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# How to explain the difference between the fraction a / b and the ratio a : b?

I found it difficult to explain the difference between the fraction a / b and the ratio a : b. This subject is for pupils of grade 5. So is there a real difference between them and how to explain the difference in simple way ?!

• Could you explain the difference to me (Student of Mathematics)? – Jasper Jan 24 '15 at 12:19
• @Jasper If you consider a:b as the number z such that bz=a, then writing it in fraction form gives what you want. I consider a:b as doing fractions without using fractions. This way, it actually generalizes quite nicely in commutative algebra. – Chris C Jan 24 '15 at 13:54
• But is the ratio a:b is a number ? – Abdallah Abusharekh Jan 24 '15 at 18:24
• Suppose there really is a difference between these two alleged concepts. Does having a clear distinction between these concepts make it easier to solve any real-world problems? Especially if you are careful to use units? – Jasper Jan 25 '15 at 21:20
• @AbdallahAbushrekh It can be. I found the wikipedia page to be quite useful: en.wikipedia.org/wiki/Ratio – Chris C Jan 25 '15 at 21:28

This paragraph from Adding It Up is a good overview of the context of this question. Rational numbers are complex because there are multiple interpretations (meanings) as well as multiple forms for expressing a rational number.

As we said at the beginning of the chapter, rational numbers can be expressed in various forms (e.g., common fractions, decimal fractions, percents), and each form has many common uses in daily life (e.g., a part of a region, a part of a set, a quotient, a rate, a ratio). One way of describing this complexity is to observe that, from the student’s point of view, a rational number is not a single entity but has multiple personalities. The scheme that has guided research on rational number over the past two decades identifies the following interpretations for any rational number, say 3/4: (a) a part-whole relation (3 out of 4 equal-sized shares); (b) a quotient (3 divided by 4); (c) a measure (3/4 of the way from the beginning of the unit to the end); (d) a ratio (3 red cars for every 4 green cars); and (e) an operation that enlarges or reduces the size of something (3/4 of 12). The task for students is to recognize these distinctions and, at the same time, to construct relations among them that generate a coherent concept of rational number. Clearly, this process is lengthy and multifaceted.

I think it is important for students to understand the connection as well as the distinction between the fraction and the ratio. They can both be written as 3/4. The point to make to students is that the expression 3/4 can be interpreted in different ways. It has different meanings that are mathematically equivalent. So the students should realize that the ratio 3:4 has the same meaning as the ratio 3/4. A second point to make is that there are different forms for writing rational numbers. Using the colon and the bar are just different ways of writing the ratio. As a ratio it is a correspondence of 3 things to 4 things. The students should also realize that the expression 3/4 can be interpreted as a fraction, 3 out of 4 equal portions of some whole. The fraction is a part-to-whole ratio. There are also part-to-part ratios, like one blue dot to 2 yellow dots. Since there are three dots in total, I can also compare the blue dots to the total, giving 1 to 3 as a ratio, or as a fraction 1/3.

Give lots of examples to make it concrete. Answer their questions to clarify the connection and the difference. Then ask them questions and let them explain their thinking. For example, “There are 12 boys and 14 girls in this class. What fractions and ratios can you make from these numbers?”

• I don't understand this answer, so it's hard for me to imagine how children could understand this approach. – Ben Crowell Apr 30 '16 at 3:09
• This is not correct. A 3 to 4 ratio, as in cooking, means for every 7 cups of stuff used, 4 are of one type, and 3, the other. Each ingredient is then 4/7 or 3/7 of the total. The G/B ratio is 12/14 or 6/7 but there are 12/26 or 6/13 students are girls. The example doesn't make this clear. – JTP - Apologise to Monica May 7 '16 at 10:20
• @JoeTaxpayer "3 to 4 ratio" is not a ratio, it is a proportionality. 4/7 and 3/7 are ratios. – Rusty Core Aug 30 '19 at 18:54

I find this diagram helpful when relating the two:

It comes from a model curriculum unit on Rates and Ratios for 6th graders (you can see them all here after registering), and I have found this particular graphic very helpful with math content professional development with 6th grade math teachers.

