 2*x1*x3 + 2*x2*x3 + x3^2 + 2*x1 + 2*x2 + 2*x3 + 1
First approach: summing monomials
Summing monomials using the monomials method:
sage: sum(P.monomials())
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1 + x2 + x3 + 1
Summing monomials iterating through the polynomial's coefficients and monomials:
sage: sum(m for c, m in P)
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1 + x2 + x3 + 1
Second approach: applying a map to coefficients
Using map_coefficients and plain 1:
sage: P.map_coefficients(lambda _: 1)
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1 + x2 + x3 + 1
Using map_coefficients and QQbar's version of 1.
sage: QQbar_one = QQbar.one()
sage: to_QQbar_one = lambda _: QQbar_one
sage: P.map_coefficients(to_QQbar_one)
x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1 + x2 + x3 + 1
Summing the monomials one way or another takes on the order of 700 µs.
sage: timeit('sum(P.monomials())')
625 loops, best of 3: 678 µs per loop
sage: timeit('sum(m for c, m in P)')
625 loops, best of 3: 682 µs per loop
Using map_coefficients is roughly ten times faster.
sage: QQbar_one = QQbar.one()
sage: to_QQbar_one = lambda _: QQbar_one
sage: timeit('P.map_coefficients(to_QQbar_one)')
625 loops, best of 3: 68.1 µs per loop
Note that the more naïve version wastes time since 1 gets converted to QQbar's 1 for each coefficient.
sage: timeit('P.map_coefficients(lambda _: 1)')
625 loops, best of 3: 139 µs per loop
Further remarks
Depending on how the reduced polynomials are used later on, it might make more sense to change their base ring to some version of the boolean ring. | {"url":"https://ask.sagemath.org/question/35871/reduce-multivariate-polynomial-coefficients-to-1/?sort=votes","timestamp":"2024-11-10T15:14:01Z","content_type":"application/xhtml+xml","content_length":"73738","record_id":"<urn:uuid:91db0784-ec27-4b7c-89ec-6dad303690e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00754.warc.gz"} |
Cite as
Yangjing Dong, Honghao Fu, Anand Natarajan, Minglong Qin, Haochen Xu, and Penghui Yao. The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise. In 39th Computational Complexity
Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 30:1-30:71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
Copy BibTex To Clipboard
author = {Dong, Yangjing and Fu, Honghao and Natarajan, Anand and Qin, Minglong and Xu, Haochen and Yao, Penghui},
title = {{The Computational Advantage of MIP^∗ Vanishes in the Presence of Noise}},
booktitle = {39th Computational Complexity Conference (CCC 2024)},
pages = {30:1--30:71},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-331-7},
ISSN = {1868-8969},
year = {2024},
volume = {300},
editor = {Santhanam, Rahul},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.30},
URN = {urn:nbn:de:0030-drops-204263},
doi = {10.4230/LIPIcs.CCC.2024.30},
annote = {Keywords: Interactive proofs, Quantum complexity theory, Quantum entanglement, Fourier analysis, Matrix analysis, Invariance principle, Derandomization, PCP, Locally testable code, Positivity testing} | {"url":"https://drops.dagstuhl.de/search?term=Yuen%2C%20Henry","timestamp":"2024-11-06T01:30:01Z","content_type":"text/html","content_length":"180409","record_id":"<urn:uuid:e7d12856-f32b-494b-8134-2bb74eb4143c>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00836.warc.gz"} |
Translate each statement to a linear inequality in two variables. 1. The total amount of 1-peso coins and 5-peso coins of Pedro in his - DocumenTVTranslate each statement to a linear inequality in two variables. 1. The total amount of 1-peso coins and 5-peso coins of Pedro in his
Translate each statement to a linear inequality in two variables. 1. The total amount of 1-peso coins and 5-peso coins of Pedro in his
Translate each statement to a linear inequality in two variables.
1. The total amount of 1-peso coins and 5-peso coins of Pedro in his bag is
Activity 1
more than P150.00.
2. Emily bought two blouses and a pair of pants. The total amount she paid
for the items is not more than P 980.00.
3. The sum of 20-peso bills (t) and 50-peso bills (1) is greater than P 420.00.
4. The difference between the weight of Hazel (h) and John () is at least 26
5. Jimuel planted his small garden with okra and talong. The number of
okra crops (k) is twice more than the number of talong crops (i) and their
total is greater than 24.
in progress 0
Mathematics 3 years 2021-09-05T05:57:47+00:00 2021-09-05T05:57:47+00:00 1 Answers 31 views 0 | {"url":"https://documen.tv/question/translate-each-statement-to-a-linear-inequality-in-two-variables-1-the-total-amount-of-1-peso-co-21469679-44/","timestamp":"2024-11-12T15:37:00Z","content_type":"text/html","content_length":"79627","record_id":"<urn:uuid:c42b83d8-ca0f-41bc-8b89-49af0ee0fe67>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00578.warc.gz"} |
Quantitative Aptitude Quiz For IBPS RRB PO, Clerk Prelims 2021- 26th May
Q2. Vessel A contains mixture of milk, water and wine in the ratio 2 : 3 : 5 and vessel B contains mixture of milk and wine in the ratio 3 : 5. If quantity of milk in vessel A is equal to quantity of
milk in vessel B and sum of water in A and wine in B is 95L then find the total quantity of mixture in vessel A.
(a) 75L
(b) 100 L
(c) 125 L
(d) 135 L
(e) 150 L
Q3. An amount of (P + 3000) is invested on C.I. at the rate (R + 2)% for two years. If total interest obtained on principal is
(a) 25%
(b) 23%
(c) 28%
(d) 30%
(e) 18%
Q4. A man goes 28 km upstream in 7 hours. If he goes the same distance by train then he takes 2 hours.If ratio of speed of boat in still water to speed of train is 3 : 7 then find the time in which
boat will cover 40 km in downstream.
(a) 3.5 h
(b) 4 h
(c) 6 h
(d) 4.5 h
(e) 5 h
Q5. Area of a rectangle is ‘r²’ where r is the radius of circle with circumference equal to 88 cm. If length of rectangle is 100% more than the radius of circle then find the breadth of rectangle.
(a) 8 cm
(b) 9 cm
(c) 7 cm
(d) 14 cm
(e) 10 cm
Q6. Abhishek lent Satish Rs.12000 on C.I. at the rate of 20% per annum and at the end of first year Satish borrowed Rs.x more from Abhishek on C.I. at the same rate. If at the end of second year,
Satish paid total amount of Rs.20400 to Abhishek then find how much extra amount Satish borrow at the end of first year?
(a) Rs.2400
(b) Rs.2000
(c) Rs.3600
(d) Rs.2600
(e) Rs.4000
Q7. In a River there are two boats A and B, where boat A covers 30 km downstream and boat B covers 30 km upstream. Boat B takes 2 hours more than boat A in covering the given distance. If sum of
speed of boat A in still water and boat B in still water is 16 km/hr and speed of water current is 1 km/hr then find the speed boat B in still water?
(a) 8 km/hr
(b) 4 km/hr
(c) 5 km/hr
(d) 6 km/hr
(e) 7 km/hr
Q8. Veer and Subham entered into partnership. Veer invested Rs.3x for first four month and Rs.5x for next six months and Subham invested Rs.1800 for 12 months. If Veer and Subham got profit share in
the ratio of 7 : 9 then, find the value of ‘5x’ ?
(a) 2000 Rs.
(b) 1600 Rs.
(c) 2400 Rs.
(d) 3600 Rs.
(e) 4000 Rs.
Q10. Retailer mark up an article 35% above its cost price and earn Rs 96 by giving 20% discount on the marked price. If he sells article at 15% discount on marked price then, find retailer’s profit
on selling one article.
(a) 118
(b) 177
(c) 236
(d) 214
(e) 154
Q11. Sum of age of A & B is 12 years more than sum of age of B, C & D. Average age of C & D is 29 yrs. Find average age of A & D if D is 12 years elder than C.
(a) 52.5 yrs
(b) 47.5 yrs
(c) 46.5 yrs
(d) 55.5 yrs
(e) 64 yrs
Q12. Mohit borrowed Rs. X on C.I. at the rate of 20% for three years. He paid at the end of first year 1/8th of amount and at the end of second year 1/6th of amount, if Mohit paid Rs. 15120 at the
end of third year to complete his debt, then find the amount borrowed by Mohit?
(a) 24000 Rs.
(b) 16000 Rs.
(c) 12000 Rs.
(d) 36000 Rs.
(e) 42000 Rs.
Q14. 8 kg of type A wheat, at the rate of Rs. 60 per kg mixed with 12 kg of type B wheat, at the rate of Rs. X per kg. If the price of resulting mixture is Rs. 67.2 per kg, then find at what price
should per kg type B wheat sold to make a profit of 25%?
(a) 84 Rs.
(b) 96 Rs.
(c) 104 Rs.
(d) 90 Rs.
(e) 78 Rs.
Q15. A bag contains 6 Red, 4 blue and 8 white ball, if three balls are drawn at random, find probability that one is Red and two are blue ?
(a) 3/68
(b) 5/68
(c) 7/68
(d) 9/68
(e) None of these
Practice More Questions of Quantitative Aptitude for Competitive Exams:
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Pressure on obstacles induced by granular snow avalanches
The increasing population and the consequential demand for dwelling and recreation areas lead to more frequent conflicts between humans and natural hazards. In mountainous regions snow avalanches
are, therefore, still a major threat to humans and infrastructure, with a significant impact on the economy and tourism. For the development of design criteria for infrastructure it is crucial to
obtain a thorough understanding on the pressure exerted by avalanches, so that they can withstand avalanche impact. Although the differences between avalanche flow regimes reportedly play a crucial
role for the avalanche-obstacle interaction, to date the impact pressure is often calculated similarly to the dynamic pressure in inviscid fluids proportional to velocity square and using empirical
drag coefficients. Indeed, in the inertial flow regime, which is typical of powder avalanches, the impact pressure is proportional to velocity square. However, in the gravitational regime, which is
typical of wet avalanches, the pressure is proportional to the flow depth. The empirical proportionality factor in the gravitational regime is referred to as the amplification factor. Field
measurements indicate that the amplification factor and the drag coefficient may range within considerable boundaries. Thus, in the absence of a physics-based framework to make the crucial choice of
the drag coefficient and the amplification factor for the impact pressure calculation, engineers need vast knowledge and experience in constructing in avalanche terrain and snow avalanche dynamics.
Even for experienced experts it is often unclear how to calculate the impact pressure adequately according to the expected avalanche flow regime or how to consider the obstacle geometry in the
calculation. The aim of this project is to develop a physics-based framework for the calculation of avalanche pressure on obstacles. In particular, we want to evaluate drag coefficients and
amplification factors as a function of snow properties and avalanche flow regimes. To reach this goal we develop a numerical Discrete Element Method model to investigate the interaction of avalanche
flows and obstacles, using a cohesive bond contact law. We test the relevance of the model by comparing simulated impact pressures with field measurements from the Vallée de la Sionne experimental
site. By varying avalanche flow velocity and cohesion in the simulations, we show that the impact pressure can be interpreted as the superposition of an inertial, a frictional and a cohesive
contribution. Further, we find a novel scaling law, reducing the problem of calculating the pressure induced by cohesive flows, to the calculation of cohesionless flows. We provide evidence that in
the cohesionless case the compression inside the influenced flow domain around the obstacle, the mobilized domain, governs the impact pressure of granular flows in the gravitational regime. If the
cohesion is high, we find that the cohesive bonds further enhance the stress transmission in the compressed mobilized domain, leading to an increase in impact pressure. Considering an inertial and a
gravitational contribution, we quantitatively link the properties of the mobilized domain to the pressure. Finally, the knowledge from previous research and the findings of this thesis allow us to
propose a physics-based framework to estimate the impact pressure by applying simple geometrical considerations and fundamental avalanche flow characteristics. | {"url":"https://graphsearch.epfl.ch/en/publication/287472","timestamp":"2024-11-06T05:20:35Z","content_type":"text/html","content_length":"112322","record_id":"<urn:uuid:55a91512-ebcf-4a92-ac93-ac83ee87c1ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00586.warc.gz"} |
Correlation - Ellistat
Ellistat Data Analysis offers the Correlation submenu, which contains several statistical tools. These tools can be used to perform correlation studies of multiple responses in a dataset. Or to
reduce the size of a dataset or monitor processes with several variables simultaneously.
In the examples below we present the tools:
• Correlation matrix
• ACP
• T² card
The dataset used in these examples can be found on the following page.
Independent Data 🇺🇸/ Données indépendantes🇫🇷
Example 1: Find the correlation between several Y responses, using the correlation matrix
Visit correlation matrix is an essential statistical tool used to understand the relationships between several variables in a data set.
• Place quantitative data from several Y columns in the grid. In the example, we want to find the correlation between responses Y1="Delta", Y2="Force" and Y3="Pressure".
• Click on the "Inferential statistics".
• In the zone 1Select the Y columns Y1="Delta", Y2="Force" and Y3="Pressure".
• In zone 2, select your data type. By default, if the selected columns contain quantitative values, Ellistat will plot the correlation curves between all the responses in pairs. In addition to the
Correlation sub-menu, you can also choose the "Proportion" or "Population" sub-menus. 📝: select "Correlation matrix".
• In zone 3In the half above the diagonal, we obtain the correlation matrix containing all the correlation graphs of two answers in pairs. The diagonal of this matrix shows the names of the
answers. And in the lower half we find the coefficients of determination R² and the significance level (P-value).
The diagram below shows the correlation graph, R² and P-value for both Delta and Pressure responses.
💡 When you click on a graph, you'll find a report of an XY Analysis of the two correlated responses:
💡 In the lower half of the correlation matrix there are two values :
R² (P-value).
Example 2: Find the correlation between several Y variables, using PCA.
Principal Component Analysis (PCA) is a statistical method used to reduce the dimensionality of a dataset while retaining as much information as possible. This technique is particularly useful when
working with multivariate data (i.e. data containing several variables).
• Place a set of quantitative data with several Y columns in the grid. In the example, we want to perform a PCA analysis on the data: Y1="Delta", Y2="force", Y3="Pressure", Y4="Pressure 2", Y5=
"Pressure 3".
• Click on the "Inferential statistics".
• In the zone 1Select the Y columns Y1="Delta", Y2="Force" and Y3="Pressure", Y4="Pressure 2", Y5="Pressure 3".
• In zone 2, select your data type. Press the "correlation" sub-menu and select "correlation".ACP". In addition to this sub-menu, you can also select the "Proportion" or "Population" sub-menus. 📝:
select "ACP"
• In zone 3we obtain the projection of the various responses in the plane composed of the principal vectors C1 (x-axis) and C2 (ordinate).
💡 In the upper part of the zone 3There are two tools used to select one of the spreadsheet factors:
With the "Label" tool :
It is possible to see the variation of individuals according to the chosen factor. This would make it possible to color-code individuals according to the chosen variable. The following case study
shows the results obtained for the label = "Delta". We can see that individuals with a strong delta are in orange/yellow. The individuals with a low delta are in blue.
The "other variable" tool
It is used to plot a factor without taking it into account when determining the main vectors. Please note! For this feature to work, the factor must not be ticked in zones 1 and 3 at the same time.
It must only be ticked in zone 3. Here, the example of the "Delta" factor (see figure below). This variable can be either a quantitative or a qualitative variable.
Whether you choose the "Label" option or the "Other variable" option, the variables selected can be quantitative or qualitative.
💡 In the middle part of the zone 3you can select several tabs:
The "Summary" tab:
This tab contains the graph, menus for displaying individuals in the graph, classification settings and the table of main vectors.
Pareto tab:
This tab shows the Pareto diagram, which expresses the contribution of each main vector.
The "variable" tab:
This tab shows the degree of correlation significance between the variables and the various main axes (C1, C2,...). A P-value<0.05 means that the correlation between the variable and the main vector
is significant. (see table below)
Individual values" tab:
This tab shows the coordinates of individuals in the space of principal vectors.
Example 3: Hotelling's T² card
Hotelling's T² chart is a statistical tool used to calculate multivariate quality control and data analysis. They enable processes with several variables to be monitored simultaneously. It is a
multivariate extension of Shewhart control charts, which focus on a single variable. Hotelling's T² is often used in contexts where several quality characteristics need to be monitored at the same
time. Examples include manufacturing, biology and engineering.
• Place quantitative data from several Y columns in the grid. In the example, we want to monitor the following data simultaneously: Y1="Delta", Y2="Force", Y3="Pressure", Y4="Pressure 2", Y5=
"Pressure 3".
• Click on the "Inferential statistics".
• In the zone 1Select the Y columns Y1="Delta", Y2="Force" and Y3="Pressure", Y4="Pressure 2", Y5="Pressure 3".
• In zone 2, select your data type. Press the "correlation" sub-menu and select "correlation".T²". In addition to this sub-menu, you can also select the "Proportion" or "Population" sub-menus. 📝:
select "T²"
• In zone 3we obtain the control chart T² with individual values and control limits.
• In the zone 4Here you'll find options such as general map settings, display options and control limit calculation.
💡 In the middle part of the zone 4Several options can be set:
• "GeneralWith this option, you can choose the type of control chart calculation (classic, Sullivan and Chi-2). You can also choose the alpha risk level and determine the data for training.
• "Display : With this option, you can transform the ordinate into a logarithmic scale and apply a Label to the data.
• "Limits With this option, you can change the control limit by setting it manually. | {"url":"https://ellistat.com/en/users-guide/05-correlation/","timestamp":"2024-11-03T16:02:10Z","content_type":"text/html","content_length":"201478","record_id":"<urn:uuid:03fc129e-502f-4951-b1a0-4fd519a1be79>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00123.warc.gz"} |
Optimal Chunk-Size for Large Document Summarization
Published on
1. Introduction
Large Language Model (LLM) is good at text summarization. However, the limited context window of LLM poses a challenge when summarizing large documents. We discuss a limitation of the commonly
used summarization strategy and introduce a straightforward improvement.
2. The Common Strategy of Large Document Summarization
The common strategy for summarizing large documents involves the following steps:
1. Determine a $\text{chunk}\_\text{size}$ based on the LLM context window and the prompt size.
2. Divide the document into $K$ chunks, where the first $K-1$ chunks each have a size of $\text{chunk\_size}$ and the last chunk has a size
$\text{last\_chunk\_size}=\text{document\_size}-(K-1)\times \text{chunk\_size},$
$K= \lceil\text{document\_size}/\text{chunk\_size}\rceil.$
We use $\lceil x \rceil$ to represent the smallest integer that is equal or greater than x, e.g. $\lceil 0.5 \rceil=1$.
3. Summarize the $K$ chunks independently using LLM.
4. Combine the $K$ chunk summaries into one global summary of the document using LLM.
The subsequent figure illustrates this general summarization strategy. The implementation of this strategy can be found in common LLM libraries like LangChain or LlamaIndex.
3. Problem: Biased Global Summary Due to Poor Chunk Size Selection
When applying the same $\text{chunk\_size}$ across all documents, there's a risk that the last chunk in some documents could be significantly smaller than the preceding chunks in the same
Example 1: For a document of size 11 and a chosen $\text{chunk\_size}=5$, the resulting chunk sizes would be $[5,5,1]$.
This disparity can lead to the following problem:
In the combination stage, the summary of the final chunk could be erroneously weighted as equally important as the others, thereby introducing a bias in the global summary of the original
See our Jupyter Notebook for a practical example of how this problem occurs in large document summarization.
Intuitively, we want all chunks to have roughly the same amount of information, so the equally weighted combination in the final step will not introduce any bias to the global summary. We can
further simplify this requirement by assuming that chunks of comparable sizes contain roughly the same amount of information. Therefore, our goal is to ensure that all chunks are of similar size.
To measure the "similarity" of the chunk sizes, we can calculate the maximum size difference between the longest chunk and the shortest chunk within the same document. For example, for the
summary strategy described in Section 2, the worst case scenario is when $\text{document\_size}=(K-1)\times \text{chunk\_size}+1$, so the last chunk has the smallest size of 1 and
This disparity can be substantial for larger values of $\text{chunk\_size}$. While intuitively reducing the $\text{chunk\_size}$ can diminish this gap, it simultaneously increases the value of
$K$, which in turn elevates the cost of the LLM. We then discuss how to reduce the chunk difference gap with a minimum chunk number $K$.
4. Optimal Automatic Chunk Size Determination
Firstly, we decide a $\text{maximum\_chunk\_size}$ based on the LLM context window and the prompt size, so the minimum chunk number $K$ is
$K= \lceil\text{document\_size}/\text{maximum\_chunk\_size}\rceil.$
We then calculate the 'average' integer chunk size
$\text{average\_chunk\_size}= \lceil\text{document\_size}/K\rceil,$
which is used to divide the document into $K$ chunks: the first $K-1$ chunks are of size $\text{average\_chunk\_size}$ and the last chunk has size
As shown in the following example, this simple trick has already improved the common strategy.
Example 2: Given $\text{maximum\_chunk\_size}=5$ and $\text{document\_size}=11$. We derive $K=\lceil 11/5\rceil=3$ and $\text{average\_chunk\_size}=\lceil 11/3\rceil=4$. The document is then
divided into three chunks of sizes $[4,4,3]$, a noticeable improvement over the prior method which resulted in sizes $[5,5,1]$.
However, this method is still not optimal in some cases.
Example 3: Given $\text{maximum\_chunk\_size}=5$ and $\text{document\_size}=13$. We derive $K=\lceil 13/5\rceil=3$ and $\text{average\_chunk\_size}=\lceil 13/3\rceil=5$. This divides the document
into three chunks of sizes $[5,5,3]$, which is worse than an obvious division of $[5,4,4]$.
To further reduce the gap between the $\text{average\_chunk\_size}$ and $\text{last\_chunk\_size}$, we can incrementally transfer one token from the preceding chunks to the last chunk until it
reaches the size of $\text{average\_chunk\_size}-1$. In total, we need to move $D$ tokens where
\begin{align*}D&=\text{average\_chunk\_size}-1-\text{last\_chunk\_size}\\\\&=K\times \text{average\_chunk\_size}-\text{document\_size}-1. \end{align*}
As a result, $D+1$ chunks now have the size $\text{average\_chunk\_size}-1$, while the remaining $K-D-1$ chunks maintain their original size of $\text{average\_chunk\_size}$.
The illustration below demonstrates how, by redistributing a token from one preceding chunk to the last chunk in Example 3, we achieve a more balanced chunk size list $[5,4,4]$.
In this approach, we only encounter two potential chunk sizes: $\text{average\_chunk\_size}$ and $\text{average\_chunk\_size}-1$. Consequently, the $\text{maximum\_chunk\_size\_difference}$
between chunks is just a constant of 1 even in the worst case. Thus, this strategy is optimal in practice while maintaining the smallest possible number of chunks.
5. Implementation
We provide a basic implementation below.
Visit our Github Repo for hands-on examples and practical comparisons!
import tiktoken
import math
def auto_chunker(document, max_chunk_size, model):
tokenizer = tiktoken.encoding_for_model(model)
document_tokens = tokenizer.encode(document)
document_size = len(document_tokens)
# total chunk number
K = math.ceil(document_size / max_chunk_size)
# average integer chunk size
average_chunk_size = math.ceil(document_size / K)
# number of chunks with average_chunk_size - 1
shorter_chunk_number = K * average_chunk_size - document_size
# number of chunks with average_chunk_size
standard_chunk_number = K - shorter_chunk_number
chunks = []
chunk_start = 0
for i in range(0, K):
if i < standard_chunk_number:
chunk_end = chunk_start + average_chunk_size
chunk_end = chunk_start + average_chunk_size - 1
chunk = document_tokens[chunk_start:chunk_end]
chunk_start = chunk_end
assert chunk_start == document_size
return chunks
In practice, an improved chunking strategy should also consider the boundaries of sentences or paragraphs. We will discuss how to integrate this consideration into this method in our upcoming
blog post. Subscribe below to stay updated and ensure you never miss enhancements for your AI product!
6. Want to Improve the Chunking for Your Domain-specific Documents?
In this blog post, we propose an optimal chunking strategy under the assumption that chunks of similar sizes hold comparable amounts of information. This proposed method could also benefit other
LLM tasks, such as retrieval-augmented question answering. In real-world situations, this assumption might not apply to domain-specific documents. For instance, in certain scientific papers,
references at the end may not necessarily reflect the document's core semantics.
Get connected
and discover how we can improve the chunking, summarization or retrieval in your domain-specific LLM pipelines! | {"url":"https://vectify.ai/blog/LargeDocumentSummarization?utm_source=gptweekly.beehiiv.com&utm_medium=referral&utm_campaign=openai-is-going-to-kill-startups","timestamp":"2024-11-03T16:51:43Z","content_type":"text/html","content_length":"208241","record_id":"<urn:uuid:b4a56b33-7b7c-45b5-8045-ec67bd492eed>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00470.warc.gz"} |
Multiplication 0-12 Worksheets | Multiplication Worksheets
Multiplication 0-12 Worksheets
The Multiplying 0 To 12 By 3 B Math Worksheet From The Multiplication
Multiplication 0-12 Worksheets
Multiplication 0-12 Worksheets – Multiplication Worksheets are a wonderful method to teach kids the twelve times table, which is the holy grail of primary mathematics. These worksheets work in
teaching trainees one factor each time, but they can likewise be utilized with two aspects. Frequently, these worksheets are grouped into support groups, and trainees can start discovering these
facts one at a time.
What are Multiplication Worksheets?
Multiplication worksheets are a helpful method to assist students discover mathematics realities. They can be made use of to show one multiplication fact at once or to assess multiplication truths
approximately 144. A worksheet that reveals a student one truth at once will certainly make it much easier to bear in mind the fact.
Using multiplication worksheets to instruct multiplication is a wonderful means to link the learning void and also offer your students effective technique. Lots of on the internet sources provide
worksheets that are both fun and easy to use. For instance, Osmo has a variety of complimentary multiplication worksheets for children.
Word troubles are one more way to connect multiplication with real-life scenarios. They can enhance your youngster’s understanding of the idea while boosting their calculation speed. Lots of
worksheets include word troubles that mimic real-life circumstances such as time, cash, or purchasing estimations.
What is the Purpose of Teaching Multiplication?
It’s important to begin teaching youngsters multiplication early, so they can appreciate the procedure. It’s also handy to give trainees plenty of practice time, so they can come to be well-versed in
Among one of the most reliable understanding aids for youngsters is a multiplication table, which you can publish out for each and every child. Youngsters can exercise the table by counting and also
repeating additions to get the answer. Some kids discover the multiples of 2, 5, and also 10 the most convenient, but once they understand these, they can proceed to harder reproductions.
Multiplication 0-12 Worksheets
Multiplication Worksheets 0 12 Printable Math Multiplication Table
Printable Blank Multiplication Table 0 12 PrintableMultiplication
Multiplication Test 0 12 Times Tables Worksheets
Multiplication 0-12 Worksheets
Multiplication 0-12 Worksheets are a wonderful way to review the times tables. They likewise help youngsters develop flexibility as they are revealed to the multiple means they can carry out
computations. Pupils might additionally discover worksheets with photos to be valuable. These worksheets can be adapted for any style or degree, as well as are complimentary to download.
These worksheets are great for homeschooling. When downloaded, you can also share them on social media or email them to your child.
Many kids battle with multiplication. They include multiplication issues at various levels of trouble.
Related For Multiplication 0-12 Worksheets | {"url":"https://multiplication-worksheets.com/multiplication-0-12-worksheets/","timestamp":"2024-11-08T15:21:38Z","content_type":"text/html","content_length":"41510","record_id":"<urn:uuid:456ba64b-1eb5-4246-b6ba-e2f2acbafa01>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00430.warc.gz"} |
The Learner Will Use Relations and Functions to Solve Problems
GOAL 4
The learner will use relations and functions to solve problems.
4.01 Use linear functions or inequalities to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
b) Interpret constants and coefficients in the context of the problem.
4.02 Graph, factor, and evaluate quadratic functions to solve problems.
4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results.
4.04 Graph and evaluate exponential functions to solve problems.
4.01 Use linear functions or inequalities to model and solve problems; justify results.
Process for Solving Linear Equations
1. Identify the ______
2. Distribute.
3. Combine ______.
4. All ______on one side.
5. Add/Subtract to ______the ______.
6. ______all fractions.
The cost of renting a cab is $3.00 plus twenty-five cents per mile.
A. Define your variables.
B. Write an equation to represent the cost (C) of renting a cab.
C. Substitute 25 in for ______.
D. Substitute _____ in for C and solve for m.
1. Solve each equation for n or substitute -2 into each equation until you find one that is true.
2. Follow the process above and solve the equation. Do not forget to distribute all values in the parenthesis and combine like terms.
3. Use the key words to set up an equation. Check to make sure you know what you are solving for.
Model the Process
Solve the equation for the variable.
Step / Justify
1. / 1.
2. / 2. Combine like terms. (7x and -2x)
3. / 3.
4. / 4. Add 6.
5. / 5.
A. C → ______
m → ______
B. C = ______m + ______
C. What would a twenty-five mile trip cost?
D. Suppose you were charged $12.75. How many miles did you ride?
1. Which equation has a solution of -2?
A. 4n + 3 = 11 B. 4 = 3n – 2
C. 5(1 + n) = -5 D. 3(n + 1) = 2
2. Solve 2(b – 3) + 5 = 3(b – 1).
A. -2 B. 2
C. -3 D. 3
3. A telephone company offers two long distance calling plans. Plan A charges 0.10 per minute and Plan B charges 0.07 per minute plus a monthly fee of $3.95. When, in terms of minutes, are the two
calling plans equivalent?
4.01 Use linear functions or inequalities to model and solve problems; justify results.
a) Solve using tables, graphs, and algebraic properties.
Writing the equation of a line
1. Find the slope.
2. Use the slope and one point to write the equation of the line using the:
A. slope-intercept form:
B. point-slope form:
C. ratio:
(Note: If m is not a fraction, make it a fraction.)
Using the graphing calculator
1. Press (STAT), (ENTER).
2. Enter the data in L1 & L2.
3. Press (STAT) → CALC→ 4:LinReg(ax + b), (ENTER)
4. Write the equation.
In 1980, the average price of a home in Greensboro was $40,000. By 2002, the average price of a home was $120,000.
A. Create a linear model based on this data.
B. Interpret the slope.
C. Predict using your equation. Remember is the years.
1. Find the value of the slope.
2. Predict using a model. The graphing calculator is good for this problem.
Model the Process (No calculator)
Write the equation of the line passing through the points (-2, 3) and (-4, 5).
Slope → ______
Equation →
Solve for y.
Now use the calculator to confirm your answer.
L1 / L2
-2 / 3
-4 / 5
Press (STAT), (ENTER) →
a = ______
b = ______
A. y = ______x + ______
B. How much does the cost of a home change annually?
C. Estimate the price of a home in:
a. 1991: ______
b. 2006: ______
1. Denisha bought a car for $15,000 and its value depreciated linearly. After 3 years the value was $11,250. What is the amount of yearly depreciation?
A. $2000 B. $1,500
C. $1,250 D. $750
2. In 1994, the average price of a new domestic car was $16,930. In 2002, the average price was $19,126. Based on a linear model, what is the predicted average price for 2008?
A. $22,969 B. $21,322
C. $20,773 D. $18,577
4.01 Use linear functions or inequalities to model and solve problems; justify results.
b) Interpret constants and coefficients in the context of the problem.
1. y → Dependent variable.
2. x → Independent variable.
3. m → Rate of change.
4. b → Initial value.
For the line, where a > 0 and b > 0:
A. The y-intercept gives the value when x = 0. Look at where the line is on the x-axis as you move the line up.
B. The slope gives the rate at which the line is increasing or decreasing. Look at where the line adjusts on the x-axis as you increase the steepness of the line.
C. Vertical shift in the line.
D. A change in steepness of the line.
1. Write an equation in slope-intercept form. Then double the slope and y-intercept and look at the value of the x-intercept.
2. Determine whether the slope and y-intercept were increased or decreased.
3. Interpret the slope of the line. For each value of x, y increases or decreases by the value of the slope.
Model the Process
The equation below represents the charge for cell phone usage on a particular company plan.
C = .10m + 3.95
Define each value.
Variable/Value / Meaning
Cost per minute
A. If b increases and a remains constant, how does the x-intercept change?
B. If a increases and b remains constant, how does the x-intercept change?
C. If b is multiplied by -1 and a remains constant, how does the line change?
D. If a decreases, getting closer to 0, and b remains constant what happens to the line?
1. If the graph of a line has a positive slope and a negative y-intercept, what will happen to the x-intercept if the slope and the y-intercept are doubled?
A. The x-intercept becomes four times larger.
B. The x-intercept becomes twice as large.
C. The x-intercept becomes one-fourth as large.
D. The x-intercept remains the same.
2. If the slope of a line changes from -4 to -1/4 and the y-intercept changes from -2 to 0, then the graph of the line will be affected in what ways?
A. Less steep; up 2 units
B. Less steep; down 2 units
C. Steeper; up 2 units
D. Steeper; down 2 units
3. In the equation, if an x-value is increased by 2, what would be the effect on the corresponding y-value?
A. The value of y will be 3 times as large.
B. The value of y will decrease to be ½ as large.
C. The value of y will increase by 6.
D. The value of y will decrease by 6.
4.02 Graph, factor, and evaluate quadratic functions to solve problems.
Always look for a GCF first!!!!!
A. Factoring
1. Find two factors of c that have a sum of b.
2. (x + ____)(x + _____)
B. Factoring
1. Multiply a × c.
2. Rewrite bx as 2 terms.
3. Solve by grouping.
C. Graphing – Look for the x-intercepts of the graph.
The function describes newspapers circulation (millions) in the United States for 1920-98 (x = 20 for 1920).
Use the graphing calculator.
A. Examine the table to see between 20 and 98 where the values of y are increasing or decreasing.
B. What is the maximum value of the graph?
C. In the table, look for the value of x when y is around 45.
1. Look for all possible y-values for the function. The graph will have a maximum value since a is a negative number (a = -3).
2. Substitute 240 for v0. Look at the highest x-intercept.
3. The number of real roots corresponds to the number of x-intercepts.
Model the Process
1. Factor
(x + ______) (x – ______)
2. Factor
2x2 – ______+ ______– 9
______( _____ – _____) + _____ (_____ – ______)
(2x – _____)(x + ______)
A. Identify periods of increasing circulation and decreasing circulation.
B. According to the function, when did newspaper circulation peak?
C. When will circulation approximate 45 million?
1. Given, what is the range of the function?
A. all real numbers less than or equal to 5
B. all integers less than or equal to 5
C. all nonnegative real numbers
D. all nonnegative integers
2. An object is fired upward at an initial velocity, v0, of 240 ft/s. The height, h(t), of the object is a function of time, t, in seconds and is given by the formula . How long will it take the
object to hit the ground after takeoff?
A. 16 seconds B. 15 seconds
C. 7.5 seconds D. 4 seconds
3. Which equation has two real roots?
A. B.
C. D.
4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results.
Solving systems of equations and inequalities by graphing
1. Graph both equations/inequalities
2. The intersection point is the solution or the common shaded region is the solution region.
Solving systems using algebra
A. Substitution
1. One equation is or can be solved for one variable.
2. Substitute the value into the other equation.
3. Solve for the remaining variable.
4. Solve for the other variable.
B. Elimination
1. Both equations are in standard form.
2. Coefficients of one of the variables are the same or opposite.
3. Add/Subtract equations together to eliminate a variable.
4. Solve for the remaining variable.
5. Solve for the other variable.
During the band’s fruit sale, five dozen oranges cost as much as four dozen grapefruits. Terry bought two dozen oranges and a dozen grapefruit spending $27.30.
A. Define your variables.
B. Write an equation that relates the number of oranges to the number of grapefruits.
C. Write an equation that shows how much was spent on fruit.
D. Solve the system.
1. Use the application as guidance. Write two equations. One for the sales and one relating cost.
2. Use the calculator.
Model the process
Solve exactly without a calculator.
Use elimination. Multiply the top equation by -3
Solve for x.
______y = ______
y = ______
(______, _____)
A. n → oranges
g → ______
B. _____ n = _____ g
C. _____n + _____ g = ______
D. What was the cost of a dozen oranges?
1. A store received $823 from the sale of 5 tape recorders and 7 radios. If the receipts from the tape recorders exceeded the receipts from the radios by $137, what is the price of a tape recorder?
A. $49 B. $68
C. $84 D. $96
2. A region is defined by this system:
In which quadrants of the coordinate plane is the region located?
A. I, II, III only B. II, III only
C. III, IV only D. I, II, III, IV
4.04 Graph and evaluate exponential functions to solve problems.
Substitute using exponential equation formulas.
· Compound Interest
A → New amount after interest
P → Initial amount deposited
r → rate (decimal)
t → time (years)
· Exponential form
a → y-intercept
b → growth (b > 0)
decay (b < 0) ratio
· Geometric Sequence
an→ nth term
a1 → 1st term
n→ number of terms
r → common ratio between consecutive terms
The average weekly food cost for a family of four in 1990 was $128.30. For the next ten years the weekly food cost increased 2.45% annually. The function represents the cost of food for that period.
A. Substitute ______in for x.
B. Find the difference between the average cost in 2006 and 1990.
1. Substitute $1,000 for ______, ______for r and ______for t.
2. Substitute half of the original price in for V. Look at the rate at which the car is depreciating or substitute each answer until you get half of the purchase price.
A. Estimate the weekly food cost for a family of four in 2006.
B. How much has food cost increased since 1990?
1. When Robert was born, his grandfather invested $1000 for his college education. At an interest rate of 4.5%, compounded annually, approximately how much would Robert have at age 18? (use the
formula , where P is the principal, r is the interest rate, and t is the time in years)
A. $1,810
B. $2,200
C. $3,680
D. $18,810
2. A new automobile is purchased for $20,000. If gives the car’s value after x years, about how long will it take for the car to be worth half its purchase price?
A. 3 years
B. 4 years
C. 5 years
D. 6 years
GOAL 3
The learner will collect, organize, and interpret data with matrices and linear models to solve problems.
3.01 Use matrices to interpret data.
3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems.
3.03 Create linear models for sets of data to solve problems.
a) Interpret constants and coefficients in the context of the data.
b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.
3.01 Use matrices to interpret data
· Read and interpret information from a matrix. | {"url":"https://docest.com/doc/420063/the-learner-will-use-relations-and-functions-to-solve-problems","timestamp":"2024-11-04T07:41:22Z","content_type":"text/html","content_length":"37016","record_id":"<urn:uuid:ea1a3bc6-129c-4608-9244-e7d14398c3f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00356.warc.gz"} |
Tanhc Function -- from Wolfram MathWorld
By analogy with the tanc function, define the tanhc function by
It has derivative
The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.
It has maximum at
which is 0.919937667... (OEIS A133919).
It has a unique real fixed point at 0.82242927726... (OEIS A133918). | {"url":"https://mathworld.wolfram.com/TanhcFunction.html","timestamp":"2024-11-06T02:25:07Z","content_type":"text/html","content_length":"56786","record_id":"<urn:uuid:5f13b33d-f3c3-4e03-a71f-f448531c91bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00829.warc.gz"} |
Question ID - 101701 | SaraNextGen Top Answer
In a row of 16 girls, when Hema was shifted by two place towards the left she became 7^th from the left and what was her earlier position from the right
(a) 7^th (b) 8^th (c) 9^th (d) 10^th
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Hawaiian’s Five-Year Downhill Slide Led it Right Into Alaska’s Arms
Alaska’s decision to buy Hawaiian Airlines and save it from a bankruptcy filing — or worse — was not something that happened overnight.Over the last five years, Hawaiian has gone through some of the
most challenging times any airline has ever faced, and the hits just kept on coming.Let’s take a look back at what this management team has had to endure.
On March 4, 2019, Southwest Airlines announced that it would begin flying to Hawaiʻi on March 17 with an inaugural flight from Oakland to Honolulu.Southwest had been planning on entering the Hawaiʻi
market for awhile but had to wait until it received ETOPS certification to begin operations, so for Hawaiian, this was like watching a slow moving freight train barreling toward the airline.While a
new competitor from the mainland was always concerning, the bigger problem was that Southwest would also start flying interisland with Honolulu – Kahului starting April 28.All the other big routes
would follow.
This wasnʻt the first time that Hawaiian had faced stiff interisland competition, but this time was different.
The interisland market has always been huge and hugely important.With the exception of Maui to Lānaʻi (and Molokai which ended in 2016), no two islands had been connected by passenger ship in
decades.For that reason, air travel is vital.Locals fly to other islands for shopping, doctor appointments, and high school sports meets.This kind of thing doesnʻt happen on the mainland, and it
means a lot more seats are needed than one might expect. Hawaiʻi is the third largest intrastate air market in the US behind much larger California and Texas.
For ages, the interisland market was a duopoly run by Hawaiian and Aloha with the occasional competitor coming in to try and disrupt the market before failing. Then, in 2008, the Great Recession
brought already weak carriers to their knees.Aloha would never recover, going out of business that year.The failure of Aloha was a lifeline for Hawaiian, as you can see by the average fares in the
market over time.