• thank you Tamisha for the diagrame .. you have just supported my undrestandind for the whole matter – Abdallah Abusharekh Feb 1 '15 at 18:06
• Glad I could assist. :) – Tamisha Thompson Feb 2 '15 at 0:18

I always get students to colour in dots. 1 red and 2 blue. Here the ratio is 1:2 red to blue. I ask, "What fraction are red?" Hopefully someone says $\frac{1}{2}$ and we can discuss the misconception. In terms of how to relate $\frac{1}{2}$ to the ratio 1:2 I do it after establishing the equivalence of ratio by scaling up both sides. I want to make green from blue and yellow in the ratio 1:2. I have 2 tins of blue how many yellow? Students often say that's easy just double the tins of yellow because you have doubled the blue. Now I ask them if I had 1000 yellow or something like they say 500 but this time because they recognise there are $\frac{1}{2}$ the number of yellow. A subtle change but one that recognise the proportion of blue and yellow is the same for equivalent ratios. Hope this helps.

• Same thing comes up with odds and probability. If the odds are 1 to 2 against you, then the probability of winning is 1/3. – Gerald Edgar Jan 24 '15 at 17:12
• Thank you. But if we have 1 red and 2 blue the fraction will equal 1/3 not 1/2 ! – Abdallah Abusharekh Jan 24 '15 at 18:27
• hence the "misconception" in saying it. – Lucas Virgili Jan 24 '15 at 18:34

I want to build a factory and I go to a bank for a loan, to finance part of the investment cost.

Bankers tend to think in $a:b$: "How many euros we will lend for every euro the compnay will invest". And they tend to have rules of thumb on the matter, say a $3:1$ rule. From the point of view of the company, this could be written $1:3$ and here, confusion may arise more easily, because "$1/3$" should be interpreted as "the compnay will invest one third of what the bank will lend" and not as "one third of the total cost of the factory".

To arrive at this last magnitude the relation is always $a:b \rightarrow \frac {a}{a+b}$

More generally, I think a fraction $a/b$ (which then can be written also as a number, a decimal, etc), is meanigfull only when $a$ and $b$ measure same entities in nature (in my example, money in the same currency). But the concept represented usually by $a:b$ can bring together items that are not alike (say, "$a$ car-accident deaths per $b$ kilometers of highways), in which case there is no meaningful interpretation of $a+b$.

The ratio $$a:b$$ and the fraction $$a/b$$ are generally not synonymous and should not be treated as such, at least without making clear their interpretations. In many contexts, $$a:b$$ corresponds to the fraction $$a/(a+b)$$ or $$b/(a+b)$$.

For example, if one cuts a pizza into $$6$$ slices, one of which has anchovies, the fraction of slices with anchovies is $$1/6$$, while the ratio of slices with anchovies to slices without anchovies is $$1:5$$.

One context, unfortunately familiar to many students, where $$a/b$$ and $$a:b$$ are not synonyms is when speaking of odds in the context of bookmaking. That the odds be given by the ratio $$3:2$$ means roughly that the bookmaker expects the bet upon side to win $$2/5$$ times and the payout in the event of a win will be $$3/5$$ of the sum of the payout and money staked. That is when speaking of odds, the ratio $$a:b$$ corresponds to the fraction $$a/(a+b)$$ or $$b/(a+b)$$ (which form is relevant depends on interpretation/use). So $$9:1$$ odds against means that the horse is not expected to win, so the payout is big if it does, $$900$$ would be paid on a $$100$$ bet, while $$9:1$$ odds in favor (usually quoted as $$1:9$$) means that the horse is expected to win, and the payout is small if it does, approximately $$11 \simeq 100/9$$ would be paid on a $$100$$ bet.

• "In many contexts, 𝑎:𝑏 corresponds to the fraction 𝑎/(𝑎+𝑏) or 𝑏/(𝑎+𝑏)" — just don't call it ratio then. "the ratio of slices with anchovies to slices without anchovies is 1:5" — no, slices with anchovies correspond to slices without anchovies in proportion 1 to 5. People just use "ratio" because it is simpler to say than a "proportional relationship". Likewise, people use "rate" instead of "speed", then they wonder why kids don't understand what rate is. – Rusty Core Aug 30 '19 at 18:58

By itself, the word ratio is indeed rather opaque and should probably be avoided. Here is what "is under the hood and usually goes without saying".

Natural (aka counting) numbers are used to measure a "quantity" (= counting number) of discrete objects such as in: the "quantity" of the set {egg, egg, egg} is 3. What do we do when we want to measure a "quantity" (= real number) of continuous stuff such as length?

In terms of proportion, the ancient Greeks would have said that the "quantity" (length) of a footstick is to the "quantity" (length) of a yardstick the same as the "quantity" (count) of the set {egg}, i.e. 1, is to the "quantity" (count) of {egg, egg, egg}, i.e. 3. In terms of ratios, they would have said that the ratio of footstick to yardstick is equal to the ratio of {egg} to {egg, egg, egg}. When we say that the "quantity" of a footstick equals $\frac{1}{3}$ the "quantity" of a yardstick, we are just writing the proportion another way.