Hawaiʻi Interisland Average Fare
When the short-lived and much-hated Mesa entered into the market with its go! brand, interisland fares plunged to around $40.Once go! killed off Aloha for good, fares began to rise. By the time it
left the market in 2014, fares had doubled to the $80 range.
At the time Southwest said it was coming to town in 2019, Hawaiian undoubtedly had a sense of déjà vu, but it also knew it would have a much tougher fight on its hands.Southwest, after all, was
unlike Aloha in that it was an enormous airline that didn’t have its world centered on Hawaiʻi. In fact, Hawaiʻi was a tiny part of the airline’s network, though a strategic one that Southwest
desperately wanted to enter. It didnʻt have to make money or run away.It could spend years funding a money-losing operation if the airline decided that was worth doing.
It didnʻt take long before interisland fares began to plummet.In Q4 2018 before Southwest arrived, the average interisland fare was $72.That had dropped to $59 by Q4 2019.
If only that was Hawaiianʻs biggest problem.As we know, a mere year after the entrance of Southwest, the COVID-19 pandemic arrived.This all but grounded the entire airline industry around the
world, but some places would recover more quickly than others. Hawaiʻi was not one of them.
Seats vs 2019 by Region
Hawaiʻi was on track for notable growth vs 2019 until the pandemic arrived, and then traffic disappeared overnight.Naturally, interisland flying continued at a higher level but comparing flying to
the mainland vs flying within the continental US shows just how dire things were.
The state of Hawaiʻi had effectively banned travel to the islands, requiring a 14-day quarantine for anyone flying in.There was no way around it until October 15, 2020 when the state allowed
travelers to visit without quarantine if they tested negative using the Safe Travels program. This program was cumbersome, but it did allow airlines to return seats to the market.Letʻs stretch that
chart above out a few years.
Seats vs 2019 by Region
The change was swift and massive.By summer of 2021, seats from Hawaiʻi to the mainland were well above pre-pandemic levels.You might think that this would be good news for Hawaiian, but it wasn’t.
With US flag carriers unable to fly their widebodies to most places internationally, they started pouring those airplanes into Hawaiʻi, adding a ton of seats and depressing fares.There was an
increase in demand, but the state’s infrastructure couldnʻt handle it.Maui County asked visitors to stop coming.They were tired of people resorting to renting U-Hauls since there were no other cars
Though these people were paying a lot once in the islands, part of the reason they could do that is because fares were cheap to get there. The average fare between the mainland and Hawaiʻi in Q2 2019
was $309.By Q2 2021, it was down 15 percent to $264.
This capacity would stabilize eventually as international markets reopened, but there was still more pain coming for Hawaiian.
By 2022, the mainland operation was getting back closer to normal.Demand was up, and March brought the end of the Safe Travels system which required pre-registration and testing.While the domestic
market was doing well, however, the international world was still locked up.
Unfortunately for Hawaiian, its focus on the Pacific Rim left it serving some of the most restrictive countries toward the tail end of the pandemic.Australia and New Zealand opened in 2022, but they
had a hard closure until that point.It was good for Hawaiian to get back into this market, but the biggest and most important international market for Hawaiian by far is Japan.
In 2019, six percent of Hawaiian’s total seats touched Japan.That, however, was more than 20 percent of total seats offered on the A330 fleet.Hawaiian had service to five airports in Japan and was
getting closer and closer with Japan Airlines until the government shot down a joint venture just as the pandemic began.There aren’t a lot of Hawaiians flying to Japan.This is almost entirely about
Japanese tourists visiting Hawaiʻi.
Japan kept its doors closed until later in 2022, but even after that there were rules about testing that helped deter Japanese travelers from coming to the islands.According to ARC/BSP data via
Cirium, for the first nine months of 2019, there were 3,600 travelers per day coming from Japan to the Hawaiian islands.Looking at the first nine months of 2022, that had plunged to about 350 per
day.This was a huge increase compared to the 43 per day during that period in 2021, but it was still almost nothing for an airline flying several widebodies a day to Japan.
As the final restrictions went away in Japan, it was believed that traffic would come back very quickly.It didn’t.Looking at the first nine months of 2023, there were only about 1,350 travelers per
day originating in Japan.What happened?
There are plenty of theories including Japan being a more cautious culture and wanting to stay closer to home post-pandemic, but the reality is probably much simpler.It’s economics.
Inflation made things a lot more expensive in the US.Hotel rates skyrocketed, but food costs were up as well.As if that wasn’t enough, the yen lost a lot of value compared to the dollar.
Japanese Yen per US Dollar Over Time
Back in 2019, the yen was fairly steady against the dollar with 1 USD equaling about 110 JPY.In 2022, the climb began.It has gone up and down, but the range has been between 125 and 150.It’s been
closer to 150 as of late.So, expenses go up in real dollars but then yen aren’t worth as much either.A vacation to Hawaiʻi is significantly more expensive for the Japanese right now, and that makes
it much less appealing.
At least Hawaiian had the mainland business running ok… until the summer of 2023 hit.It was then that Pratt & Whitney started sounding ominous messages about its engines that powered Hawaiian’s
A321neo aircraft.
The upshot of that mess was that Hawaiian didn’t have enough airplanes to fly its bread-and-butter west coast routes.It had to cancel several flights, and going into 2024 the expectation was of more
pain before things got better.
The only saving grace here — in a horribly twisted way — is that Hawaiian didn’t need all those flights thanks to a massive decline in demand for tourism on Maui when fire destroyed Lāhainā in
August. All of West Maui remained closed for two months before slowly reopening, and there has been real concern about when demand will fully recover. Some airlines have already reduced capacity well
into next summer.
Eventually, the Japanese will return, Hawaiian will get paid by Pratt & Whitney and Maui demand will bounce back, but with all these problems hitting the airline from all sides for a sustained period
of time, it was questionable how long the airline could make it without resorting to a bankruptcy filing.
Investors knew this was all bad news, and the share price suffered.Hawaiian was backed into a corner, so if it’s true that Alaska was the one that actually approached Hawaiian, it must have seemed
like a blessing from Kāne that would allow the airline to survive in this new, combined creation.I’d imagine that nobody from Hawaiian wanted to sell, but after such a difficult few years, this had
to have seemed like the best possible option by far.
37 comments on “Hawaiian’s Five-Year Downhill Slide Led it Right Into Alaska’s Arms”
1. I guess it’s sort of on topic, sort of not but I’d be curious to see what you think AS will do with the Hawaiian wide bodies, Cranky?
Unless AA wins the NEA on appeal, it’s hard to imagine AS would ever be allowed to enter any AA JV which is basically every JV Alaska would want to join, especially from SEA. Too much pricing
coordination would be required for AS to join a JV with a domestic competitor after the legal precedent the court set with the NEA, for better or worse.
I guess they could just keep doing the HA model of flying wide bodies on long US routes like BOS/JFK then Japanese and Oceania destinations but they have competition on most of those routes and
you’d think AS would want some longhaul flying out of SEA with the widebodies.
With all the competition HA faces to/from Hawaii, I’ve wondered if they could do a better job creating better connecting banks to connect most of the mainland destinations to their Oceania and
Pacific network. It could provide some better pricing and unique one stop connections if they tried to provide some unique one stop connections to places they serve and don’t serve today like
RAR, PPT, CNS, ADL, MEL, CHC, APW, PPG, TBU, NAN, etc (some of these islands may obviously a bit too small and more connected to AU/NZ vs the US for traffic patterns). I may be alone in this, but
when I flew HNL-BNE a few years ago, I just loved how much nicer the flight was to divide up the flying time to Australia. Perhaps bring back more of this old route network (not all with wide
bodies) with a true connecting back to the Mainland: https://crankyflier.com/2022/12/06/hawaiian-tests-its-neos-with-new-cook-islands-flight/
The better money seems like it would be with longhaul flying out of SEA but I just wonder how worthwhile that will be when AS would be competing against DL and the AA JVs, most likely.
Probably an interesting article for another time vs a reply to me today… but it is something I’ve wondered.
1. I would expect them to keep the A330s focused on Hawaii initially. They have an established formula for marketing those flights, so it’s much less risky than developing demand from scratch
for a new destination pair.
Given the weakness in the Japanese market, one interesting option would be to redeploy some of the A330s to serve large cities in the eastern US where Alaska already has a station. They could
potentially command a fare premium by offering nonstops to travelers who would otherwise have to connect. They’ve run nonstops to HNL for a long time to NYC and BOS, but there are several
major airports that no nonstop to Hawaii. These fall into two general categories:
– AA eastern hubs (PHL, CLT, MIA). AA doesn’t currently run any nonstops east of DFW, but instead routes all passengers through DFW, PHX, or LAX. There’s an opportunity to pick off some of
those passengers who would prefer a nonstop, but that’s complicated by Alaska’s partnership with AA.
– Airports that aren’t a hub for a legacy network carrier. Most of these are dominated by Southwest. The largest examples include BWI, BNA, STL, TPA. There are also many mid-sized examples in
the midwest, including PIT, CVG, MCI, CMH, IND, and CLE. Alaska/Hawaiian would have a structural advantage at these airports, because no other carrier could feasibly offer a nonstop. Legacy
network carriers aren’t going to fly a widebody for a random spoke-to-spoke flight, and Southwest doesn’t have any aircraft with the range to get to Hawaii from there. Alaska is already
paying some of the fixed costs of maintaining an operation and marketing in these places, though I’m not sure which of these airports could accommodate an A330 without additional investment.
It’s not obvious that there is enough demand to regularly fill up an A330, but maybe there is enough Allegiant-style 1x-2x/week frequency, with the planes serving different destinations each
More speculatively, they could try adding flights to OGG (Maui) and KOA (Big Island) to legacy hubs that currently only have nonstops to HNL (e.g. ATL, IAD, IAH). However, they haven’t run
nonstops to OGG or KOA from their existing stations in BOS or JFK, so my guess is that the demand just isn’t there to support this.
Trans-Pacific flights have weak demand right now for reasons that are idiosyncratic to each major destination. Japanese travel demand to the US is weak for the reasons mentioned in the post,
US-China travel demand is low due to geopolitics and other concerns, and US airlines are handicapped on flights to India by the ban on use of Russian airspace. NRT would be the obvious choice
to try, but it’s already served daily by OneWorld partner JAL. Maybe they try SEA-HKG to connect with Cathay Pacific? Doesn’t seem that promising in the current environment, but it would open
up onward connections to Southeast Asia.
1. I can certainly see the A330s staying in HNL for a bit given the cargo agreement with Amazon. It would make sense for HA to pool passenger a330 flights with Amazon cargo a330 flights for
a while.
Which then does make you wonder if the 787s would see a new home other than HNL.
I could see AS using the oneworld strengths in places like BNA, AUS (already there), RDU, or some other major cities (even Dulles given the DCA strength of OneWorld) but I wonder how much
AA would like that since it would dilute the DFW yields, to your point
1. Max – The A330-300s that fly for Amazon are mainland-based. I believe the base is in Cincinnati, but right now the one airplane operating is just going back and forth between there
and San Bernardino. Maybe that hasn’t opened yet with only 1 airplane, but point being the A330-300s are already not going to be HNL-based.
1. I don’t disagree that the planes are based there but is Hawaiian opening a new pilot base in SBD too? Just seems needlessly expensive to open a new pilot base when every plane
would likely hit HNL anyway and there would be better synergy to use the existing HNL pilot base.
I just assumed Amazon was going to use the HNL pilot base to fly the Amazon cargo a330s, regardless of where the planes are based?
1. Max – The only cargo base is a mainland base. It won’t be in HNL. I think it’s CVG but I’m not 100% sure.
2. Apologies
Just know something about this topic…
The synergies of the Amazon work came from HA’s HNL pilot base being able to fly both pilot and cargo planes out of HNL — same pilot base (didn’t matter where the planes went
on the mainland or whether Amazon wanted to fly an a330 to sbd or the Amazon hub at Cvg)
So it surprised me to see you suggest the pilot base for any Hawaiian a330 flying would be anywhere but HNL.
That’s where the cargo goes to Hawaii and where it would make sense to base HA pilots if they’re flying cargo or pax.
But apologies. Don’t mean to be argumentative on the topic
But that was the basis of my conjecture: that HA would want an a330 pilot base in HNL for a few years while flying amazon cargo and their own passengers
But I appreciate the clarifications if things have changed
3. Sorry
Passenger and cargo flights from HNL was the synergy :)
4. From what I’ve read the Amazon A330 operation is sort of its own thing. Own pilot base, own dispatch facility.. it’s not as integrated into the rest of the HA system as
everything else.
5. Always good to learn something. Thanks Cranky & Nick
6. Cranky is correct – Hawaiian has two pilot bases: HNL and CVG: https://pilotcareercenter.com/Air-Carrier-PCC-Profile/517/Hawaiian-Airlines
2. Sincere question: what do you mean by “I could see AS using the oneworld strengths in places like BNA, AUS (already there), RDU, or some other major cities”?
I think it’s very unlikely that Alaska would fly the widebodies on any route that isn’t anchored by one of their hubs (realistically just SEA or one of the Hawaii airports). For
example, I don’t think they’d try flights to Europe from BNA or RDU, because they have no feed from their own network for those flights.
But I’m probably just misunderstanding you.
1. Alex – I think he means from Honolulu.
1. All conjecture from me
Wish I was cooler to provide facts :)
But yes. I meant speculation from HNL to strong OneWorld cities east of dfw (or even within the catchment area like MCI, OKC, or DSM)
2. That makes sense, thanks. I agree that there might be opportunities to add some long-haul flying from Hawaii to destinations in the eastern US that aren’t legacy hubs, like
BNA, RDU, and others. See my reply to Tim below.
I don’t think the oneworld presence in BNA or RDU makes much of a difference though. AA will prefer to route connections to DFW or PHX to fly on their own metal, while BA
would route passengers through SEA, which is much closer to the LHR-HNL great circle route than any other major airport in the lower 48. SEA is actually very well-positioned
as a scissor hub between Europe and Hawaii. Unfortunately there is just not that much demand in either direction.
3. The 330 freighter (A333F) and 330 passenger (332) are not interchangeable. The freighters have Amazon livery and can’t fly pax. They are also entirely based mainland (CVG) and don’t
touch Hawaii.
2. Very accurate and detailed challenges that have been facing HA for years.
It should be noted that WN is the only one of the big 4 that could have done what they did in entering not just the mainland-HA market but also the intrastate HA market which anyone could see
would crush HA on its own. WN has a privileged position in Washington DC in its ability to convince lawmakers it is all about lowering prices and helping consumers.
The government of Hawaii helped bring about HA’s demise with their covid policies but AS has been “kind enough” to carry the brand on and make commitments to not do with HA what it did w/ Virgin
America – which is to clean house, dump routes and simplify the fleet and operations. AS will increase its own costs through a lot more complexity in the name of diversifying its own network –
and mainland US to Hawaii fares will go up to the joy of the big 4. AS will be a higher cost competitor across their system w/ HA “bolted on” and that will help other legacies, including DL.
As for adding longhaul SEA flying, you need only look at BOS and NYC where B6 and DL directly compete. While part of the reason DL’s growth has been so strong in the NE is B6’ persistent
operational mess, B6 is seeing that DL and other legacies on both sides of the Atlantic will fight hard to keep B6 small in the international arena. AS is run by smarter people than B6 (as
evidenced by the two carrier’s financials) and AS will think twice about jumping into the longhaul international market esp. given that they will use widebodies (if they add longhaul flying)
while the A321NEO for B6 is much less of a threat to other legacies at BOS and JFK. A whole lot of other carriers besides DL are adding longhaul international service to SEA and rely on AS to
feed their flights so AS would also be stepping on their own partners to add AS longhaul international flights.
and, Max, the US has never allowed 2 US carriers be in a JV w/ the same foreign carrier on the same routes. AA routes from SEA and other west coast hubs are part of JVs which means AA would have
to give up JV on its own network in order for AS to get in unless the US changes policy – which didn’t happen with the NEA
1. thanks for repeating what I already said, Tim.
1. @Max Power – but you didn’t work effluent praise of Delta and a blast at JetBlue into your post, so he had to join in , :-)
2. @TimDunn AS commitment to the HA brand maybe similar to DL’s commitment to maintaining CVG as a hub when it merged with Northwest. Once the merger was approved and a few years later, DL
decided to optimize its operations. AS may do the same in a few years to Its HA assets, regardless of what it is currently telling politicians and regulators.
1. Brian W
There is a big difference between maintaining two brands and maintaining hubs. Brands are relatively cheap to maintain and it was a given that AS had to placate Hawaii’s interests while
it was a given that AS would never walk away from the intra-island and longhaul international flying which is core to what the state of Hawaii needs out of an airline.
It is more notable that HA is apparently telling its pilots that it will keep the 717s until the end of the decade; unless AS decides to start dumping the A330s or A321s, both of which
would be costly to do, AS could end up w/ a more complex fleet than AA.
The DL/NW merger was the first of the big 4 megamergers which meant that legislators and regulators had no idea what consolidation would do. Whether Delta knew CVG and MEM would not
survive, they found cover to close those hubs because of high fuel prices that were the result of the 2008-9 financial crisis. It was actually the FL/WN merger that provide a solution to
DL’s problem of a huge 50 seat RJ fleet, giving DL the ability to pick up the 717s – which DL and now HA apparently want to use as long as possible.
It was subsequent mergers that came w/ commitments in writing to maintain hubs or to divest assets; the DL/NW merger resulted in neither.
And while MAX above notes that AA and AS would end up competing w/ each other, it is still very real that AS would compete against its own international partners since it provides feed to
practically every longhaul operator from SEA other than DL.
And it is still possible that regulators will require AA and AS to shrink if not eliminate their cooperation or AA itself might decide that the value of AS is competing against it is
larger than the value from the codesharing that AA and AS does, esp. since AA has apparently decided to not build a SEA longhaul hub.
The big 3 are likely to ask and the DOJ will likely agree to require AS and HA to maintain codeshare services on intrastate Hawaii flights to ensure strong competition remains across AS/
HA’s network.
3. International partner airlines also often have structurally lower costs than US airlines: Crew compensation is lower, the overhead of their HQ staff is lower, etc. To the extent possible, it
might be better for Alaska to encourage as many oneworld partner airlines as possible to run flights to SEA, and then focus on providing seamless connections within the US for those
Everyone likes the idea of a Seattle scissor hub between Asia and the US for geographic regions, but I think the core problem is that there’s just not enough of a local demand base to support
it. The Seattle MSA itself is not that big (~4m people in the 2020 Census), so it’s not clear that it needs much more international service than it already has. Most big cities already have
direct service to either Tokyo or Seoul, so they can just make any ongoing connections at those hubs. Seattle could compete for passengers traveling from smaller metro areas, but the low
population density of the northwestern US means that there just aren’t that many potential passengers in SEA’s “catchment zone”. Most passengers would overfly an existing international hub
(e.g. ORD, DTW, DEN, SFO) “on their way” to SEA.
1. SEA is more than just a scissor hub but a true hub – many more domestic flights for each int’l flight.
While foreign carriers other than in western Europe and Japan have lower labor costs than US carriers on longhaul service, US carriers and esp. at SEA benefit from joint ventures and, in
DL’s case, being able to financially link the benefit of domestic flights to the international operation.
When a carrier like AA, DL or UA carry a domestic connection to a longhaul international flight they operate or a foreign carrier connects a passenger that originated from the US via
their overseas gateway to a shorter gateway, they can justify the shorthaul segment based on the total value of the passenger to the network. Joint ventures are intended to duplicate the
same concept between two carriers.
Because it does not participate in joint ventures, AS has to charge foreign carriers enough to generate comparable revenue to what it can get if it sells seats solely for US O&Ds. Foreign
carriers can offer high quality service because of their lower labor costs but they pay more for connecting service esp. for lower fare international passengers. It isn’t a given that
foreign carriers end up making more money than a US carrier on the same international passenger, esp. if it involves a connection that is not under a joint venture.
AS could shift into the category as AA, DL and UA in being able to “subsidize” its international operation w/ its already existing domestic system esp. at SEA but they face a very
competitive environment in SEA. As you note, SEA has a lot of international service but it is in a favorable geographic position which helps carrier costs across the Pacific. SEA is a
lower cost airport than SFO and the handling costs per passenger at SFO are expected to grow faster than at SEA. SEA does have challenges but the airport and airlines do have a viable
niche. And most international carriers, including DL, have large and growing positions at LAX which is much more about the local market. LAX and SEA work together from a network
perspective more than SFO does with either LAX or SEA.
Let’s also not forget that there are limited barriers to entry for longhaul international service to major HNL markets where HA provides the only longhaul service other than to Tokyo
which AA DL and UA all serve directly or via a JV. If AS decides to get into the longhaul international business from the mainland, DL and UA esp. might have reason to consider longhaul
international service from HNL.