In other words, the idea of proportion was meant to reduce the measure of "quantity" of stuff (which they couldn't do) to the measure of "quantity" of sets (which they could do).

From my (by now very remote) experience with 5 graders, I think that the above distinction, measuring sets versus measuring continuous stuff (which, by the way, requires the introduction of units). is quite within their reach.

Warning: what follows may be a bit off-topic but is tightly related to the question.

The trouble comes when we want to look at a fraction as indicative of a measure the same way as when we look at a natural number as indicative of a measure. For instance, when we look at 3084385 and 47975 we immediately see that the first is larger than the second. Not so immediately with fractions.

So, of course, we look at $\frac{1}{3}$ as code for "divide 3 into 1" but the question now is "where to stop the division?" and that of course depends on the real-world situation. That we have rules for dealing with the code, e.g. $\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$ is nice and fortunate but not really necessary: as engineers are wont to put it, "The real real numbers are the decimal numbers".

See Gowers Mathematics, A Very Short Introduction. While spending a whole chapter (7) on infinite decimals, Gowers never even mentions real numbers. And, appropriately recast, the content of Gowers' Chapter 7 should be accessible to 5 grader.

First of all, I would leave out ratios entirely, if possible. They aren't as expressive as fractions are, and fractions are more widely used.

Why are ratios less expressive?

With (binary) ratios, you can compare only two things: $a:b$ (scores in a game, male-to-female distribution in the class, ...), while with fractions, you can have $a/x$, $b/x$, $c/x$, the fraction of several things from a larger group $x$ (color of hair, election results, ...).

Of course you could also do $a:b:c:...$, but I think that's too complicated?

How to explain the difference?

You could emphasize the difference in operations on both: adding ratios is simple, adding fractions is not. Multiplying fractions, or a fraction with a number makes sense, multiplying ratios doesn't. Another key is proper use of language to make clear what is being compared: The ratio of the number of one thing to another number of things, versus the fraction of a number of things of a total number of things.

• I don't think that $a:b:c$ is too complicated. It nicely points out the fact that ratios written this way aren't technically numbers. – DavidButlerUofA Jan 26 '15 at 0:06
• You can't leave out ratios altogether. For one thing, fractions are ratios. Rational numbers are numbers that can be expressed as ratios. The ratio a:b can be written as a/b. – Burt Furuta Jan 26 '15 at 20:10

Cooking examples work great here.

Rice is made with 1 part rice 2 parts water. The ratio is 1:2 rice:water. Or we can say 1/2 the water is the volume of rice to use. Or, as is often the case, you have an odd amount of rice, use twice that amount of water. I once saw my sister fill a 1 cup measure with rice, throw away the rest, and then add 2 cups of water. I asked why she threw out what looked like another 1/2 cup of rice. It wasn't enough for a "recipe". I don't know if she was a bad cook or bad at math, but she could have just measured 1-1/2 cups rice and double to 3 cups water to use.

We use 2:1 for Margarita's as well. 2 parts Mix to 1 part Tequila. Mix to Tequila is 2:1, and the final drink has 1/3 Tequila, 2/3 Margarita Mix.

It's key to understand that (ratio) 2:1 results in 3 parts of stuff, made up of (fraction) 1/3 one ingredient, 2/3 the other.

• I loved your example using cooking examples! Great job! Going to use it in my lesson. – user7088 Aug 28 '16 at 13:41
• 1 fresh squeeze lime juice : 1 orange liqueur (I prefer Cointreau, but Triple Sec is more traditional) : 2 tequila. On the rocks. Mix is teh debil (yes, "teh"; not "the"). :P – Xander Henderson Jul 25 '19 at 21:33

Ratio is just a single fraction. Whereas proportion is a ratio between 2 fractions.

Example - 5 boys and 3 girls in a class. Ratio of boys is 5/8 Ratio of girls is 3/8 But proportion of boys to girls is 5:3

The basic difference is Ratio is always with respect to some large quantities or superior quantities, whereas proportion is between same kind of quantities.