Those that hold onto hope of AS longhaul international service from SEA or other west coast gateways might consider that AS has considered all of the competitive game theories and is not
interested in repeating what happened with Virgin America which was to challenge one or more big 4 carriers that essentially led to the failure of much of that merger.
3. Here’s an interesting thought. Alaska Airlines with their frequent flier program has always had the ability to get their customers on other international airlines without a JV. I wonder with this
expansion with Hawaiian if the international partners will be able to get more of their customers on a combined Alaska/Hawaiian operation?
Compared to the Virgin America merger with its higher costs to operate in SFO, Virgin brand licensing (actually used or not) and aircraft leases, I can see Alaska and Hawaiian as successful
separate brands with integrated back end systems.
4. Yes, Brian W, I don’t trust ANYTHING which an airline executive tells a regulator. They lie about everything from cabin comfort to baggage service, to labor relations, to helping passengers with
physical disabilities, to protecting the public from terrorism (World Trade Center on 9/11), to keeping hubs open (Delta gutting its 2nd largest hub Cincinnati-Northern Kentucky).
Here in California, we used to have Pacific Southwest Airlines as a regional low-cost airline. Eventually it merged with American. Today, there is no trace of PSA in California.
1. You forgot about Air Cal and Reno Air too………
2. Wasn’t PSA bought by US Air?
1. southbay – Yes. I didn’t bother correcting it since technically it’s all part of American now! But American bought AirCal and USAir bought PSA back in the day.
1. *It’s all part of America West now
1. @mrysmf – lol, too true!
2. I think I was surprised to learn that US Air bought PSA since US Air was a very east coast airline at the time and this purchase seemed really out of place.
But, then again AA bought Reno Air and dismantled that pretty quickly.
5. How does one start merger talks?
Does AS send HA a DM on Instagram, saying, “Hey, wanna hook up?” Is it a request sent to legal counsel or a cold call to a board member or c-suite executive?
I am genuinely curious; I’ve always wondered how any merger ball gets rolling.
1. I would think that executives at major corporations know each other and already have lines of communication open. They may have met at industry conferences, been on boards together or worked
together on joint ventures or whatever. There are also clubs (think Rotary but at a higher level) for them to socialize.
If a company wants to sell itself, typically it would hire an investment bank to market it to potential acquirers. If a company wants to make an unsolicited offer, my guess is they would
first talk to the large shareholders and then go to management.
But this is all based on limited info, I have no direct experience with any of this.
2. Remington – It can start in a variety of ways, but in this case it sounds like Alaska’s CEO reached out to Hawaiian’s, at least that’s the story they’re giving.
6. The 787s at a combined HA/AS are the equivalent of the 747 at Eastern, Delta, National in 1970. They too big, expensive, and not suited for Hawaii-Mainland service, really. They make sense for
Asia flying. They don’t for AS’s domestic network. The A330s are young. The first one was delivered in 2010. The issue will be what route map AS envisions for a combined airline. The HA Asia and
Oceania network is significant, if smaller than it was pre-pandemic.
It doesn’t seem to make much sense for AS to launch TPAC or TATL on the 787 out of SEA. They could certainly do it, but it would come at a huge expense and get them little beyond what they get
now with JVs/Alliances.
1. Could AS had a deal to sell their 787 slots to another carrier?
7. It’s becoming increasingly difficult for smaller, niche carriers to survive in the more and more consolidated airline industry,
Captain Obvious
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Based on the model by Bejan et al., natural convection heat transfer, from three attached horizontal isothermal cylinders immersed in quiescent air in an inverted triangular array, has been
numerically investigated over 10 ≤ Ra ≤ 106. The representative results made up of streamlines and isothermal contours, local and average Nusselt number have been showed. It can be seen there exist
several vortexes in the wake region of the downstream cylinders because of the strong interaction of two merging plumes. Additionally, with the effect of the vortex and the preheating among three
attached cylinders, the average heat transfer rate for the whole configuration has been reduced by about 38.7~58.5% compared with single cylinder. Finally, average Nusselt number for this
configuration has been correlated with Rayleigh number, which provides a valid prediction for engineering calculation including this kind of fundamental configuration
PAPER SUBMITTED: 2021-12-26
PAPER REVISED: 2022-07-01
PAPER ACCEPTED: 2022-07-18
PUBLISHED ONLINE: 2022-10-08
, VOLUME
, ISSUE
Issue 6
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7.7 Input and Output Functions
Functions that perform input from a stdio stream, and functions that output to a stdio stream, of mpf numbers. Passing a NULL pointer for a stream argument to any of these functions will make them
read from stdin and write to stdout, respectively.
When using any of these functions, it is a good idea to include ‘stdio.h’ before ‘gmp.h’, since that will allow ‘gmp.h’ to define prototypes for these functions.
See also Formatted Output and Formatted Input.
Print op to stream, as a string of digits. Return the number of bytes written, or if an error occurred, return 0.
The mantissa is prefixed with an ‘0.’ and is in the given base, which may vary from 2 to 62 or from -2 to -36. An exponent is then printed, separated by an ‘e’, or if the base is greater than 10
then by an ‘@’. The exponent is always in decimal. The decimal point follows the current locale, on systems providing localeconv.
For base in the range 2..36, digits and lower-case letters are used; for -2..-36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that
significance order) are used.
Up to n_digits will be printed from the mantissa, except that no more digits than are accurately representable by op will be printed. n_digits can be 0 to select that accurate maximum.
Read a string in base base from stream, and put the read float in rop. The string is of the form ‘M@N’ or, if the base is 10 or less, alternatively ‘MeN’. ‘M’ is the mantissa and ‘N’ is the
exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if base is negative, in decimal. The decimal point expected is taken from the current
locale, on systems providing localeconv.
The argument base may be in the ranges 2 to 36, or -36 to -2. Negative values are used to specify that the exponent is in decimal.
Unlike the corresponding mpz function, the base will not be determined from the leading characters of the string if base is 0. This is so that numbers like ‘0.23’ are not interpreted as octal.
Return the number of bytes read, or if an error occurred, return 0.
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Muslim Contributions to Mathematics
When we talk about Muslim contributions to mathematics we are usually referring to the years between 622 and 1600 ce. This was the golden era of Islam when it was influential both as a culture and
religion, and was widespread from Anatolia to North Africa, from Spain to India.
Mathematics, or "the queen of the sciences" as Carl Friedrich Gauss called it, plays an important role in our lives. A world without mathematics is unimaginable. Throughout history, many scholars
have made important contributions to this science, among them a great number of Muslims. It is beyond the scope of a short article like this one to mention all the contributions of Muslim scholars to
mathematics; therefore, I will concentrate on only four aspects: translations of earlier works, and contributions to algebra, geometry, and trigonometry. In order to understand fully how great were
the works of scholars in the past, one needs to look at them with the eye of a person of the same era, since things that are well-known facts today might not have been known at all in the past.
There has never been a conflict between science and Islam. Muslims understand everything in the universe as a letter from God Almighty inviting us to study it to have knowledge of Him. In fact, the
first verse of the Qur'an to be revealed was:
Read! In the Name of your Lord, Who created… (Alaq 96:1).
Besides commanding us to read the Qur'an, by mentioning the creation the verse also draws our attention to the universe. There are many verses which ask Muslims to think, to know, to learn and so on.
Moreover, there are various sayings of the Prophet Muhammad, peace be upon him, encouraging Muslims to seek knowledge. One hadith says, "A believer never stops seeking knowledge until they enter
Paradise" (al-Tirmidhi).
In another hadith, the Prophet said, "Seeking knowledge is a duty on every Muslim" (Bukhari). Hence it is no surprise to see early Muslim scholars who were dealing with different sciences.
Prophet Muhammed (pbuh) said, “Knowledge is the lost property of a Muslim; whoever finds it must take it” [1]; hence Muslims started seeking knowledge. One way they did this was to start translating
all kinds of knowledge that they thought to be useful. There were two main sources from which Muslim scholars made translations in order to develop the field of science, the Hindus and the Greeks.
The Abbasid caliph al-Mamun (804–832) had a university built and ordered its scholars to translate into Arabic many works of Greek scholarship. Between 771 and 773 CE the Hindu numerals were
introduced into the Muslim world as a result of the translation of Sithanta from Sanskrit into Arabic by Abu Abdullah Muhammad Ibrahim al-Fazari. Another great mathematician, Thabit ibn Qurra, not
only translated works written by Euclid, Archimedes, Apollonius, Ptolemy and Eutocius, but he also founded a school of translation and supervised many other translations of books from Greek into
Arabic. While Hajjaj bin Yusuf translated Euclid’s Elements into Arabic, al-Jayyani wrote an important commentary on it which appears in the Fihrist (Index), a work compiled by the bookseller Ibn
an-Nadim in 988. A simplified version of Ptolemy’s Almagest appears in Abul-Wafa’s book of Tahir al-Majisty and Kitab al-Kamil. Abu’l Wafa Al-Buzjani commented on and simplified the works of Euclid,
Ptolemy and Diophantus. The sons of Musa bin Shakir also organized translations of Greek works.
These translations played an important role in the development of mathematics in the Muslim world. Moreover, the ancient Greek texts have survived thanks to these translations.
Algebra and geometry
The word "algebra" comes from "Al-Jabr", which is taken from the title of the book Hisab Al-Jabr wal Muqabala by Muhammad ibn Musa al-Khwarizmi (780–850). Al-Khwarizmi, after whom the "algorithm" is
named, was one of the great mathematicians of all times. Europe was first introduced to algebra as a result of the translation of Khwarizmi's book into Latin by Robert Chester in 1143. The book has
three parts. The first part deals with six different types of equations:
(ax^2 = bx) ; (ax^2 = b) ; (ax = b) ; (ax^2 + bx = c) ; (ax^2 + c = bx) ; (bx + c = ax^2)
Khwarizmi gives both arithmetic and geometric methods to solve these six types of problems [2]. He also introduces algebraic multiplication and division. The second part of Hisab Al-Jabr deals with
mensuration. Here he describes the rules of computing areas and volumes. Since Prophet Muhammad, peace be upon him, said, “Learn the laws of inheritance and teach them to people, for that is half of
knowledge,”[3] the last and the largest part of this section concerns legacies, which requires a good understanding of the Islamic laws of inheritance. Khwarizmi develops Hindu numerals and
introduces the concept of zero, or “sifr” in Arabic, to Europe. The word “zero” actually comes from Latin “zephirum,” which is derived from the Arabic word “sifr.”
The three sons of Musa bin Shakir (about 800–860) were perhaps the first Muslim mathematicians to study Greek works. They wrote a great book on geometry, Kitab Marifat Masakhat Al-Ashkal (The Book of
the Measurement of Plane and Spherical Figures), which was later translated into Latin by Gerard of Cremona. In the book, although they used similar methods to those of Archimedes, they move a step
further than the Greeks to consider volumes and areas as numbers, and hence they developed a new approach to mathematics. For example, they described the constant number pi as “the magnitude which,
when multiplied by the diameter of a circle, yields the circumference.”[4]
A well-known poet, philosopher and astronomer Omar Khayyam (1048–1122) was at the same time a great mathematician. His most famous book on algebra is Treatise on the Demonstration of Problems of
Algebra. In his book besides giving both arithmetic and geometric solutions to second degree equations he also describes geometric solutions to third degree equations by the method of intersecting
conic sections. He also discovered binomial expansion [26]. His work later helped develop both algebra and geometry.
Thabit bin Qurra (836–901) was an important mathematician who made many discoveries in his time. As mentioned in the Dictionary of Scientific Biography [5] he “played an important role in preparing
the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and
non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.”
To give an idea of his importance, we will just give here, without details, one of his theorems on amicable numbers. Two natural numbers m and n are called “amicable” if each is equal to the sum of
the proper divisors of the other:
for n > 1, let pn=3.2^2n–1 and qn=9.2^2n–1–1. If p[n–1] , p[n] and q[n] are prime numbers, then a=2n p[n–1] p[n] and b=2^nq[n] are amicable. [6]
Abu Kamil (about 850–930), an Egyptian mathematician, wrote the Book on Algebra which consists of three parts:
(1) Solutions of quadratic equations,
(2) Application of algebra to geometry,
(3) Diophantine equations.[7],[8]
He improved the work of Khwarizmi and applied algebraic methods to geometry. His research was on quadratic equations, multiplication and division of algebraic quantities. His work also includes
addition and subtraction of radicals. He found the following formulas:
ax.bx=abx^2; a(bx)=(ab)x; (10–x)(10–x)=100+x^2–20x
Abu Kamil also wrote the Book On Surveying and Geometry, which was intended for government land surveyors. There, he stated the nontrivial rules for calculating areas, volumes, perimeters, and
diagonals of different objects in geometry.[9]
Ibrahim ibn Sinan (908–946), a grandson of Thabit bin Qurra, was both an astronomer and a mathematician. Fuat Sezgin writes, "He was one of the most important mathematicians in the medieval Islamic
world." [10] He studied geometry, and his work on calculation of the area under the graph of a parabola is especially appreciated. Going further than Archimedes, he introduced a more general method
of integration. [11]
Abu Bakr ibn Muhammad ibn al-Husayn al-Karaji (953–1029), also known as al-Karkhi, is regarded as the first person to have developed algebraic operations without using geometry. One of his major
works was Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra). Historian Woepcke recognizes Al-Fakhri as the beginning of the theory of algebraic calculus. [12] Here, al-Karkhi introduced the
monomials x, x^2, x^3, ... and 1/x, 1/x^2, 1/x^3, ... and explained product rules among them. Moreover, he was the first to find the solutions of the equations ax^2n+bx^n=c. [13] Al-Karkhi proved the
sum formula for integral cubes by using the method of proof by induction, and hence became the first to use this method. [14]
Abu'l Hasan ibn Ali al-Qalasadi (1412–1486) was an Andalusian Muslim mathematician. His main contribution was to introduce algebraic symbolism, and he used short Arabic words for his symbols. For
example, he used the symbol for the sound "sh" from the Arabic word meaning "thing" to represent what we call x, the unknown. [15]
Khwarizmi also contributed to trigonometry. He established accurate trigonometric tables for sine and cosine, and he was the first to introduce tangent tables. [16] In 1126, these works were
translated into Latin by Adelard of Bath.
Al-Battani or Albetagnius (about 850–929) was a Muslim astronomer and mathematician. In his research on astronomy he used trigonometric methods which were a lot more advanced than the geometric
methods used by Ptolemy. [17] He introduced trigonometric ratios. For example, for a right triangle with adjacent sides a and b, he gives the formula b sin(A) = a sin(90^0 – A), which is equivalent
to tan A = a/b. He was the first to introduce the cotangent function. [18]
Muhammad Abu'l Wafa (940–998), born at Buzjan in Khorasan, introduced the use of secant, cosecant and tangent functions. He gave a new method of constructing sine tables. He calculated sin(30^0) with
an accuracy of up to eight decimal digits. He improved spherical trigonometry and proved the law of sines for general spherical triangles. [19] In particular, he developed the half/double angle
2 sin^2 (x/2)=1–cos x; sin 2x=2sin x cos x
He was the first to introduce the notion of secant and cosecant, and hence completed the list of all six trigonometric functions. [20]
Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (989–1079) was an Arab mathematician from Andalus. He was the author of The Book of Unknown Arcs of a Sphere which was "the first treatise on spherical
trigonometry." [21] Here he mentioned formulas for right handed triangles and law of sines. He also stated the formula for the solution of a spherical triangle in terms of the polar triangle. [21] He
had a strong influence on the West.
Another outstanding mathematician Nasir al-Din al-Tusi (1201–1274) wrote Treatise On The Quadrilateral, considered the best book on trigonometry written in medieval times, [25] later translated into
French by Alexandre Carathéodory Pasha in 1891. In his book al-Tusi made enormous advances in plane and spherical trigonometry. The Dictionary of Scientific Biography [22] states, "This work is
really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth." The well-known sine
law is also stated in this work: a/sin A = b/sin B = c/sin C.
Ghiyath al-Din al-Kashi (1393–1449) produces sine tables of up to eight decimal places. In 1424, he computed 2π to an accuracy of sixteen decimal digits. He wrote a very impressive book on
mathematics: Miftah al-Hussab (Key to Arithmetic). His main purpose in this book is to provide sufficient knowledge of mathematics for those who are working on astronomy,surveying, architecture,
accounting and trading. [23] He also describes how to find the fifth root of any number. [24]
Unfortunately, the contributions of Muslims often go unrecognized. Muslim scholars contributed to science in many aspects such as mathematics, astronomy, geography, philosophy, medicine, art,
architecture and so on. However, today few realize that in that era Islam played an important role in all aspects of life. Europe faced losing the works of major scholars, but as a result of their
translations into Arabic most of this scholarship not only survived, but was further developed. Inspired by the Qur'an and hadiths, Muslims sought knowledge for the benefit of humankind. As the
Qur'an says, "Are those who know equal to those who know not?"(Zumar 39:9). We should appreciate the scholars of all eras for their contributions to science.
Shirali Kadyrov is a PhD candidate at the Ohio State University, Mathematics Department.
1. Tirmidhi, `Ilm, 19.
2. B.L. van der Waerden, A History of Algebra.
3. Ibn Maja, Hadith No: 2719.
4. D. El-Dabbah, The geometrical treatise of the ninth-century Baghdad mathematicians Banu Musa (Russian), in History Methodology Natur. Sci., No. V, Math. Izdat. (Moscow, 1966), 131–139.
5. Y. Dold-Samplonius, A. T. Grigorian, B. A. Rosenfeld, Biography in Dictionary of Scientific Biography (New York 1970–1990).
6. For more, see S. Brentjes and J. P. Hogendijk, Notes on Thabit ibn Qurra and his rule for amicable numbers, Historia Math. 16 (4) (1989), 373–378.
7. R. Lorch, Abu Kamil on the pentagon and decagon, Vestigia mathematica (1993), 215–252.
8. J. Sesiano, La version latine medievale de ‘l'Algebre d'Abu Kamil, in Vestigia mathematica (Amsterdam, 1993), 315–452.
9.J. Sesiano, Le Kitab al-Misaha d'Abu Kamil, Centaurus 38 (1996), 1–21.
10. F. Sezgin, History of Arabic literature (German) Vol. 5 (Leiden, 1974), 292–295.
11. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Ibrahim.html
12. F. Woepcke, Extrait du Fakhri, traite d'Algebre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853).
13. Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 239. ISBN 0471543977.
14. Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255–259. Addison-Wesley. ISBN 0321016181.
15. J. Samso, Las ciencias de los antiguos en al-Andalus (Madrid, 1992).
16. http://en.wikipedia.org/wiki/History_of_trigonometry
17. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Battani.html
18. http://www.unhas.ac.id/~rhiza/saintis/battani.html
19. http://www.britannica.com/EBchecked/topic/2127/Abul-Wafa
20. http://www.bhatkallys.com/article/article.asp?aid=3442
21. O'Connor, John J. & Robertson, Edmund F., Abu Abd Allah Muhammad ibn Muadh Al-Jayyani.
22. S. H. Nasr, Biography in Dictionary of Scientific Biography (New York 1970–1990).
23. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Al-Kashi.html
24. http://www.bhatkallys.com/article/article.asp?aid=3442
25. http://members.tripod.com/worldupdates/newupdates10/id142.htm
26. Heinrich Dorrie, David Antin (1965). 100 Great Problems of Elementary Mathematics: Their History and Solution, p.34–36. ISBN 0486613488. | {"url":"http://staging6.fountainmagazine.com/all-issues/2009/issue-67-january-february-2009/muslim-contributions-to-mathematics","timestamp":"2024-11-06T02:49:25Z","content_type":"text/html","content_length":"110176","record_id":"<urn:uuid:d1d90037-3b68-4dbb-8a2f-08df6c7e1e56>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00103.warc.gz"} |
The Locked Boxes: A Math BreakoutEDU Digital
— BreakoutEDU, Math, Technology —
The Locked Boxes: A Math BreakoutEDU Digital
September 16, 2017
At the beginning of this school year, I introduced my new students to BreakoutEDU and they loved it! The first BreakoutEDU Digital that they completed was the “Catch the Bus” breakout that is
available on the BreakoutEDU website. So, I decided it was time to create a new BreakoutEDU Digital for my students to review integers and fractions, which is a large portion of our Number Systems
Enter…”The Locked Boxes” BreakoutEDU Digital!
The Storyline:
Since this was only the second BreakoutEDU that my students have completed, I did not have a complex storyline. The storyline is simple:
You are presented with five boxes – four small boxes and one large box. The large box has a hidden secret but you must break into the other boxes first.
Can you unlock the box and find what is inside?!
As my students move through the year, the storylines tend to get more complicated and require the students to pay careful attention to detail. In this one, they will work through the “locked boxes”
that will lead them to questions that they must answer to unlock the codes.