• I have never seen this distinction made. Out of curiosity, can you link to a source that does this? – Joonas Ilmavirta Apr 28 '16 at 14:01
• A proportion is an equation stating that there are two equal ratios 3:5=6:8. If you google "define proportion mathematically", you will find this definition and the definition on many math sites you used. – Amy B Apr 28 '16 at 14:21
• @AmyB "A proportion is an equation stating that there are two equal ratios 3:5=6:8" — Indeed. The problem is, these two particular ratios are not equal ;-) – Rusty Core Jul 23 '19 at 23:39
• @RustyCore It shouldn't have taken 2 years for someone to notice that! I tried to edit but wasn't able to so I will rephrase: A proportion is an equation stating that there are 2 equal ratios 3:5=6:10. If you google "define proportion mathematically" you will find this definition on many math sites. – Amy B Jul 25 '19 at 5:33

Relation between division and fraction have to be understood to understand the relationship between fraction and ratio: 1. Division: finding how many times of divisor in the dividend, or the value of a part 1/5 means 5 is 0.20 times in 1; value of one part is 0.20. 2. Fraction: It is how many times the denominator in numerator or how many parts of denominator in numerator. It means multiplying the value of one part with numerator: 2/5 = 1/5 X 2 = 0.40 3 Ratio: Ratio is a division finding how many times one entity in to other. And it is a fraction for one in numerator. 4. All are written in fraction form with same value

Division and Fractions have applications in daily life where as ratio being an another form of division and fraction does not find place in daily life applications. Therefore ratio is not an independent entity but a name to refer how many times in a division in context. Probably it is good idea not to teach ratio as an independent entity. Therefore ratio is an application of fractions like percentage, rebate, loss and profit. They all use the properties of equivalent fractions. Proportion too is an equivalent fraction.

There are several possible fractions one could associate with a given ratio.

Say that a recipe for lemonade calls for $$2$$ cups of lemon juice and $$5$$ cups of water. This could be expressed with the ratio $$2:5$$. Here are some different associated fractions:

1. $$2/7$$ of the lemonade is lemon juice.
2. $$5/7$$ of the lemonade is water.
3. $$2/5$$ of a cup of lemon juice is the amount which needs to be added to 1 cup of water.
4. $$5/2$$ of a cup of water is the amount which needs to be added to 1 cup of lemon juice.

I don't think any of these fractions has a right to be called "the fraction" associated with the ratio.

Some additional evidence that ratio and fraction are distinct concepts:

The ratios 0:5 and 5:0 both make sense. For instance, you could have 0 parts lemon juice to 5 parts water.

One can have ratios between more than 2 quantities. The ratio 1:2:5, as in 1 part sugar, 2 parts lemon juice, and 5 parts water makes perfect sense. This ratio would be equivalent to 2:4:10 (since the proportions are the same), but there is no single associated fraction.

For a mathematical definition one could use the following:

An ratio of $$n$$ quantities is an equivalence class of elements of $$\mathbb{R}^n-\{\mathbf{0}\}$$ under the equivalence relation $$\mathbf{v} \sim \mathbf{w}$$ iff there exists a $$c \in \mathbb{R}-\{0\}$$ such that $$\mathbf{v} = c \mathbf{w}$$.

In other words, ratios are just elements of real projective spaces!

I'm not sure how you would translate the following into the language of 10-year-olds, but the teacher should first understand this much:

The variables x and y are respectively in the ratio a:b means only that x/y = a/b (for non-zero values of a, b, x and y).

Similarly, the variables x, y and z are respectively in the ratio of a:b:c means only that x/y = a/b and y/z =b/c (for non-zero values of a, b, c, x, y and z).

Above while representing ratio in a flow diagram above one branch is represented by part/part. There is no such concept as part/part in fraction form. The denominator is always a 'whole'. While comparing two items one of them should be taken as whole and other as part of the whole.

The diagram in the present form violates the fundamental principle of fraction.

• I don't understand why fractions cannot be used to describe part/part relations. For example, in the flow chart, 13 boys and 11 girls is represented by the ratio $13 : 11$. As a fraction, you may regard this as either $$\frac{13}{11} \qquad\text{or}\qquad \frac{13/T}{11/T} = \frac{13/24}{11/24},$$ where $T$ is the total number of students (which is $24$, hence the second expression). – Xander Henderson Jul 25 '19 at 21:37
• @XanderHenderson I do not see the meaning of the fraction $13/11$ in this context. It could mean something like For every 1 girl, there is $13/11$ of a boy'', but that is awkward. – Steven Gubkin Aug 27 '19 at 12:18
• Or... for every 13 boys, there are 11 girls. This seems to be a question of dimensional analysis, which was taught in my high school chemistry and physics classes. For example: there are approximately 3.3 feet per meter, so we can use the conversion factor $$\frac{33 \ \text{feet}}{10\ \text{meters}}$$ to convert from meters to feet. We could also use $3.3\ \text{feet}/\text{meter}$, but working with whole numbers is sometimes easier. – Xander Henderson Aug 27 '19 at 13:25


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[This article was first published on R – Predictive Hacks, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
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Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. Another less common measures are the skewness (third moment) and the kurtosis (fourth moment). Today, we will try to give a brief explanation of these measures and we will show how we can calculate them in R.

## Skewness

The skewness is a measure of the asymmetry of the probability distribution assuming a unimodal distribution and is given by the third standardized moment.

We can say that the skewness indicates how much our underlying distribution deviates from the normal distribution since the normal distribution has skewness 0. Generally, we have three types of skewness.

• Symmetrical: When the skewness is close to 0 and the mean is almost the same as the median
• Negative skew: When the left tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the right tail. In this case, we can use also the term “left-skewed” or “left-tailed”. and the median is greater than the mean.
• Positive skew: When the right tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the left tail. In this case, we can use also the term “right-skewed” or “right-tailed”. and the median is less than the mean.

The graph below describes the three cases of skewness. Focus on the Mean and Median.

### Skewness formula

The skewness can be calculated from the following formula:

$$skewness=\frac{\sum_{i=1}^{N}(x_i-\bar{x})^3}{(N-1)s^3}$$

where:

• σ is the standard deviation
• $$\bar{x }$$ is the mean of the distribution
• N is the number of observations of the sample

### Skewness values and interpretation

There are many different approaches to the interpretation of the skewness values. A rule of thumb states that:

• Symmetric: Values between -0.5 to 0.5
• Moderated Skewed data: Values between -1 and -0.5 or between 0.5 and 1
• Highly Skewed data: Values less than -1 or greater than 1

### Skewness in Practice

Let’s calculate the skewness of three distribution. We will show three cases, such as a symmetrical one, and one positive and negative skew respectively.

We know that the normal distribution is symmetrical.

set.seed(5)

# normal
x = rnorm(1000, 0,1)
hist(x, main="Normal: Symmetrical", freq=FALSE)
lines(density(x), col='red', lwd=3)
abline(v = c(mean(x),median(x)),  col=c("green", "blue"), lty=c(2,2), lwd=c(3, 3))

The exponential distribution is positive skew:

set.seed(5)
# exponential
x = rexp(1000,1)
hist(x, main="Exponential: Positive Skew", freq=FALSE)
lines(density(x), col='red', lwd=3)
abline(v = c(mean(x),median(x)),  col=c("green", "blue"), lty=c(2,2), lwd=c(3, 3))

The beta distribution with hyper-parameters α=5 and β=2

set.seed(5)
# beta
x= rbeta(10000,5,2)
hist(x, main="Beta: Negative Skew", freq=FALSE)
lines(density(x), col='red', lwd=3)
abline(v = c(mean(x),median(x)),  col=c("green", "blue"), lty=c(2,2), lwd=c(3, 3))

Notice that the green vertical line is the mean and the blue one is the median.

Let’s see how we can calculate the skewness by applying the formula:

set.seed(5)
x= rbeta(10000,5,2)

sum((x-mean(x))^3)/((length(x)-1)*sd(x)^3)

We get:

3.085474

Notice that you can also calculate the skewness with the following packages:

library(moments)
moments::skewness(x)

# OR

library(e1071)
e1071::skewness(x)

There are some rounding differences between those two packages. Also at the e1071 the formula is without subtracting the 1from the (N-1).

## Kurtosis

In statistics, we use the kurtosis measure to describe the “tailedness” of the distribution as it describes the shape of it. It is also a measure of the “peakedness” of the distribution. A high kurtosis distribution has a sharper peak and longer fatter tails, while a low kurtosis distribution has a more rounded pean and shorter thinner tails.

Let’s see the main three types of kurtosis.

• Mesokurtic: This is the normal distribution
• Leptokurtic: This distribution has fatter tails and a sharper peak. The kurtosis is “positive” with a value greater than 3
• Platykurtic: The distribution has a lower and wider peak and thinner tails. The kurtosis is “negative” with a value greater than 3

Notice that we define the excess kurtosis as kurtosis minus 3

### Kurtosis formula

The kurtosis can be derived from the following formula:

$$kurtosis=\frac{\sum_{i=1}^{N}(x_i-\bar{x})^4}{(N-1)s^4}$$

where:

• σ is the standard deviation
• $$\bar{x }$$ is the mean of the distribution
• N is the number of observations of the sample

### Kurtosis interpretation

Kurtosis is the average of the standardized data raised to the fourth power. Any standardized values that are less than 1 (i.e., data within one standard deviation of the mean, where the “peak” would be), contribute virtually nothing to kurtosis, since raising a number that is less than 1 to the fourth power makes it closer to zero. The only data values (observed or observable) that contribute to kurtosis in any meaningful way are those outside the region of the peak; i.e., the outliers. Therefore, kurtosis measures outliers only; it measures nothing about the “peak”.