In this BreakoutEDU, students will:
• Add, subtract, multiply, and divide positive and negative integers.
• Add, subtract, multiply, and divide positive and negative mixed numbers.
• Solve complex world problems.
Digital Tools Used:
In this BreakoutEDU, I have used the following digital tools and resources:
• Google Forms (of course!)
• Google Sites
• Google Drawings
• Hidden/Embedded Links
• Google Slides
• Letter/Number Ciphers
Most of the BreakoutEDU requires students to answer problems on Google Forms but they do have to find the forms and/or break into the forms to solve the problems.
To access the “The Locked Boxes” Breakout EDU Digital, click HERE.
Final Thoughts:
I was very excited to see how much my new students enjoyed BreakoutEDU. Every group of students is different and, as a teacher, it is important to know what will engage and inspire them. I’m
looking forward to them participating in more BreakoutEDU activities and, eventually, creating their own!
Discover more from i ❤ edu
Subscribe to get the latest posts sent to your email.
12 thoughts on “The Locked Boxes: A Math BreakoutEDU Digital”
1. These are amazing! Thanks for the info! When you create digital breakouts through google sites, how do you get around the google form resetting when the kids click on a link on the home page? Do
you have them open the google form in a new window before they start?
1. Hi Megan,
Exactly! I have them open multiple tabs. 🙂
2. Do you have an answer key or anything of this?? I’ve worked through the problems but can’t figure out the final code since there are only 4 boxes listed at the top. What about the last box?
1. Hi Nicole!
If you start typing in the last box, it will give you a hint! 🙂
3. So can not figure out the last box…hint?
1. Hi Michele,
The last box is a hidden link that has been jumbled. You just have to find the correct order! 🙂
4. This was awesome! I collaborated with a teacher and WE ESCAPED!!
1. Congrats!!
5. Box #2 is asking for the name of a twelve sided figure, but the answer is supposed to be 12 letters long?? Shouldn’t it be a dodecagon?
1. That is correct! Check whether or not it should be in all caps or not.
6. I am really struggling finding the answer for the 6 letter lock using Box #1… any hints?
1. Hello! Did you find the lock you could click on?
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Sparse Matrix Techniques
A key component of the HPMIXED procedure is the use of sparse matrix techniques for computing and optimizing the likelihood expression given in the section Model Assumptions. There are two aspects to
sparse matrix techniques, namely, sparse matrix storage and sparse matrix computations. Typically, computer programs represent an matrix in a dense form as an array of size , making row-wise and
column-wise arithmetic operations particularly efficient to compute. However, if many of these numbers are zeros, then correspondingly many of these operations are unnecessary or trivial. Sparse
matrix techniques exploit this fact by representing a matrix not as a complete array, but as a set of nonzero elements and their location (row and column) within the matrix. Sparse matrix techniques
are more efficient if there are enough zero-element operations in the dense form to make the extra time required to find and operate on matrix elements in the sparse form worthwhile.
The following discussion illustrates sparse techniques. Let the symmetric matrix be the matrix of mixed model equations of size .
There are 15 elements in the upper triangle of , though eight of them are zeros. The row and column indices and the values of seven nonzero elements are listed as follows:
i 1 1 2 2 3 4 5
j 1 4 2 3 3 4 5
8.0 2.0 4.0 3.0 5.0 7.0 9.0
The most elegant scheme to store these seven elements is to store them in a hash table with row and column indices as a hash key. However, this scheme is not efficient as the number of non-zero
elements gets very large. The classical and widely used scheme, and the one the HPMIXED procedure employs, is the format, in which the nonzero elements are stored contiguously row by row in the
vector c. To identify the individual nonzero elements in each row, you need to know the column index of an element. These column indices are stored in the vector ; that is, if , then . To identify
the individual rows, you need to know where each row starts and ends. These row starting positions are stored in the vector . For instance, if is the first nonzero element in the row i and , then .
The row i ending position is one less than . Thus, the number of nonzero elements in the row i is , these elements in the row i are stored consecutively starting from the position
and the corresponding columns indices are stored consecutively in
For example, the seven nonzero elements in matrix are stored in format as
c 8.0 2.0 4.0 3.0 5.0 7.0 9.0
Note that since matrices are stored row by row in the format, row-wise operations can be performed efficiently but it is inefficient to retrieve elements column-wise. Thus, this representation will
be inefficient for matrix computations requiring column-wise operations. Fortunately, the likelihood calculations for mixed models can usually avoid column-wise operations.
In mixed models, sparse matrices typically arise from a large number of levels for fixed effects and/or random effects. If a linear model contains one or more large CLASS effects, then the mixed
model equations are usually very sparse. Storing zeros in mixed model equations not only requires significantly more memory but also results in longer execution time and larger rounding error. As an
illustration, the example in the Getting Started: HPMIXED Procedure has 3506 mixed model equations. Storing just the upper triangle of these equations in a dense form requires elements. However,
there are only 60,944 nonzero elements—less than 1% of what dense storage requires.
Note that as the density of the mixed model equations increases, the advantage of sparse matrix techniques decreases. For instance, a classical regression model typically has a dense coefficient
matrix, though the dimension of the matrix is relatively small.
The HPMIXED procedure employs sparse matrix techniques to store the nonzero elements in the mixed model equations and to compute a sparse Cholesky decomposition of these equations. A reordering of
the mixed model equations is required in order to keep the minimum memory consumption during the factorization. This reordering process results in a different g-inverse from what is produced by most
other SAS/STAT procedures, for which the g-inverse is defined by sequential sweeping in the order defined by the model. If mixed model equations are singular, this different g-inverse produces a
different solution of mixed model equations. However, estimable functions and tests based on them are invariant to the choice of g-inverse, and are thus the same for the HPMIXED procedure as for
other procedures. | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_hpmixed_details04.htm","timestamp":"2024-11-11T10:57:29Z","content_type":"application/xhtml+xml","content_length":"32172","record_id":"<urn:uuid:1bc9efec-e07a-4fb3-9ce0-2a7d141ef6f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00222.warc.gz"} |
Probabilistic / Approximate Counting [Complete Overview]
Open-Source Internship opportunity by OpenGenus for programmers. Apply now.
In this article, we will be introducing and exploring the idea of Probabilistic algorithms in depth with the different algorithms like Morris Algorithm, HyperLogLog, LogLog and others in this domain.
Table of content:
1. Overview of Probabilistic/ Approximate Counting algorithms
2. Problem statement of counting
3. Approximate counting algorithm (Morris Algorithm)
3.1 Overview
3.2 Working
3.3 Applications
3.4 Complexity
4. Hyperloglog algorithm
4.1 Overview
4.2 Loglog algorithm
4.3 Working
4.4 Improvements & Differences
4.5 Applications
4.6 Complexity
5. References
Overview of Probabilistic/ Approximate Counting algorithms
The probabilistic/approximate counting algorithms allow one to count a large number of events using a small amount of memory.
These type of algorithms are especially useful when the memory aspect for a program, application, etc. in terms of usage and complexity has to be minimal.
In this article, we will be exploring two Probabilistic/Approximate Counting algorithms, which are:
1. Approximate counting or Morris's algorithm
2. LogLog algorithm
3. Hyperloglog algorithm
4. Further improvements
Problem statement of counting
In most of the probabilistic algorithms, the counting procedure is usually similar.
Let us assume that we have a hash function denoted by:
fun hash(x: records): scalar range[0 ... 2ᴸ-1]
where length is denoted by L.
The function transforms the records (passed into it) into integers uniformly distributed over the scalar range of binary strings of length L.
It can be observed that if the values of hash(x) are distributed uniformly, the pattern 0ᵏ1 ... appears with a probability of 2⁻ᵏ⁻¹. The main idea is to record observations on the occurrence of the
said patterns in a vector BITMAP of length L (BITMAP[0 ... L-1]).
Let us assume that the list/multi-set whose cardinality is to be determined is M, then we the following operations have to be performed to accomplish the objective:
1. for i from 0 to L-1 do:
1.1. put BITMAP[i] = 0
2. for all x in M do:
2.2. put index = ρ(hash(x))
2.3. if BITMAP[index] = 0 do:
2.3.1. put BITMAP[index] = 1
Another function/method that has been referenced in the above algorithm is ρ(i), which basically represents the position of the least significant 1-bit in the binary representation of i. It is
essentially going to return the index of the least significant 1-bit present in the value that is returned by the hash(x) function.
The final value obtained in the BITMAP vector is used as an anchor point for the counting operation.
Keep in mind that most of the advanced algorithms have modifications of their own to the algorithm, however, the base mostly remains the same.
3. Approximate counting algorithm (Morris Algorithm)
3.1 Overview
The approximate counting algorithm was invented by Robert Morris in 1977.
The algorithm makes use of probabilistic techniques to increment the counter in order for it to keep count of the events and because of this, its exactness isn't really absolute, although it does
provide a fairly good estimate of the true value while providing a minimal and yet fairly constant relative error.
This algorithm stemmed from the time when once Morris had to count a large number of events in an 8-bit register. Seeing that the count would probably exceed the 255 mark, he (Morris) decided that it
would be better if he developed a probabilistic algorithm with a quantifiable margin of error first and then deployed it.
The algorithm is considered as one of the foundational algorithm or base for streaming algorithms.
3.2 Working
A simple and obvious way to create a working probabilistic/approximate counter would be to count every alternate event.
In Morris's algorithm, however, instead of keeping track of the total number of events n or some constant multiple of n, the value we store in the register is the logarithmic value of (1 + n).
How does the counter make the decision of as to which event should be counted or not? One of ways is to make use of the coin flip technique, which is relatively simple. Here is how it works:
We flip a coin and if it comes out to be head, then we increment the counter else we let it be.
The counter will represent an order of magnitude estimate, i.e., an approximation of the log of a value relative to some understood reference value, of the actual count. When the counter is
operating, only the exponent is stored of the probabilistic technique that is used to increment the counter. This is done to save space.
For example, with base as 2, the counter can estimate the count to be 1, 2, 4, 8, 16, .... and so on. With base as 3, the counter can estimate the count to be 1, 3, 9, 27, 81, .... and so on.
To increment from 9 to 27, a pseudo-random number is to be generated such that a probability of ¼ (0.25) generates a positive change in the counter. If it does, then the counter is incremented, else
the counter will remain at 9.
The following table, with its base as 3, demonstrates the working and the potential values of a counter:
Binary value of counter Estimate Values range for actual count Expectation
0 1 0, or the initial value 0
1 3 1, or more 6
10 9 2, or more 24
11 27 3, or more 78
100 81 4, or more 240
101 243 5, or more 726
When the counter holds the value of 100 (binary value), which equates to an exponent of 4 (the decimal equivalent of 100), then the estimated count is 3^4, or 81. There is a fairly low probability
that the actual count of increment events was 4 (1 * ⅓ * ⅑ * ¹⁄₂₇ = ¹⁄₇₂₉). The actual count of increment events is likely to be somewhere around 81, but it could be arbitrarily high.
3.3 Applications
The algorithm can be used for investigating large data sets/streams for patterns. It is particularly useful in applications of data compression, sight and sound recognition, and other artificial
intelligence applications.
3.4 Complexity
Let's assume that we have to count up to n by masking use of the Morris's algorithm. We know that it will go up to O(logn) (remember that the counter only keeps the exponent to save space), and so
the total number of bits that would be required to count from 0 up to O(logn) would become O(log(logn)).
Hence, the overall space complexity of Morris's algorithm is O(log(logn)).
4. Hyperloglog algorithm
4.1 Overview
The Hyperloglog algorithm is derived and is a further extension of the earlier loglog algorithm. The main objective of both these algorithms remains the same, that is, to estimate to number of unique
/distinct items in a given list/set/stream. The hyperloglog algorithm determines the cardinality or the uniqueness (number of unique elements) in a list/set/stream.
Like most probabilistic algorithms, the hyperloglog algorithm also uses significantly less memory or space to accomplish the same task as its counterpart algorithms, mainly because of the fact that
the algorithm stresses mostly on saving memory and space without compromising its performance and main objective.
Let's have a brief look at the loglog algorithm first in order for us to have a better understanding of the Hyperloglog algorithm.
4.2 Loglog algorithm
Loglog is a probabilistic algorithm that makes use of a hashing function in order to randomize data and then convert them to a form that resembles random binary data. The hashed data set is then
reformed into cardinality estimates by the algorithm.
In general, the loglog algorithm makes use of n small bytes of auxiliary memory to estimate the number of unique elements of a list in a single pass with an accuracy that is of the order of 1/√n.
It can be useful in many scenarios, such as to count the number of different words and their cardinality from a whole book very quickly, etc.
In terms of space complexity, the loglog algorithm consumes O(log(logn)) bits of storage if n is the range till which the algorithm has to operate.
There are many optimized versions of this algorithm like super-loglog, hyperloglog, etc.
4.3 Working
Much like the algorithm from which hyperloglog originated (loglog algorithm), it also makes use of a hash function that is applied to every element that is present in the given set/list to acquire a
set of uniformly distributed random numbers with the same cardinality as that of the original set.
However, only doing this to obtain the cardinality of a set will result in a large variance. To counter this, the hyperloglog algorithm further sub-divides the set into various subsets, after which
the maximum number of leading zeros in the numbers for each subset is calculated. After this operation, a harmonic mean is used to combine the calculated estimates for all the subsets into an
estimate of the cardinality of the combined subsets.
The hyperloglog algorithm has three primary operations and its data is stored in an array P of counters called registers with size p that are set to 0 in their initial state.
1> Add operation: This operation is used to add a new element to the set. It consists of computing the hash of the input data d with a hash function h, getting the first b bits (b = log2p). The
address of the register that is to be modified is obtained by adding 1 to them. The bits that are remaining are used to compute ρ(w), which returns the position of the 1 that is present at the
left-most position in terms of index (in binary form). The maximum value between the current value of the register and ρ(w) is chosen as the new value of the register.
2> Count operation: This operation is used to obtain the cardinality of the set. The harmonic mean of the p registers is calculated and then the estimate E of the count (cardinality) is derived using
a constant.
3> Merge operation: This operation is used to merge or join two sets together. For example, if we have two different sets U1 and U2 that are using the hyperloglog algorithm, then the merge operation
will obtain the maximum for each pair of registers, i.e., j from 1 to p.
Merge(U1, U2)[j] = max(U1[j], U2[j])
4.4 Improvements & Differences
Although the overall complexity of both the Hyperloglog and the Loglog algorithm is roughly the same, they still have some significant differences between them as Hyperloglog is simply an improved
and more efficient version of the loglog algorithm.
• The hyperloglog algorithm further divides the sets into sub-sets and then performs further operations on them. This is done mainly to ensure that the variance (when the cardinality is being
obtained) remains minimal. This increases the efficiency and accuracy of this algorithm.
• The hyperloglog uses the harmonic mean to estimate the cardinality of a given list/set, whereas the loglog algorithm uses the normal mean.
• The standard error estimation for the Hyperloglog algorithm is (1.04/√n), whereas it is (1.30/√n) for the loglog algorithm, where n are the number of buckets. The difference, although not very
significant, does come into play when the algorithms are to be used on big data sets (which they usually are).
4.5 Applications
The main application of the hyperloglog algorithm is to estimate accurate cardinality for a given set while being extremely space-sufficient.
It can be used in applications or softwares that have very large data sets and can also be utilized for various forms of graph statistics.
4.6 Complexity
The relative error of Hyperloglog algorithm is generally 1/√p with a space complexity of
O(ε⁻²log(logn) + logn), where n is the cardinality of the set and p is the number of registers.
Taking the primary operations into consideration, the add operation is dependent upon the size of the output of the main hash function, and hence its running time is O(1), because of its fixed
nature. Since the count and merge operations are actually dependent upon the size of the registers (p), they have a running time of O(p). However, if the number of registers are fixed, then the
running time of both these operations will also become constant.
Hence, the hyperloglog algorithm has a overally space complexity of O(log(logn))
• "Probabilistic counting algorithms for database applications" by Philippe Flajolet and G. Nigel Martin in "Journal of Computer and System Sciences" October 1985. | {"url":"https://iq.opengenus.org/probabilistic-approximate-counting/","timestamp":"2024-11-02T20:35:41Z","content_type":"text/html","content_length":"72184","record_id":"<urn:uuid:57211757-054c-4502-abe6-b43f26d9e627>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00526.warc.gz"} |
Decomposition algebras and axial algebras
We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras:
(1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3)
They admit a natural notion of homomorphisms, making them into a nice category.
We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial
under some mild assumptions.
We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large
number of examples.
We also take the opportunity to fix some terminology in this rapidly expanding subject.
• 17A99, 20F29
• Association schemes
• Axial algebras
• Decomposition algebras
• Fusion laws
• Griess algebra
• Majorana algebras
• Norton algebras
• Representation theory
• math.GR
• math.RA
Dive into the research topics of 'Decomposition algebras and axial algebras'. Together they form a unique fingerprint. | {"url":"https://research.birmingham.ac.uk/en/publications/decomposition-algebras-and-axial-algebras","timestamp":"2024-11-01T22:59:15Z","content_type":"text/html","content_length":"57491","record_id":"<urn:uuid:5b841283-9dbe-4862-bda2-b4f7cdde68d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00001.warc.gz"} |
Troubleshooting: don't be too quick to zoom in
Troubleshooting: don’t be too quick to zoom in
Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number
2, for 3, for 5, for 7, for 11… What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…
…the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the
answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.
However, beyond awareness of this tendency, the real challenge is knowing when to zoom in and when to zoom out. There are no easy answers, which emphasises the importance of intuition, and that only
comes with practice and knowledge. | {"url":"https://wordspeak.org/posts/dont-be-too-quick-to-zoom-in.html","timestamp":"2024-11-03T21:50:16Z","content_type":"text/html","content_length":"14461","record_id":"<urn:uuid:622646a7-8d78-4eb7-abe7-0bb75960e7c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00032.warc.gz"} |
Mueller matrices for common optical components such as polarizers, phase retarders, and attenuating filters. The matrices are built using StaticArrays.jl for speed and can be arbitrarily rotated.
Import the library like any other Julia package
Mueller.jl provides building blocks for common components. Here I generate the Mueller matrix for an optical system comprising three linear polarizers, each rotated 45 degrees from the one prior.
Notice the matrix multiplication is inverse the order of the optical components.
julia> M = linear_polarizer(π/2) * linear_polarizer(π/4) * linear_polarizer(0)
4×4 StaticArrays.SMatrix{4, 4, Float64, 16} with indices SOneTo(4)×SOneTo(4):
0.125 0.125 0.0 0.0
-0.125 -0.125 0.0 0.0
-1.53081e-17 -1.53081e-17 0.0 0.0
0.0 0.0 0.0 0.0
you'll notice some roundoff due to the finite precision of π/4, you can avoid this by using Unitful.jl
julia> using Unitful: °
julia> M = linear_polarizer(90°) * linear_polarizer(45°) * linear_polarizer(0°)
4×4 StaticArrays.SMatrix{4, 4, Float64, 16} with indices SOneTo(4)×SOneTo(4):
0.125 0.125 0.0 0.0
-0.125 -0.125 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
let's see what happens when completely unpolarized light passes through these filters. We can represent light using the Stokes vector
julia> S = [1, 0, 0, 0] # I, Q, U, V
4-element Vector{Int64}:
julia> Sp = M * S
4-element StaticArrays.SVector{4, Float64} with indices SOneTo(4):
the output vector has 1/8 the total intensity of the original light, and it is 1/8 polarized in the -Q direction (vertical). This demonstrates the somewhat paradoxical quantum behavior of light (
Bell's Theroem, inspired by this video): even though the light passes through two orthogonal linear polarizers (the 0° and 90° ones) because the wave equation operates probabilistically, 50% passes
through the first polarizer, 50% of that light passes through the 45° polarizer, and then 50% of the remaining light passes through the final polarizer, combining to 1/8 of the original light.
If you would like to contribute, feel free to open a pull request. If you want to discuss something before contributing, head over to discussions and join or open a new topic. If you're having
problems with something, please open an issue. | {"url":"https://juliapackages.com/p/mueller","timestamp":"2024-11-14T17:18:23Z","content_type":"text/html","content_length":"119450","record_id":"<urn:uuid:47ae1316-25c3-487e-af35-f6107dcae423>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00361.warc.gz"} |
Class 10 Real NumbersClass 10 Real Numbers NCERT Solutions & CBSE Notes - Leverage Edu
Class 10th Maths syllabus aims to impart students with the basic and fundamental concepts of this discipline to further build upon in the upcoming grades. Amongst the most interesting of these
concepts are Real Numbers which form the foundation of several imperative Mathematical topics. In your previous classes, you must have studied about multiple types of numbers such as Natural, Whole,
Rational, Irrational and Integers. But do you know that all these types of numbers fall under Real Numbers? Amazing! Right? As we are here to walk you through the topic, we will keep on stating some
witty facts like these. So, stay tuned and let’s start our discussion of class 10 real numbers.
What are Real Numbers?
All numbers, including positive, negative, and zero, come under the broad category of real numbers. In general terms, real numbers represent all those numbers which are not imaginary numbers.
Therefore, a real number combines the properties of rational and irrational numbers.For example: 4, 43, -34, -5, 3/4, 5/9, π(3.14), etc.
Before we move further with the concept of real numbers, let us quickly recall the major types of numbers which you must have previously studied.
Natural Numbers
Natural numbers represent all the positive numbers in nature from 1 to infinity. The set of natural numbers is represented as ‘N’ and does not include decimals, fractions, or any negative numbers. N
= 1, 2, 3, 4, 5, 6, …
Whole Numbers
Whole numbers represent all positive numbers, including zero, and are represented as ‘W’. W = 0, 1, 2, 3, 4, 5, 6, …
Integers include all positive and negative numbers with zero, not representing any decimal or fractional component. For instance, 2, 3, 51, 0, -1, -5, while 2.45, √2, and 10 ¾ are not integers. They
are presented as ‘I’. I = …., -2, -1, 0, 1, 2, …
Rational Numbers
Rational Numbers are numbers that can be presented as fractions and decimals in the form of a/b of two integers with ‘a’ as the numerator and ‘b’ as a non-zero denominator. For example, 2/3, 4/5, 1/
2, 10 ¾, etc. They are represented as ‘R’. R = ½ , ¾ , 5.24, 0.5432, etc.
Irrational Numbers
Irrational Numbers represent all numbers that are not a rational number or can not be expressed in the ratio of any integers. For example, √2, √3, √5, √7, and π are irrational numbers.
Presentation Of Real Number
All types or forms of a real number as per real numbers chapter (integers, fractions, and decimals) can be precisely placed on the Number line. For instance, real number 4 as 4.0, -3 as -3.0,
fractions in the form of 2/5 as 0.40, 1/2 as 0.50, and so on.
Euclid Division Lemma (Algorithm)
Euclid Division Lemma algorithm follows the conceptual formula [dividend = (divisor x quotient) + remainder]. As per this theorem, if there are two positive integers let’s say p and q, then there
exists a parallel integers r and s that satisfies the statement as p = qr + s.(where s represents a number equal to more than 0 and less than q).
We can easily find the HCF using Euclid’s algorithm. Let’s study a simple example to get the HCF of two integers 455 and 42, respectively.
Then, by Euclid’s Lemma, we have:
455 = 42 × 10 + 35
Similarly, if we use the dividend method (divisor 42, remainder 35) and then as per the Euclid’s Lemma,
42 = 35 × 1 + 7
Now, moving on here the divisor becomes 35 and remainder 7, and using the Euclid lemma to find:
35 = 7 x 5 + 0
As the remainder becomes zero, we can stop the process to conclude the HCF of two integers 455 and 42 as 7. Thus, using Euclid’s algorithm mentioned in Class 10 Real Numbers, you can calculate the
HCF of two numbers in a simple manner.
Decimal Expansion
While studying class 10 chapter on real numbers, you will also get to know about the expansion of a decimal rational number in two ways. Terminating decimal expansion are those when decimal ends with
finite numbers. For instance, ½ as 0.50 in decimals, and vice versa.
0.375 = 375/1000 = 3 x 5^ 3 / 2^3 x 5^3 = ⅜
Non-terminating decimal expansions are with those when decimals are non-terminating. They can be either repeating (rational numbers) or non-repeating (irrational numbers).
For Example:
7/12 = 0.58333… = 0.583
9/11 = 0.8181… = 0.81(repeating)
Source: NCERT
So, decimal expanses of rational numbers will be either terminating or non-terminating.
Fundamental Theorem of Arithmetic
The theorem states that any composite number is presentable as factors or products of primes. Therefore, irrelevant of the order, the prime factorization of a natural number will always be unique in
itself. For instance, 3 × 5 × 7 x 11 is taken the same as 5 × 7 x 11 x 3 or in any order that they are written in.
Class 10 Maths Real Numbers Practice Questions
Here are a few practice questions for class 10 maths real numbers.
• The decimal expansion of the rational number 43/ 2^45^3 will terminate after how many places of decimals?
• Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and 15 respectively.
• Express 98 as a product of its primes.
• HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, find the other number.
• Find HCF and LCM of 13 and 17 by prime factorisation method.
• Find the HCF (865, 255) using Euclid’s division lemma.
• Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively. (
• Show that 3√7 is an irrational number.
• Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number?
Class 10 Maths Real Numbers MCQs with Answers PDF
Thus, we hope that this blog has provided you with useful and informative study notes on class 10 real numbers. Being a class 10th student, it is necessary to sort out your interests and select a
stream accordingly. If you are struggling in choosing the right stream, connect with Leverage Edu experts and they will help you select the right one as per your career aspirations! Hurry Up! Book an
e-meeting now. | {"url":"https://leverageedu.com/blog/class-10-real-numbers/","timestamp":"2024-11-11T17:55:38Z","content_type":"text/html","content_length":"333017","record_id":"<urn:uuid:3aab10bf-0c5f-4b54-8b8a-57d4dcf103c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00577.warc.gz"} |
On the Strict Singularity and Finitely Strict Singularity of Non-Compact Embeddings Between Besov Spaces
On the Strict Singularity and Finitely Strict Singularity of Non-Compact Embeddings Between Besov Spaces
Core Concepts
This note provides a complete classification of the "quality" of non-compactness for embeddings between Besov spaces, characterizing when they are finitely strictly singular and strictly singular,
which could indicate the existence of "most optimal" target spaces.
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Note about non-compact embeddings between Besov spaces
Chuah, C. Y., Lang, J., & Yao, L. (2024). Note about non-compact embeddings between Besov spaces. arXiv preprint arXiv:2410.10731.
This note investigates the "quality" of non-compactness for embeddings between Besov spaces, aiming to classify when these embeddings are finitely strictly singular and strictly singular.
Deeper Inquiries
Can the techniques used in this note be extended to classify the "quality" of non-compactness for embeddings between other function spaces beyond Besov spaces?
It's certainly possible to explore extending the techniques used in the note to other function spaces. Here's a breakdown of the key techniques and potential challenges: Key Techniques Used: Wavelet
Decomposition: This is a central tool for Besov spaces, allowing for their characterization in terms of sequence spaces. This simplifies the analysis of embeddings. Reduction to Sequence Spaces: By
using wavelet isomorphisms, the authors translate the problem of studying embeddings between Besov spaces to studying embeddings between sequence spaces like ℓq(ℓp) spaces. Analysis of Bernstein
Numbers: Bernstein numbers are used as a measure of "quality" of non-compactness. The authors analyze these numbers for the sequence space embeddings. Diagonal Theorem: This theorem helps to lift
results about finite strict singularity from individual embeddings between sequence spaces to embeddings between spaces of sequences. Extending to Other Function Spaces: Spaces with Similar
Decompositions: The techniques could be applicable to function spaces that admit decompositions analogous to wavelet decompositions. Examples include: Triebel-Lizorkin spaces: These spaces are
closely related to Besov spaces and share similar wavelet characterizations. Modulation spaces: These spaces use a different type of decomposition based on the short-time Fourier transform, but the
principle is similar. Challenges for Other Spaces: Lack of Suitable Decompositions: Spaces without convenient decompositions would require different approaches. Complexity of Embeddings: The analysis
of Bernstein numbers might become significantly more complex for spaces with less tractable structures. Finding Analogous "Optimal" Embeddings: The concept of "most optimal" embeddings, linked to
non-strict singularity in the note, might need to be redefined or adapted for different function space settings. In summary, while the techniques hold promise for spaces with similar decomposition
properties, extending them to a broader class of function spaces would require careful consideration of the specific structures and challenges posed by those spaces.
Could there be alternative characterizations of "most optimal" target spaces for embeddings between function spaces that do not rely on the notion of strict singularity?
Yes, there are alternative ways to characterize "most optimal" target spaces that don't solely depend on strict singularity. Here are a few possibilities: Interpolation Theory: Optimal Interpolation
Spaces: Given an embedding X ↪ Y, one can seek the "smallest" or "most optimal" space Z such that X ↪ Z ↪ Y, where Z is an interpolation space between X and Y. This often involves finding the
interpolation space with the smallest possible parameters. K-Functionals and Interpolation Inequalities: The optimality of Z can be related to the behavior of the K-functional associated with the
interpolation couple (X, Y). Sharp interpolation inequalities can also provide insights into the optimality of embeddings. Rearrangement-Invariant Spaces: Optimal Target within a Class: As mentioned
in the note, the embedding into Lp∗,p(Ω) is optimal within the class of rearrangement-invariant spaces. One could explore optimality within other specific classes of function spaces relevant to the
problem at hand. Rearrangement Inequalities: Sharp rearrangement inequalities can be used to establish the optimality of embeddings within classes of rearrangement-invariant spaces. Entropy Numbers
and Other Approximation Quantities: Minimal Decay of Entropy Numbers: Instead of strict singularity, one could focus on the decay rate of entropy numbers. An embedding with the slowest possible decay
of entropy numbers (within a given context) could be considered "most optimal." Other s-Numbers: Similar considerations could be applied to other s-numbers, such as approximation numbers or Gelfand
numbers, to characterize optimality. Geometric Properties of Target Spaces: Minimizing Distortion: The "most optimal" target space could be the one that minimizes the distortion of the image of the
embedding. This could involve studying properties like Banach-Mazur distances or other measures of geometric similarity between spaces. The choice of the most appropriate characterization would
depend on the specific function spaces involved and the desired properties of the "most optimal" target space in the given application.
What are the practical implications of understanding the "quality" of non-compactness in applications involving Besov spaces, such as in image processing or partial differential equations?
Understanding the "quality" of non-compactness, particularly in the context of strict singularity, has significant practical implications in various applications involving Besov spaces: 1. Image
Processing: Image Compression: Non-compact embeddings are closely related to the ability to approximate functions (or images) in one space by elements of another space. Finitely strictly singular
embeddings suggest the possibility of achieving good compression rates with relatively few coefficients. Denoising and Regularization: In image denoising and regularization problems, Besov spaces are
often used as regularization terms. The "quality" of non-compactness can influence the effectiveness of these regularization methods and the smoothness properties of the reconstructed images. 2.
Partial Differential Equations (PDEs): Existence and Regularity of Solutions: The compactness of embeddings between function spaces is crucial in proving the existence and regularity of solutions to
PDEs. Understanding the precise nature of non-compactness can provide insights into the types of solutions that can be expected. Numerical Analysis of PDEs: In numerical methods for PDEs, such as
finite element methods, the "quality" of non-compactness can affect the convergence rates of the numerical schemes. Finer distinctions beyond simple compactness can lead to more accurate error
estimates and better algorithm design. 3. Other Applications: Approximation Theory: The study of non-compact embeddings is fundamental in approximation theory, where the goal is to approximate
functions in one space by elements of another. Strict singularity and related concepts provide tools for analyzing the efficiency of such approximations. Statistical Learning Theory: Besov spaces are
used in statistical learning theory to characterize the complexity of function classes. The "quality" of non-compactness can influence the generalization properties of learning algorithms and the
rates of convergence. In essence, understanding the "quality" of non-compactness allows for: More Refined Analysis: It goes beyond simply knowing whether an embedding is compact or not, providing a
deeper understanding of the relationship between the spaces involved. Improved Algorithm Design: It can guide the development of more efficient algorithms for image processing, numerical analysis,
and other applications. Sharper Theoretical Results: It leads to more precise theoretical results, such as improved error estimates and convergence rates. | {"url":"https://linnk.ai/insight/scientificcomputing/on-the-strict-singularity-and-finitely-strict-singularity-of-non-compact-embeddings-between-besov-spaces-dghmwEhp/","timestamp":"2024-11-03T18:32:24Z","content_type":"text/html","content_length":"292566","record_id":"<urn:uuid:a379d244-8691-4717-bc99-6e10e8663153>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00694.warc.gz"} |
embodied cognition
When I was nearly 18 I was part of the British team to the International Mathematical Olympiad (IMO) in Bucharest (see my account of the experience). The US team were Jewish^1, all eight of them.
While this was noteworthy, it was not surprising. There does seem to be a remarkable number of high achieving Jewish mathematicians, including nearly a quarter of Fields Medal recipients (the Maths
equivalent of the Nobel Prize) and half of the mathematics members of the US National Academy of Sciences^2.
Is this culture or genes, nature or nurture?
As with most things, I’d guess the answer is a mix. But, if of culture, what? There is a tradition of Biblical numerology, but hardly widespread enough to make the substantial effects. Is it to do
with the discipline of learning Hebrew, maybe just discipline, or perhaps is it that mathematics is one of the fields where there has been less prejudice in academic appointments^3.
I have just read a paper, “Disembodying Cognition” by Anjan Chatterjee, that may shed a little light on this. The paper is an excellent overview of current neuroscience research on embodiment and
also its limits (hence ‘disembodying’). One positive embodiment result related to representations of actions, such as someone kicking a ball, which are often depicted with the agent on the left and
the acted upon object on the right. However, when these experiments are repeated for Arab participants, the direction effects are reversed (p.102). Chaterjee surmises that this is due to the
right-to-left reading direction in Arabic.
In mathematics an equation is strictly symmetrical, simply stating that two thinsg are equal. However, we typically see equations such as:
y = 3x + 7
where the declarative reading may well be:
y is the same as “3x + 7”
but the more procedural ‘arithmatic’ reading is:
take x, multiple by three, add seven and this gives y
In programming languages this is of course the normal semantics … and can give rise to confusion in statements such as:
x = x + 1
This is both confusing if read as an equation (why some programming languages have := read as “becomes equal to”), but also conflicts with the left-to-right reading of English and European languages.
COBOL which was designed for business use, used English-like syntax, which did read left to right:
ADD Tiree-Total TO Coll-Total GIVING Overall-Total.
Returning to Jewish mathematicians, does the right-to-left reading of Hebrew help in early understanding of algebra? But if so then surely there should be many more contemporary Arab mathematicians
also. This is clearly not the full story, but maybe it is one contributory factor.
And, at the risk of confusing all of us brought up with the ‘conventional’ way of writing equations, would it be easier for English-speaking children if they were introduced to the mathematically
equivalent, but linguistically more comprehensible:
3x + 7 = y
1. Although they did have to ‘forget’ while they were there otherwise they would have starved on the all-pork cuisine[back]
2. Source jews.org “Jews in Mathematics“.[back]
3. The Russians did not send a team to the IMO in 1978. There were three explanations of this (i) because it was in Romania, (ii) because the Romanians had invited a Chinese team and (iii), because
the Russian national mathematical Olympiad had also produced an all Jewish team and the major Moscow university that always admitted the team did not want that many Jewish students. Whether the
last explanation is true or not, it certainly is consonant with the levels of explicit discrimination in the USSR at the time. [back]
tales from/for Berlin – appropriation, adoption and physicality
A few weeks ago I had a short visit to Berlin as a guest of Prometei, a PhD training program at the University of Technology of Berlin focused on “prospective engineering of
human-technology-interaction”. While there I gave an evening talk on “Designing for adoption and designing for appropriation” and spent a very pleasant afternoon seminar with the students on
“Physicality and Interaction”.
I said I would send some links, so this is both a short report on the visit and also a few links to appropriation and adoption and a big long list of links to physicality!
matterealities and the physical embodiment of code
Last Tuesday morning I had the pleasure of entertaining a group of attendees to the Matterealities workshop @ lancaster. Hans and I had organised a series of demos in the dept. during the morning
(physiological gaming, Firefly (intelligent fairylights), VoodooIO, something to do with keyboards) … but as computer scientists are nocturnal the demos did not start until 10am, and so I got to talk
with them for around an hour beforehand :-/
The people there included someone who studied people coding about DNA, someone interested in text, anthropologosts, artists and an ex-AI man. We talked about embodied computation^1, the human body as
part of computation, the physical nature of code, the role of the social and physical environment in computation … and briefly over lunch I even strayed onto the modeling of regret … but actually a
little off topic.
physicality – Played a little with sticks and stones while talking about properties of physical objects: locality of effect, simplicity of state, proportionality and continuity of effect^2.
physical interaction – Also talked about the DEPtH project and previous work with Masitah on natural interaction. Based on the piccie I may have acted out driving when talking about natural inverse
ubiquity of computation – I asked the question I often do “How many computers do you have in your house” … one person admitted to over 10 … and she meant real computers^3. However, as soon as you
count the computer in the TV and HiFi, the washing machine and microwave, central heating and sewing machine the count gets bigger and bigger. Then there is the number you carry with you: mobile
phone, camera, USB memory stick, car keys (security codes), chips on credit cards.
Firefly demo later in the morning they got to see what may be the greatest concentration of computers in the UK … and all on a Christmas Tree. Behind each tiny light (over 1000 of them) is a tiny
computer, each as powerful as the first PC I owned allowing them to act together as a single three dimensional display.
embodiment of computation – Real computation always happens in the physical world: electrons zipping across circuit boards and transistors routing signals in silicon. For computation to happen the
code (the instruction of what needs to happen) and the data (what it needs to happen with and to) need to be physically together.
The Turing Machine, Alan Turing’s thought experiment, is a lovely example of this. Traditionally the tape in the Turing machine is thought of as being dragged across a read-write head on the little
machine itself.
However … if you were really to build one … the tape would get harder and harder to move as you used longer and longer tapes. In fact it makes much more sense to think of the little machine as moving
over the tape … the Turing machine is really a touring machine (ouch!). Whichever way it goes, the machine that knows what to do and the tape that it must do it to are brought physically together^4.
This is also of crucial importance in real computers and one of the major limits on fast computers is the length of the copper tracks on circuit boards – the data must come to the processor, and the
longer the track the longer it takes … 10 cm of PCB is a long distance for an electron in a hurry.
brain as a computer – We talked about the way each age reinvents humanity in terms of its own technology: Pygmalion in stone, clockwork figures, pneumatic theories of the nervous system, steam
robots, electricity in Shelley’s Frankenstein and now seeing all life through the lens of computation.
This withstanding … I did sort of mention the weird fact (or is it a factoid) that the human brain has similar memory capacity to the web^5 … this is always a good point to start discussion 😉
While on the topic I did just sort of mention the socio-organisational Church-Turing hyphothesis … but that is another story
more … I recall counting the number of pairs of people and the number of seat orderings to see quadratic (n squared) and exponential effects, the importance of interpretation, why computers are more
than and less than numbers, the Java Virtual Machine, and more, more, more, … it was very full hour
Single-track minds – centralised thinking and the evidence of bad models
Another post related to Clark’s “Being there” (see previous post on this). The central thesis of Clark’s book is that we should look at people as reactive creatures acting in the environment, not as
disembodied minds acting on it. I agree wholeheartedly with this non-dualist view of mind/body, but every so often Clark’s enthusiasm leads a little too far – but then this forces reflection on just
what is too far.
In this case the issue is the distributed nature of cognition within the brain and the inadequacy of central executive models. In support of this, Clark (p.39) cites Mitchel Resnick at length and
I’ll reproduce the quote:
“people tend to look for the cause, the reason, the driving force, the deciding factor. When people observe patterns and structures in the world (for example, the flocking patterns of birds or
foraging patterns of ants), they often assume centralized causes where none exist. And when people try to create patterns or structure in the world (for example, new organizations or new
machines), they often impose centralized control where none is needed.” (Resnick 1994, p.124)^1
The take home message is that we tend to think in terms of centralised causes, but the world is not like that. Therefore:
(i) the way we normally think is wrong
(ii) in particular we should expect non-centralised understanding of cognition
However, if our normal ways of thinking are so bad, why is it that we have survived as a species so long? The very fact that we have this tendency to think and design in terms of centralised causes,
even when it is a poor model of the world, suggests some advantage to this way of thinking.
1. Mitchel Resnik (1994). Turtles Termites and Traffic Jams: Explorations in Massively Parallel Microworlds. MIT Press.[back]
multiple representations – many chairs in the mind
I have just started reading Andy Clark’s “Being There”^1 (maybe more on that later), but early on he reflects on the MIT COG project, which is a human-like robot torso with decentralised computation
– coherent action emerging through interactions not central control.
This reminded me of results of brain scans (sadly, I can’t recall the source), which showed that the areas in the brain where you store concepts like ‘chair’ are different from those where you store
the sound of the word – and also I’m sure the spelling of it also.
This makes sense of the “tip of the tongue” phenomenon, you know that there is a word for something, but can’t find the exact word. Even more remarkable is that of you know words in different
languages you can know this separately for each language.