Kurtosis in Practice

Let’s try to calculate the kurtosis of some cases:

Normal Distribution

set.seed(5)
# normal
x = rnorm(1000, 0,1)
sum((x-mean(x))^4)/((length(x)-1)*sd(x)^4)

[1] 3.058924

As expected we got a value close to 3!

Exponential distribution

set.seed(5)
# exponential
x = rexp(1000)
sum((x-mean(x))^4)/((length(x)-1)*sd(x)^4)

[1] 10.13425

As expected we get a positive excess kurtosis (i.e. greater than 3) since the distribution has a sharper peak.

Beta distribution

set.seed(5)
# beta
x = rbeta(1000,5,5)
sum((x-mean(x))^4)/((length(x)-1)*sd(x)^4)

[1] 2.634339

As expected we get a negative excess kurtosis (i.e. less than 3) since the distribution has a lower peak.

Notice that you can also calculate the kurtosis with the following packages:

library(moments)
moments::kurtosis(x)

# OR

library(e1071)
e1071::kurtosis(x)

## Conclusion

We provided a brief explanation about two very important measures in statistics and we showed how we can calculate them in R.

To leave a comment for the author, please follow the link and comment on their blog: R – Predictive Hacks.

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• Corpus ID: 119255419

# Parametric Resonance and Dark Matter Axion-Like Particles

@article{Arza2015ParametricRA,
title={Parametric Resonance and Dark Matter Axion-Like Particles},
author={Ariel Arza and Paola Arias and Jorge Gamboa},
journal={arXiv: High Energy Physics - Phenomenology},
year={2015}
}
• Published 8 June 2015
• Physics
• arXiv: High Energy Physics - Phenomenology
We study the local effects of an external time-dependent magnetic field on axion-like particles assuming they are all the dark matter of the universe. We find that under suitable conditions the amplitude of the dark matter field can resonate parametrically. The resonance depends on the velocity of the axion-like particles and scales quadratically with the strength} of the external magnetic field, $\frac{\rho}{\rho_{DM}} \sim {B_0}^3$. By considering typical experimental benchmark values, we…
4 Citations
• Physics
• 2016
When ultralight axion dark matter encounters a static magnetic field, it sources an effective electric current that follows the magnetic field lines and oscillates at the axion Compton frequency. We
• Physics
• 2015
We present a brief overview of a new generation of high-precision laboratory and astrophysical measurements to search for ultralight (sub-eV) axion, axion-like pseudoscalar and scalar dark matter,
• Physics
• 2015
We present an overview of recent developments in the detection of light bosonic dark matter, including axion, pseudoscalar axion-like and scalar dark matter, which form either a coherently

## References

SHOWING 1-3 OF 3 REFERENCES

• Physics
• 2012
Very weakly interacting slim particles (WISPs), such as axion-like particles (ALPs) or hidden photons (HPs), may be non-thermally produced via the misalignment mechanism in the early universe and
Among the prominent low-mass dark matter candidates is the QCD axion but also other light and weakly interacting particles beyond the Standard Model. We review briefly the case for such dark matter
• Physics
• 2013
Dark sectors, consisting of new, light, weakly-coupled particles that do not interact with the known strong, weak, or electromagnetic forces, are a particularly compelling possibility for new


text:
What measure does Lebesgue measure induce on the fat Cantor set?

I know that the fat Cantor set under the subspace topology is homeomorphic to Cantor space $\{0,1\}^{\mathbb N}$ under the product topology induced by the discrete topology on $\{0,1\}$. Call the natural homeomorphism $f$.

What about the measure induced by Lebesgue measure on Cantor space via $f$? Is it the same (up to a constant) as the usual product measure, i.e. the probability measure used to describe an infinite sequence of coin tosses? A reference would be helpful.

• All the uncountable Polish spaces have the same Borel sets, so their measure theory is fairly the same. Apr 30 '13 at 1:11
• Are you saying that the Borel measure on the fat Cantor set (probably) induces the product measure on Cantor space? Or that Lebesgue measure does? Apr 30 '13 at 1:21
• I guess you're saying that the Borel measure induces the product measure and Lebesgue measure induces its completion. The product measure on Cantor space isn't complete, right? Apr 30 '13 at 1:50
• I don't remember all the gory details, in fact I don't even remember the exact proof that the Borel sigma algebras of all the uncountable Polish spaces are isomorphic. But the point is that the result might not be a scalar multiplication of the usual product measure, but it should be some very nice measure with respect to the product measure. Apr 30 '13 at 1:56
• @AsafKaragila: What do you mean by nice? It could very well be singular. Apr 30 '13 at 19:24

Yes, it is.

Note that the Cantor set is in fact a compact group, and the product measure $\mu$ is the Haar measure, that is, the invariant finite, regular Borel measure, which is by Haar's theorem unique up to a scaling factor.

That Lebesgue measure restricted to the fat Cantor set is invariant is an easy exercise (it follows easily from the fact that the Lebesgue measure is translation invariant).

It is also regular, because it is a restriction of a regular measure, so by Haar's theorem it is a scalar multiple of the product measure.

Let $C$ be the fat Cantor set; $\mu$ is the restriction of the Lebesgue measure to $C$, normalized so that $\mu(C)=1$. For every $n\ge 1$, the pre-Cantor set of generation $n$ consists of $2^{-n}$ intervals $I_{n, k}$ of equal length, and the sets $C\cap I_{n,k}$ are congruent to one another. Therefore, $\mu(C\cap I_{n,k})=2^{-n}$. Observe that this is the same measure that $I_{n,k}$ gets from coin-tossing. Also, the intervals $I_{n,k}$ generate the Borel $\sigma$-algebra because every open subset of $C$ is a union of such intervals. Conclusion: yes, the measures are the same up to normalization.

• Looks good, except $\mu(I_{n,k})$ doesn't really make sense, you should write $\mu(C\cap I_{n,k})$ instead. Apr 30 '13 at 19:54
• @tomasz Thanks, I made the corrections. Apr 30 '13 at 21:04
• This answer is also helpful, thanks. Apr 30 '13 at 22:18

Caution: Thanks to tomasz, I realize my "answer" is almost surely irrelevant to OP's question. Yet I decide not to delete it in case it might interest some others. Perhaps more importantly, it's kind of difficult for me to delete something that took me quite a while to finish.

This is an elaboration of Asaf Karagila's comment, "all the uncountable Polish spaces have the same Borel sets".All the stuff is borrowed from Probability measures on metric spaces, Parthasarathy(1967).

Main result: Two Borel sets of Polish spaces are isomorphic, iff they have the same cardinality.

Defination: Given Polish spaces $$X_1$$ and $$X_2$$, Borel sets, $$B_1$$ and $$B_2$$ and $$B_1 \subseteq X_1$$, $$B_2 \subseteq X_2$$, $$B_1$$ is isomorphic to $$B_2$$, which is denoted as $$B_1 \sim B_2$$, iff there is a bijection $$\phi : B_1 \to B_2$$, and both $$\phi$$ and its inverse, $$\phi^{-1}$$ are measurable functions.

Theorem $$1$$: Let $$M = 2^{\omega}$$ endowed with product topology, there exists a Borel subset $$E \subseteq M$$ such that $$E \sim [0,1]$$.

Proof: Consider the well-known continuous surjective function $$\tau$$ from the Cantor space onto the interval $$[0,1]$$: $$(x_n)\mapsto\sum_{n\in\omega}\frac{x_n}{2^{n+1}}$$ The problem for this function is that it's not injective, which means, we need to make a choice between eventual constant sequences that ends with $$1$$ and those ending with $$0$$. One such set $$E$$ is the set of sequences in which $$0$$ occurs infinite times and the sequence that only comsists of $$1$$. It's not difficult to see the function $$\tau$$ restricted on $$E$$ and its inverse are both continuous which implies that they're also measurable.

Theorem $$2$$: There exists a Borel set $$E_1 \subseteq M$$,$$(M = 2^{\omega})$$ such that $$E_1 \sim [0,1]^{\omega}$$.

Proof: With the help of the bijection defined in Theorem $$1$$, we have the induced bijection from $$2^{E}$$ to $$[0,1]^{\omega}$$.Since $$2^{\omega} = (2^{\omega})^{\omega}= 2^{\omega \times \omega}$$, $$M$$ is homeomorphic to $$2^{M}$$ by $$\iota$$, hence $$M \sim 2^{M}$$. Thus there's a subset of $$M$$,$$E_1 = \iota^{-1}(E)$$. So we have $$\tau \circ \iota$$ that guarentee the isomorphism between $$E_1$$ and $$[0,1]^{\omega}$$.

Theorem 3: Let $$X$$ be a Polish space, and $$G \subseteq X$$ be a Borel set. Then there exists a Borel set $$E' \subseteq M$$ such that $$G \sim E'$$.