So, musing on this, there seem to be very good reasons why, even within our own mind, we hold multiple representations for the “same” thing, such as chair, which are connected, but loosely coupled.
1. Andy Clark. Being There. MIT Press. 1997. ISBN 0-262-53156-9. book@MIT[back] | {"url":"https://alandix.com/blog/tag/embodied-cognition/","timestamp":"2024-11-11T00:04:37Z","content_type":"text/html","content_length":"63401","record_id":"<urn:uuid:29e7a5eb-311c-4eda-9c40-cf4045f669bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00784.warc.gz"} |
Is it possible to hire someone for assistance with quantum algorithms for solving problems in quantum pattern recognition and machine perception for my assignment? Computer Science Assignment and Homework Help By CS Experts
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Alexander Grothendieck 1928-2014
I just heard that Alexander Grothendieck passed away today, at the age of 86, in Saint-Girons. For a French news story, see here.
Grothendieck’s story was one of the great romantic stories of modern mathematics, and many would consider him the greatest mathematician of the twentieth century. For some blog entries about him
here, see for example this and this. I’ll add other links as I see them or think of them.
Update: For some blog entries about Grothendieck’s recent life, you could start here.
One of the best places to learn about Grothendieck is from his friend Pierre Cartier, in an article that can be found here, among other places.
Le Monde now has an obituary.
Steve Landsburg has a blog post.
Update: The news about Grothendieck came out in the French press a day ago, but at this point the only things I’ve seen in the English-language press are an AP wire story, and this at the Independent
. Come on science journalists, if any story about mathematics and mathematicians is worth writing about, this one is.
Update: There are now obituaries at the New York Times and the Telegraph. The IHES has a page at their website.
27 Responses to Alexander Grothendieck 1928-2014
1. A great mathematician and a great man. Too sad.
2. Adieu Shourik!
3. This are sad news. He had very unique personality, and was a great master of structure, intuition, poetry, and anarchy. But maybe his death and surrounding publicity can, ironically, stimulate
translation of his works to English? In particular of Recoltes et semailles (only some small part of it was translated). There is an ongoing crowdfunding project of translation of his biography
from German to English: http://www.gofundme.com/7ldiwo. Maybe a translation of Recoltes could be organised in the same way?
4. One of the greatest–if not the greatest–mathematicians of the 20th century has died, yet the English media has not picked up on it.
5. Grothendieck has passed away, a great man and a great human being. I think it is a matter of respect and a matter of honesty to be very careful with statements about him. The internet is full
with wrong, half-true, incomplete, sensational and misleading information about him. Do not believe everyting you read and check everything carefully.
Winfried Scharlau
6. Thanks for this nice post Peter.
Note though, that, as Cartier mentions, Grothendieck insisted on the spelling “Alexander” for his first name. A bit of political rebellion against the oppressively intrusive French State?
7. CIP,
Thanks. Given Cartier’s comment, I did change the spelling.
I was somewhat tempted to leave the French version though, since it’s quite sad to see that so far virtually the only new coverage of his death is coming from the French press, where it’s a big
8. The point is that only a few mathematicians can really connect to his work .
He was the purest form of mathematicians ,the best mathematician in the 20th century no doubt about it .
It would take world leading mathematicians to talk about his maths,for every mathematician it would be the same as some child talking about seashells ,fantasmagoric divination.
The truth is that what distinguishes him is that he answers the Why questions , mathematicians live in a universe which is bounded in the same manner the physical world at some era ,R.Feynman
talked about this impossibility in physics. But in maths it is possible … the why is generally untouchable
His maths reflect the universe .
9. The last forty years of Grothendieck’s life were a long goodbye to mathematics but his “broken dream […] to develop a theory of motives” (to quote P. Cartier in his famous article “A mad day’s
work…”) seems to me a silent hello to physics thanks to the works of Kontsevitch and Connes…
And here is his forecast about a unification theory :
<blockquote cite=
[…] Toujours est-il que de trouver un modèle “satisfaisant” […], que celui-ci soit “continu”, “discret” ou de nature “mixte” — un tel travail mettra en jeu sûrement une grande imagination
conceptuelle, et un art consommé pour appréhender et mettre à jour des structures mathématiques de type nouveau. Ce genre d'imagination ou de “flair” me semble chose rare, non seulement parmi les
physiciens […], mais même parmi les mathématiciens (et là je parle en pleine connaissance de cause). Pour résumer, je prévois que le renouvellement attendu (s’il doit encore venir…) viendra
plutôt d’un mathématicien dans l’âme ; bien informé des grands problèmes de la physique, que d’un physicien . Mais surtout, il y faudra un homme ayant “l’ouverture philosophique” pour saisir le
nœud du problème. Celui-ci n’est nullement de nature technique, mais bien un problème fondamental de “philosophie de la nature
in Récoltes et semailles
(Chapitre 2. Promenade à travers une œuvre ou l’Enfant et la Mère. § 2.20. Coup d’œil chez les voisins d’en face, p. 80 (transcription d’Yves Pocchiola))
10. English translation* of the last Grothendieck’s quote from “Harvests and Seeds”
<blockquote cite=
It nevertheless remains true that the finding of a ‘satisfactory’ model […] – whether this model was ‘continuous’, ‘discrete’ or of a ‘mixed’ nature – would require a great conceptual
imagination, and a consummate art for apprehending and updating mathematical structures of a new type. This kind of imagination or ‘flair’ is rare indeed, not only amongst physicists (Einstein
and Schrödinger seem to be notable exceptions), but even amongst mathematicians (and there I am speaking in full knowledge of the facts).
To sum up, I predict that the long-awaited renewal (if it is still coming…) will come from a born mathematician well-informed about the big questions of physics rather than from a physicist. But
above all, we will need a man with the kind of ‘philosophical openness’ necessary to take hold of the heart of the problem. This problem is by no means a technical one, but is rather a
fundamental question of ‘natural philosophy’.
11. Here is a wonderful biography of Grothendieck by Pierre Cartier:
12. black fire on white fire…salut
13. Winfried Scharlau, I am impressed that you were able to find him some years ago. He must have told you some interesting things; he wasn’t communicating with many people by that time. I hope he
had a pleasant “retirement.”
14. RIP
15. Sad. A truly great, great mathematician. I hope Grothendieck did not burn all his papers as, I believe, he one time threatened to do. What a loss that would be.
17. Alexandre Grothendieck in his own handwriting.
18. French television went to Lassere, the village where Grothendieck spend the last decades of his life:
I’m bit shocked by the complete lack of reaction in the US media.
20. Dear Als
Thanks for putting the address of this video in your comment.
Grothendieck wanted to be alone and I think he recived his gift.
21. Peter, unsolicited advice: why don’t you contribute to cover this yourself in the US media, by writing a piece, for e.g. the Wall Street Journal ?
22. It is not so easy trying to explain modern mathematics to the public. Several years ago I tried to cover the 2002 Field Medals for Salon.com. I was so frustrated I ended up taking a completely
different angle:
“Math = beauty + truth / (really hard),” Salon, September 5, 2002.
Several people have told me they like it, including an editor at Forbes, and one persobn saying it was the best article about mathematics she’d ever read. (But she wasn’t a mathematician.)
23. Grothendieck
Brilliance and bizarrerie —
Inextricably intertwined:
No subject that he plumbed
Took the measure of his mind.
A soul forever questing
For the Great Beyond —
A heart forever testing
The limits of despond.
24. I was surprsied that the BBC did not pick up on this and still hasn’t! The only English press that has finally covered this is the Telegraph on 14th Nov. I myself only heard of Alexander
Grothendiek through one of Peter Woit’s recent blogs (i work in a completely different scientific field). I got intrigued by the fact that a genius can abandon his amazing career. it became clear
to me that this guy was not only a pure mathematecian but also carried his purity of reasoning to other areas of life from scientific funding issues to the environment and the human condition.
ok, maybe some describe his reaction to scientific corruption and social injustices as insanity but actually it’s totally admirable and inspiring to all kinds of scientists.
25. Are there any mathematical notes or manuscripts that can now become available to be examined by experts?
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Rest Frequency and Observing Frequency
Next: Setting the Observing Frequency Up: Spectral Line Observations Previous: Spectral Lines Contents
The rest frequency of a spectral line of interest can be calculated if it is not already tabulated. The apparent frequency (or, the observing frequency), however, needs to be calculated for each
source since it depends on the relative velocity between the source and the observer. The observed frequency (
It is more useful, and common to define velocities w.r.t. the
In principle, the apparent frequency of a spectral line from a source is always changing due to the change in the radial velocity between the source and the observer. In a given observing session
during a day the source can be observed from rise to set. During this period the radial component of the velocity between the source and the earth due to the rotation of the earth can (in an extreme
case) change from -0.465 to +0.465 km s
Next: Setting the Observing Frequency Up: Spectral Line Observations Previous: Spectral Lines Contents NCRA-TIFR | {"url":"https://www.gmrt.ncra.tifr.res.in/doc/WEBLF/LFRA/node112.html","timestamp":"2024-11-13T11:23:24Z","content_type":"text/html","content_length":"8918","record_id":"<urn:uuid:84bbbe08-5cc7-4f64-984f-2dfcca1f6175>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00177.warc.gz"} |
Get a matrix of transitions and their probabilities in
getTransitionProbabilities {BoolNet} R Documentation
Get a matrix of transitions and their probabilities in probabilistic Boolean networks
Retrieves the state transitions and their probabilities in a probabilistic Boolean network. This takes the transition table information calculated by the markovSimulation method.
markovSimulation An object of class MarkovSimulation, as returned by markovSimulation. As the transition table information in this structure is required, markovSimulation must be called with
returnTable set to TRUE.
Returns a data frame with the first n columns describing the values of the genes before the transition, the next n columns describing the values of the genes after the transition, and the last column
containing the probability of the transition. Here, n is the number of genes in the underlying network. Only transitions with non-zero probability are included.
See Also
## Not run:
# load example network
# perform a Markov chain simulation
sim <- markovSimulation(examplePBN)
# print out the probability table
## End(Not run)
version 2.1.9 | {"url":"https://search.r-project.org/CRAN/refmans/BoolNet/html/getTransitionProbabilities.html","timestamp":"2024-11-12T07:15:56Z","content_type":"text/html","content_length":"3326","record_id":"<urn:uuid:9cc2ab33-263c-4834-b23f-7b13b460cc3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00798.warc.gz"} |
Note: to open the Keynote files, you will need to install the
Computer Modern
fonts. I use these fonts so that the main text of the slide matches the font of equations copied from TeX. If you do not install these fonts, the Keynote files will open but will have incorrect fonts
so the layout of the text will be wrong.
Invited Talks
Adversarial Examples and Adversarial Training
Generative Adversarial Networks
Other Subjects
Contributed Talks | {"url":"https://www.iangoodfellow.com/slides/","timestamp":"2024-11-13T06:17:20Z","content_type":"text/html","content_length":"14696","record_id":"<urn:uuid:2c3602ef-8096-4310-b13a-97aed336420e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00620.warc.gz"} |
Library mathcomp.field.algebraics_fundamentals
The main result in this file is the existence theorem that underpins the construction of the algebraic numbers in file algC.v. This theorem simply asserts the existence of an algebraically closed
field with an automorphism of order 2, and dubbed the Fundamental_Theorem_of_Algebraics because it is essentially the Fundamental Theorem of Algebra for algebraic numbers (the more familiar version
for complex numbers can be derived by continuity). Although our proof does indeed construct exactly the algebraics, we choose not to expose this in the statement of our Theorem. In algC.v we
construct the norm and partial order of the "complex field" introduced by the Theorem; as these imply is has characteristic 0, we then get the algebraics as a subfield. To avoid some duplication a
few basic properties of the algebraics, such as the existence of minimal polynomials, that are required by the proof of the Theorem, are also proved here. The main theorem of closed_field supplies us
directly with an algebraic closure of the rationals (as the rationals are a countable field), so all we really need to construct is a conjugation automorphism that exchanges the two roots (i and -i)
of X^2 + 1, and fixes a (real) subfield of index 2. This does not require actually constructing this field: the kHomExtend construction from galois.v supplies us with an automorphism conj_n of the
number field Q[z_n] = Q[x_n, i] for any x_n such that Q[x_n] does not contain i (e.g., such that Q[x_n] is real). As conj_n will extend conj_m when Q[x_n] contains x_m, it therefore suffices to
construct a sequence x_n such that (1) For each n, Q[x_n] is a REAL field containing Q[x_m] for all m <= n. (2) Each z in C belongs to Q[z_n] = Q[x_n, i] for large enough n. This, of course, amounts
to proving the Fundamental Theorem of Algebra. Indeed, we use a constructive variant of Artin's algebraic proof of that Theorem to replace (2) by (3) Each monic polynomial over Q[x_m] whose constant
term is -c^2 for some c in Q[x_m] has a root in Q[x_n] for large enough n. We then ensure (3) by setting Q[x_n+1] = Q[x_n, y] where y is the root of of such a polynomial p found by dichotomy in some
interval [0, b] with b suitably large (such that p[b] >= 0), and p is obtained by decoding n into a triple (m, p, c) that satisfies the conditions of (3) (taking x_n+1=x_n if this is not the case),
thereby ensuring that all such triples are ultimately considered. In more detail, the 600-line proof consists in six (uneven) parts: (A) - Construction of number fields (~ 100 lines): in order to
make use of the theory developped in falgebra, fieldext, separable and galois we construct a separate fielExtType Q z for the number field Q[z], with z in C, the closure of rat supplied by
countable_algebraic_closure. The morphism (ofQ z) maps Q z to C, and the Primitive Element Theorem lets us define a predicate sQ z characterizing the image of (ofQ z), as well as a partial inverse
(inQ z) to (ofQ z). (B) - Construction of the real extension Q[x, y] (~ 230 lines): here y has to be a root of a polynomial p over Q[x] satisfying the conditions of (3), and Q[x] should be real and
archimedean, which we represent by a morphism from Q x to some archimedean field R, as the ssrnum and fieldext structures are not compatible. The construction starts by weakening the condition p[0] =
-c^2 to p[0] <= 0 (in R), then reducing to the case where p is the minimal polynomial over Q[x] of some y (in some Q[w] that contains x and all roots of p). Then we only need to construct a
realFieldType structure for Q[t] = Q[x,y] (we don't even need to show it is consistent with that of R). This amounts to fixing the sign of all z != 0 in Q[t], consistently with arithmetic in Q[t].
Now any such z is equal to q[y] for some q in Q[x] [X] coprime with p. Then up + vq = 1 for Bezout coefficients u and v. As p is monic, there is some b0 >= 0 in R such that p changes sign in ab0 =
[0; b0]. As R is archimedean, some iteration of the binary search for a root of p in ab0 will yield an interval ab_n such that |up[d]| < 1/2 for d in ab_n. Then |q[d]| > 1/2M > 0 for any upper bound
M on |v[X]| in ab0, so q cannot change sign in ab_n (as then root-finding in ab_n would yield a d with |Mq[d]| < 1/2), so we can fix the sign of z to that of q in ab_n. (C) - Construction of the x_n
and z_n (~50 lines): x n is obtained by iterating (B), starting with x_0 = 0, and then (A) and the PET yield z n. We establish (1) and (3), and that the minimal polynomial of the preimage i n of i
over the preimage R n of Q[x_n] is X^2 + 1. (D) - Establish (2), i.e., prove the FTA (~180 lines). We must depart from Artin's proof because deciding membership in the union of the Q[x_n] requires
the FTA, i.e., we cannot (yet) construct a maximal real subfield of C. We work around this issue by first reducing to the case where Q[z] is Galois over Q and contains i, then using induction over
the degree of z over Q[z n] (i.e., the degree of a monic polynomial over Q[z_n] that has z as a root). We can assume that z is not in Q[z_n]; then it suffices to find some y in Q[z_n, z] \ Q[z_n]
that is also in Q[z_m] for some m > n, as then we can apply induction with the minimal polynomial of z over Q[z_n, y]. In any Galois extension Q[t] of Q that contains both z and z_n, Q[x_n, z] = Q
[z_n, z] is Galois over both Q[x_n] and Q[z_n]. If Gal(Q[x_n,z] / Q[x_n]) isn't a 2-group take one of its Sylow 2-groups P; the minimal polynomial p of any generator of the fixed field F of P over Q
[x_n] has odd degree, hence by (3) - p[X]p[-X] and thus p has a root y in some Q[x_m], hence in Q[z_m]. As F is normal, y is in F, with minimal polynomial p, and y is not in Q[z_n] = Q[x_n, i] since
p has odd degree. Otherwise, Gal(Q[z_n,z] / Q[z_n]) is a proper 2-group, and has a maximal subgroup P of index 2. The fixed field F of P has a generator w over Q[z_n] with w^2 in Q[z_n] \ Q[x_n],
i.e. w^2 = u + 2iv with v != 0. From (3) X^4 - uX^2 - v^2 has a root x in some Q[x_m]; then x != 0 as v != 0, hence w^2 = y^2 for y = x + iv/x in Q[z_m], and y generates F. (E) - Construct conj and
conclude (~40 lines): conj z is defined as conj n z with the n provided by (2); since each conj m is a morphism of order 2 and conj z = conj m z for any m >= n, it follows that conj is also a
morphism of order 2. Note that (C), (D) and (E) only depend on Q[x_n] not containing i; the order structure is not used (hence we need not prove that the ordering of Q[x_m] is consistent with that of
Q[x_n] for m >= n). | {"url":"https://math-comp.github.io/htmldoc_1_12_0/mathcomp.field.algebraics_fundamentals.html","timestamp":"2024-11-08T20:16:26Z","content_type":"application/xhtml+xml","content_length":"30754","record_id":"<urn:uuid:e6dfb88c-94da-4b63-8868-2018d2d80156>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00043.warc.gz"} |
Algebra and Trigonometry (4th edition) – eTextBook PDF - eTextPdf
Algebra and Trigonometry (4th edition) – MindTap Course List – eTextBook PDF
In this ebook titled “Algebra and Trigonometry 4th edition“, this bestselling authors team (James Stewart, Lothar Redlin, Saleem Watson) explains concepts clearly and simply, without glossing over
difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing college students with a solid foundation in the principles of mathematical
thinking. Comprehensive and evenly paced, the etextbook provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help college
students develop insight into math ideas. The writers’ attention to detail and clarity — the same as found in any of James Stewart’s market-leading Calculus ebooks — is what makes this ebook the
market leader.
Book Features
• eTextbook (PDF) only
• Does NOT contain online access code
• Comprehensive and evenly paced
• Integrates a significant amount of graphing calculator material to help college students develop insight into mathematical ideas
1 review for Algebra and Trigonometry (4th edition) – eTextBook PDF
1. Very pleased with the quick service.
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You must be logged in to post a review. | {"url":"https://etextpdf.com/product/algebra-and-trigonometry-4th-edition-etextbook-pdf/","timestamp":"2024-11-12T06:48:06Z","content_type":"text/html","content_length":"96749","record_id":"<urn:uuid:8884e2be-2163-4626-b66a-8ae51a78c2d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00650.warc.gz"} |
Square Meters and Square Inches Converter - CalculatorBox
Square Meters and Square Inches Converter
Figures rounded to max decimal places.
Using the Square Meters and Square Inches Converter
This converter can help you when converting between square meters, a metric unit of area, and square inches, an imperial unit of area.
Start off by choosing between the American and British spelling of the word meter (which is spelled ‘metre’ if you select the British spelling).
You can choose your input unit in the ‘CONVERT FROM’ section. The choice is between square meters (m^2) and square inches (in^2).
In the ‘CONVERT TO’ section, you choose the output unit out of the same 2 options.
Another way to choose the input and output units is to simply stick with the default selection or swap the order by clicking on the icon with the 2 arrows headed in opposite directions.
The next step requires you to type in the value of your input unit as a decimal number, using the decimal dot. Write the value into the ‘VALUE TO CONVERT’ section.
Choose the number of decimal places you want your result rounded toward and click on ‘CONVERT’.
Your result will come out as a decimal number rounded toward the requested number of decimal places.
Additionally, you will also receive a conversion rate between your input and output units, as well as a convenient ‘COPY’ icon, that allows you to copy and paste the result elsewhere.
Converting Square Meters and Square Inches Manually
A key to establishing formulae for the manual conversion of these two units lies in their conversion rate.
The conversion rate is determined based on how many units of square meters are in 1 square inch, and vice versa.
Area Equivalent Area
1 square inch 0.00065 square meters.
1 square meter 1,550 square inches.
Keep in mind that both of these values are rounded for convenience. If you require more accurate results, use our converter.
The 2 formulae we can derive from these equivalent values are as follows.
The choice of the best formula is determined by the input and output of each problem you are trying to solve. For problems where the square inch is your input unit and the square meter is your
output, the first formula would work best, as the output is also the subject of the formula.
Similarly, when trying to convert square meters into inches, we would use the second formula.
The following examples will demonstrate the usage of the formulae in practice.
EXAMPLE 1: A napkin has an area of 25 in^2. What is the area of this napkin in m^2?
We have an input in square inches and an output in square meters. This calls for using the first formula, where we substitute 25 for square inches and count as follows.
m^2 = in^2 * 0.00065 \\= 25 *0.00065 \\=0.01625 ~m^2.
EXAMPLE 2: What is the area expressed in square inches of a sail that has an area of 2.33 m^2?
This problem will utilize the second formula, as it has an input in square meters and an output in square inches. We substitute 2.33 for square meters and calculate as follows.
in^2 = m^2 * 1,550 \\= 2.33 * 1,550 \\= 3,611.5 ~in^2.
Unexpected Areas
There are things around us that sometimes seem larger or smaller than they really are. Below is a list of 5 items whose areas will probably shock you, expressed in either square meters or square
inches. You can use our converters to convert them to other units.
Item Area in in^2
Skin for the average human ~2,900 in^2
14-inch pizza 154 in^2
Large towel 4 m^2
Monopoly board 400 in^2
A standard US school desk 900 in^2 | {"url":"https://calculatorbox.com/calculator/square-meters-and-square-inches-converter/","timestamp":"2024-11-10T21:04:48Z","content_type":"text/html","content_length":"148697","record_id":"<urn:uuid:5fe1fa48-293d-40f9-a664-bfa43e03e1ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00467.warc.gz"} |
A monadic reasoning around function composition in Golang
Function composition is something we as developers do every day, more or less. This concept come from Mathematics: if you search on Wikipedia, you find out that function composition is an operation
that takes two functions \(f\) and \(g\) and produces a function \(h\) such that \(h(x) = g(f(x))\). In this operation, the function \(g\) is applied to the result of applying the function \(f\) to
the input \(x\). That is, the functions \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) are composed to yield a function that maps \(x\) in \(X\) to \(g(f(x))\) in \(Z\).
There’s an entire way to develop software using this concept of composition: in this post, I will discuss how we can leverage this concept in Golang, trying to provide some pros and cons as well
about this approach.
Let’s start by saying that function composition patterns exist for a while: an ex-colleague of mine (thank you martiano) recently bet me on implementing the Monad in Golang, and that’s the reason I
recently worked on this. So, before going into details about function composition and how this is used by, let’s start by giving a definition of Monad.
A monad is just a monoid in the category of endofunctors
Ok…let’s introduce some concept required to understand this sentence.
In abstract algebra, a monoid is an algebraic structure with a single associative binary operation and an identity element. Suppose that \(S\) is a set and \(\cdot\) is some binary operation \(S \
times S \rightarrow S\), then \(S\) with \(\cdot\) is a monoid if it satisfies the following two axioms:
• Associativity: for all \(a, b\) and \(c\) in \(S\), the equation \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) holds.
• Identity element: there exists an element \(e\) in \(S\) such that for every element \(a\) in \(S\), the equations \(e \cdot a = a \cdot e = a\) hold.
You should have already realized that many of the things you’ve seen in your past respect these two axioms. Keep this concept in mind for a while. Let’s talk go ahead with the concept of
endofunctors, that means…let’s talk about functors first.
A functor is a very simple but powerful idea coming from category theory. Let’s give a simple definition:
A functor is a mapping between categories
Thus, given two categories, \(C\) and \(D\), a functor \(F\) maps objects in \(C\) to objects in \(D\) — it’s a function on objects. If \(a\) is an object in \(C\), we’ll write its image in \(D\) as
\(F;a\) (no parentheses).
But a category is not just objects — it’s objects and morphisms^1 that connect them. A functor also maps morphisms — it’s a function of morphisms. But it doesn’t map morphisms willy-nilly — it
preserves connections. So if a morphism \(f\) in \(C\) connects object \(a\) to object \(b\), like this
$$ f \colon a \to b $$
the image of \(f\) in \(D\), \(F f\), will connect the image of \(a\) to the image of \(b\):
$$ F;f \colon F;a \to F;b $$
So a functor preserves the structure of a category: what’s connected in one category will be connected in the other category. At this point, let’s just define an endofunctor:
An endofunctor is a functor that maps a category to that same category
Let’s make no other assumptions on endofunctors: just keep in mind they are functor acting on a single category.
Once more thing
There’s something more to the say. The first is about the composition of morphisms. If \(h\) is a composition of \(f\) and \(g\):
$$ h \colon g \cdot f $$
we want its image under \(F\) to be a composition of the images of \(f\) and \(g\):
$$ F;h \colon F;g \to F;f $$
Second, we want all identity morphisms in \(C\) to be mapped to identity morphisms in \(D\):
$$ F id_a = id_{Fa} $$
Here, \(id_a\) is the identity at the object \(a\), and \(id_{F;a}\) a the identity at \(F;a\).
That is, functors must preserve composition of morphisms and identity morphisms. And now we have all the instrument to better understand the real definition of a Monad.
Back to monad
I actually gave the shorter version of the definition of a Monad above, to keep things simple as much as possible. We said that a monad is a monoid in the category of endofunctors. Since we
introduced some more concepts, all told:
A monad in \(X\) is a monoid in the category of endofunctors of \(X\), with product \(\times\) replaced by a composition of endofunctors and unit set by the identity endofunctor
Ok, at this point in time you should start feeling a bit more confident about the terms and the idea behind monad than before. While hoping you’re feeling this, I just wanna present the problem from
a programming perspective since, in the end, we are replacing the same concepts of identity and composition, with two standard operations and nothing more.
A monad is composed by a type constructor (\(M\)) and two operations, unit and bind
The unit operation (sometimes also called return) receives a value of type \(a\) and wraps it into a monadic value of type \(m;a\), using the type constructor. The bind operation receives a function
\(f\) over type \(a\) and can transform monadic values \(m;a\) applying \(f\) to the unwrapped value \(a\).
We can also say the Monad pattern is a design pattern for types, and a monad is a type that uses that pattern. With these elements, we can compose a sequence of function calls (a “pipeline”) with
several bind operators chained together in an expression. Each function call transforms its input plain type value, and the bind operator handles the returned monadic value, which is fed into the
next step in the sequence.
Coding a Monad
Let’s assume we have three functions \(f1\), \(f2\) and \(f3\), each of which returns an increment of its integer parameter. Additionally, each of them generates a readable log message, representing
the respective arithmetic operation. Like the ones shown below.
func f1(x int) (int, string) {
return x + 1, fmt.Sprintf("%d+1", x)
func f2(x int) (int, string) {
return x + 2, fmt.Sprintf("%d+2", x)
func f3(x int) (int, string) {
return x + 3, fmt.Sprintf("%d+3", x)
Imagine now that we would like to chain the three functions \(f1\), \(f2\) and \(f3\) given a parameter \(x\) - i.e. we want to compute \(x+1+2+3\). Additionally, we want to have a readable
description of all applied functions. We would do something like this.
x := 0
log := "Ops: "
res, log1 := f1(x)
log += log1 + "; "
res, log2 := f2(res)
log += log2 + "; "
res, log3 := f3(res)
log += log3 + "; "
fmt.Printf("Res: %d, %s\n", res, log)
Nice. Buuuut… this solution sounds a lot like 80s code, isn’t it?
What’s the problem?
The problem is that we have repeated the so-called glue code, which accumulates the overall result and prepares the input of the functions. If we add a new function \(f4\) to the sequence, we have to
repeat this glue code again. Moreover, manipulating the state of the variables res and log makes the code less readable, and is not essential to the program logic.
Ideally, we would like to have something as simple as the chain invocation \(f3(f2(f1(x)))\). Unfortunately, the return types of \(f1\) and \(f2\) are incompatible with the input parameter types of \
(f2\) and \(f3\).
A Monad approach
To solve the problem we introduce two new functions:
func unit(x int) *Log {
return &Log{Val: x, Op: ""}
func bind(x *Log, f Increment) *Log {
val, op := f(x.Val)
return &Log{Val: val, Op: fmt.Sprintf("%s%s; ", x.Op, op)}
This is kind of similar to the functions we described before - yes… you might think “the names, maybe”. Actually, not only: lets first introduce the other new elements required to let these two
functions compile correctly.
type Increment func(y int) (int, string)
type Log struct {
Val int
Op string
We first introduced a new function type. The syntax is pretty simple:
type MyFunctionTypeName func(<params>) <returns>
This is a useful and common way to define a function as a type and identify a pattern without rewriting every time the entire signature defining the function inputs and parameters. We also introduced
a new structure, the Log that actually contains nothing more than two fields, Val and Op: this is a wrap around the results returned by our initial functions, and this is not a coincidence. Let’s
focus on the two new functions, unit and bind.
func unit(x int) *Log {
return &Log{Val: x, Op: ""}
The unit function returns a new *Log (pointer of a Log instance) for a given x value. This is the step zero needed in the composition step we are gonna implement soon, thanks to the bind function.
func bind(x *Log, f Increment) *Log {
val, op := f(x.Val)
return &Log{Val: val, Op: fmt.Sprintf("%s%s; ", x.Op, op)}
The bind function returns a *Log (pointer of a Log instance) for a given x value and function \(f\) of type Increment. That means, a function that has the Increment type signature. The bind operation
applies the provided function to the given input, of type *Log, and return a *Log instance with the computed result, and… see? What we called glue-code before, i.e. the operations of logs concat
// no more required and not repeated
log += log2 + "; "
log += log3 + "; "
is no more required and not anymore repeated. Hence, we can solve the problem with a single chained function invocation:
x := 0
fmt.Printf("%s\n", bind(bind(bind(unit(x), f1), f2), f3).ToString())
where ToString() is just a util method to pretty print the *Log object instance:
func (l *Log) ToString() string {
return fmt.Sprintf("Res: %d, Ops: %s", l.Val, l.Op)
Let’s summarize the bind chain: we actually created a common value (the first *Log instance created with the unit(x) call) starting from our primitive value x. Then, we started to delegate the each
bind to use the result of the internal bind call as input (back from unit and up to the most external bind), literally chaining the three functions \(f1\), \(f2\) and \(f3\).
Doing like this avoids the shortcomings of our previous approach because the bind function implements all the glue-code and we don’t have to repeat it. We can add a new function \(f4\) by just
including it in the sequence as bind(f4, bind(f3, ... )) and we won’t have to do other changes.
Generalize the idea behind
Let’s assume we want to generically compose the functions \(f1\), \(f2\), … \(fn\). If all input parameters match all return types, we can simply use \(fn(… f2(f1(x)) …)\). However,
often this approach is not applicable. For instance, in the *Log example, the types of the input parameters and the returned values are different. In the example we wanted to “inject” additional
logic between the function invocations and wanted to aggregate the interim values.
Before calling \(f1\), we execute some initialization code. We initialized variables to store the aggregated log and interim values. After that, we call the functions \(f1\), \(f2\), … \(fn\)
and between the invocations we put some glue code - in the example we aggregate the log and the interim values.
*Log as MonadicType
In order to compose the bind and unit functions, the return types of unit and bind, and the type of bind’s first parameter must be compatible. This is called a Monadic Type.
Last but not least, since repeating the calls to bind again and again can be tedious, we can even define a pipeline support method like the one below:
func pipeline(x *Log, fs ...Increment) *Log {
for _, f := range fs {
x = bind(x, f)
return x
and finally, compose function with a really functional style like this:
x := 0
res := pipeline(unit(x), f1, f2, f3)
fmt.Printf("%s\n", res.ToString())
I truly hope this opened a bit your mind around how you can leverage function passed as parameters in Golang, and use this approach to implementation function composition as well your code!
If you like this post, please upvote it on HackerNews here.
You can find the entire gist here.
1. A category \(C\) consists of two classes, one of objects and the other of morphisms. There are two objects that are associated to every morphism, the source and the target. A morphism \(f\) with
source \(X\) and target \(Y\) is written \(f\colon X \to Y\), and is represented by an arrow from \(X\) to \(Y\). ↩︎ | {"url":"https://madeddu.xyz/posts/reposted/go-function-composition/","timestamp":"2024-11-05T04:22:21Z","content_type":"text/html","content_length":"67963","record_id":"<urn:uuid:82dbc01b-4ff8-44cb-ac9d-63d5d62902cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00242.warc.gz"} |
Modeling Infinite Ground Plane in Antennas and Arrays
This example illustrates modeling of antennas and arrays with infinite ground plane. The main advantage of modeling the ground plane as infinite is that the ground plane is not meshed. This helps in
speeding up the solution. The structure is modelled using the method of images. Several antenna elements in Antenna Toolbox™ have a ground plane as part of the structure. For other elements, the
ground plane can be introduced by placing them in front of a reflector.
Dipole Over Finite Ground Plane
The default reflector has a dipole over a ground plane operating around 1GHz.
Looking at the radiation pattern, we see that there is some leakage below the ground. This is because of the size of the ground plane.
To prevent the leakage the ground plane must be made larger. However, by increasing the size of the ground plane the size of the mesh increases. The increase in mesh size increases the simulation
time. Below is the mesh created for the structure above.
Increasing the ground plane size will put more triangles on the ground which will result in more simulation time.
Dipole Over Infinite Ground Plane
A simple way to prevent leakage below the ground plane is to make the ground plane infinite. This can be achieved by making either the GroundPlaneLength or GroundPlaneWidth or both to be infinite. In
this case the ground plane is replaced by a blue sheet indicating that the ground plane is not made of metal.
r.GroundPlaneLength= inf;
The pattern plot shows no leakage below the ground. This result can be used as a first pass to get a general idea of the antenna. The infinite ground plane can be replaced by a large finite one at
the end to look for edge effects. Another interesting factor is the increase in maximum directivity value. As there no back lobe, all the energy is radiated above the ground plane, increasing the
maximum directivity from 7.38 to 7.5dBi.
As mentioned before, the infinite ground plane is not meshed. The mesh for this structure shows only the mesh for the dipole element.
The impedance of the reflector with infinite ground plane looks similar to the one with the finite ground plane. The resonance value is shifted slightly from 880MHz for finite ground to 890MHz for
the infinite case.
impedance(r, 850e6:2e6:950e6);
Patch Antenna Array Over Infinite Ground
The concept of infinite ground becomes even more important for arrays. As the number of elements increase the size of the ground plane increases dramatically as the space on the ground between the
elements also needs to be meshed. So choosing infinite ground plane for arrays is fairly common. Infinite ground planes in arrays are also called ground screens.
p = patchMicrostrip('GroundPlaneWidth', inf);
arr = rectangularArray('Element', p);
arr.RowSpacing = 0.075;
arr.ColumnSpacing = 0.1;
Above is the four element rectangular patch array on an infinite ground plane. The individual patches resonate around 1.75 GHz. Below is the pattern of the full array.
Again the plot shows that there is no radiation below the ground. The mesh that is used to solve the array is shown. As mentioned previously, the ground is not meshed so the overall size of the
system is reduced resulting in faster computation time.
[1] C. A. Balanis, Antenna Theory. Analysis and Design, Wiley, New York, 3rd Edition, 2005.
See Also
Metasurface Antenna Modeling | Model Infinite Ground Plane for Balanced Antennas | {"url":"https://es.mathworks.com/help/antenna/ug/modeling-infinite-ground-plane-in-antennas-and-arrays.html","timestamp":"2024-11-05T13:22:40Z","content_type":"text/html","content_length":"77113","record_id":"<urn:uuid:031f9579-2ed2-4dac-9041-9d792ef4e3d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00283.warc.gz"} |
Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd
Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd
We consider the solution of second order elliptic PDEs in Rd with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the Poisson
equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H1 / 2–stable projection, for instance, the L2–projection for p = 1 or
the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each H1 / 2–stable projection
yields convergence of the adaptive algorithm even with quasi–optimal convergence rate. Numerical experiments with the Scott–Zhang projection conclude the work.
Aurada, M., et al. "Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd." ESAIM: Mathematical Modelling and Numerical Analysis
- Modélisation Mathématique et Analyse Numérique 47.4 (2013): 1207-1235. <http://eudml.org/doc/273190>.
abstract = {We consider the solution of second order elliptic PDEs in Rd with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the
Poisson equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H1 / 2–stable projection, for instance, the L2–projection for p
= 1 or the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each H1 / 2–stable
projection yields convergence of the adaptive algorithm even with quasi–optimal convergence rate. Numerical experiments with the Scott–Zhang projection conclude the work.},
author = {Aurada, M., Feischl, M., Kemetmüller, J., Page, M., Praetorius, D.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {adaptive finite element method; convergence analysis; quasi–optimality; inhomogeneous Dirichlet data; convergence; quasi-optimality; stability; second-order elliptic equations; Poisson
equation; mixed Dirichlet-Neumann boundary conditions; Scott-Zhang projection; error estimation; numerical experiments},
language = {eng},
number = {4},
pages = {1207-1235},
publisher = {EDP-Sciences},
title = {Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd},
url = {http://eudml.org/doc/273190},
volume = {47},
year = {2013},
TY - JOUR
AU - Aurada, M.
AU - Feischl, M.
AU - Kemetmüller, J.
AU - Page, M.
AU - Praetorius, D.
TI - Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 4
SP - 1207
EP - 1235
AB - We consider the solution of second order elliptic PDEs in Rd with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the
Poisson equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H1 / 2–stable projection, for instance, the L2–projection for p
= 1 or the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each H1 / 2–stable
projection yields convergence of the adaptive algorithm even with quasi–optimal convergence rate. Numerical experiments with the Scott–Zhang projection conclude the work.
LA - eng
KW - adaptive finite element method; convergence analysis; quasi–optimality; inhomogeneous Dirichlet data; convergence; quasi-optimality; stability; second-order elliptic equations; Poisson equation;
mixed Dirichlet-Neumann boundary conditions; Scott-Zhang projection; error estimation; numerical experiments
UR - http://eudml.org/doc/273190
ER -
1. [1] M. Aurada, M. Feischl, J. Kemetmüller, M. Page and D. Praetorius, Each H1 / 2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd
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What is the solution to the inequality 2 <2(x+4)<18? | HIX Tutor
What is the solution to the inequality #2 <2(x+4)<18#?
Answer 1
Given #color(white)("XXXX")2 < 2(x+4) < 18# #rArrcolor(white)("XXXXXXXXXXXXXXX") 2 < 2x+8 <18#
Things you can do with expressions in an inequality which maintain the inequality:
#2 < 2(x+4) < 18color(white)("XXX")rArrcolor(white)("XXX") 2 < 2x+8 <18#
Given the above rules, we can subtract #8# from each expression, to get: #color(white)("XXXX")-6 < 2x < 10#
Then we can divide each expression by #2#, to get: #color(white)("XXXX")-3 < x < 5#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Lesson 15
Infinite Decimal Expansions
Lesson Narrative
In this lesson, students further explore finding decimal expansions of rational numbers as well as irrational numbers. In the warm-up, students find the decimal expansion of \(\frac37\), which starts
to repeat as late as the seventh decimal place. However, once the first repeating digit shows up, repeated reasoning allows the students to stop the long-division process (MP8). The discussion of the
warm-up is a good place to introduce students to the overline notation for repeating decimal expansions.
In the first classroom activity, students learn how to take a repeating decimal expansion and rewrite it in fraction form. The activity uses cards with the steps and explanations of the process and
asks students to put these cards in order. Once they have the correct order, they use the same steps on different decimal expansions. While the numbers are different, the structure of the method is
the same (MP7).
In the last activity of this lesson, and of this unit, students investigate how to approximate decimal expansions of irrational numbers. In an earlier lesson, students learned that \(\sqrt{2}\)
cannot be written as a fraction and they estimated its location on the number line. Now they use “successive approximation,” a process of zooming in on the number line to find more and more digits of
the decimal expansion of \(\sqrt{2}\). They also use given circumference and diameter values to find more precise approximations of \(\pi\), another irrational number students know. In contrast to
the previous lesson, students see that there is no easy was to keep zooming in on these irrational numbers. They are not predictable like a repeating decimal. Because it is not possible to write out
the complete decimal expansion of an irrational number we use symbols to name them. However, in practice we use approximations that are good enough for a given purpose.
Learning Goals
Teacher Facing
• Compare and contrast (orally) decimal expansions for rational and irrational numbers.
• Coordinate (orally and in writing) repeating decimal expansions and rational numbers that represent the same number.
Student Facing
Let’s think about infinite decimals.
Required Preparation
Prepare enough copies of the Some Numbers are Rational blackline master for each group of 2 to have a set of 6 cards.
Student Facing
• I can write a repeating decimal as a fraction.
• I understand that every number has a decimal expansion.
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