Proof:Theorem 2 makes our life easier by enabling us to turn our attention away from the Cantor space $$M$$ to Hilbert cube $$[0,1]^{\omega}$$. One important result is that every Polish Space is homeomorphic to a $$G_\delta$$ subspace of the Hilbert Cube (see here). We are done.

• Remark:Thanks to Theorem $$1$$ to $$3$$, we don't need to consider an arbitary Polish space. We only have to prove the theorem for the Cantor space.

Theorem 4: Let $$\mathscr{N} = \omega^{\omega}$$ denote the Baire space. For any Polish space $$X$$, there's a continuous surjection $$\psi$$ from $$\mathscr{N}$$ onto $$X$$.

Proof: This theorem is omnipresent in various textbooks, notes, et cetera. e.g. See here.

Theorem 5: Let $$X$$ be a Polish space and $$E \subseteq X$$ be a Borel set, then there exist a continuous function $$\psi$$ that maps $$\mathscr{N}$$ into $$X$$ such that $$\psi(\mathscr{N}) = E$$.

Proof: We show this by induction. It's obvious that it's true for closed sets, since they're complete with the same metric in $$X$$. Suppose it holds for a denumerable set of subsets of $$X$$, which is $$\{A_n\}_{n \in \omega}$$ with $$\psi_n(\mathscr{N}) = A_n$$. Define a continuous function as a cartesan product of $$\{\psi_n\}_{n \in \omega}$$, i.e. $$(\psi_0, \psi_1, \psi_2, \ldots) : \mathscr{N}^{\omega} \to X^{\omega}$$. Also define a subset of $$\mathscr{N}^{\omega}$$ as $$\mathscr{N}' = \{ x \in \mathscr{N}^{\omega} : \forall m, n \in \omega \psi_m(x) = \psi_n(x)\}$$. As a continuous preimage of a diagonal set, $$\mathscr{N}'$$ is closed in $$\mathscr{N}^{\omega}$$. Now define $$\psi :$$\mathscr{N}'$$\to X$$, for any $$x \in \mathscr{N}'$$ and let $$\pi_1$$ projects $$x$$ to its first component. $$x \mapsto \psi_1(\pi_1(x))$$ Now we have $$\psi(\mathscr{N}') = \bigcap_{n \in \omega}A_n$$. Notice that $$\mathscr{N}$$ is homeomorphic to $$\mathscr{N}^{\omega}$$. So all we need is a composition of two isomophism. To complete the proof, we also have to prove it holds for $$\bigcup_{n \in \omega}A_n$$. Let $$\mathscr{N}_j = \{ x \in \mathscr{N}: \pi_1(x) = j\}$$ for each $$j$$. $$\mathscr{N}_j$$ is isomorphic to $$\mathscr{N}$$, so there a continuous surjection $$\phi_j$$such that $$\phi_j(\mathscr{N}_j) = A_j$$. Define $$\phi : \mathscr{N} \to \bigcup_{n \in \omega}A_n$$, for each $$x$$ in $$\mathscr{N}$$: $$x \mapsto \phi_j(x) \text{iff } \pi_1(x) = j$$.

Theorem $$6$$: Given $$X$$ an uncountable seperable metric space, then there is a partition of $$X$$ with two atoms. One atom is countable, and the other, which is dense in itself, consists of all condensation points.

Proof: This is essentially Cantor–Bendixson theorem.

Theorem $$7$$: Let $$X$$ be a Polish space, $$Y$$ be an uncountable seperable metric space. Assume that there exists a continuous surjection $$\phi$$ from $$X$$ onto $$Y$$. Then there exists a subset $$C \subseteq X$$, such that $$\phi : C \to Y$$ is a homeomorphism. Moreover, $$C$$ is homeomorphic to the Cantor space, and $$\phi(C)$$ is compact.

Proof: With $$\bf{AC}$$, we can construct a subset $$A \subseteq X$$ such that $$\phi$$ ristricted to $$A$$ is a bijection. By Theorem $$6$$, we have a subset $$D \subseteq A$$ such that $$D$$ is an uncountable perfect set. Notice that $$D$$, as a Hausdorff uncountable perfect set contains a copy $$C$$ of Cantor set, which is the same as the construction in the proof of $$|\mathbb{R}| = 2^{\aleph_0}$$. $$\phi(C)$$ is compact follows from the fact it's homeomorphic to the Cantor space, which is compact.

Corrollary: Let $$X$$ be a Polish space, $$E$$ be an uncountable Borel set in $$X$$. Then there exists a compact subset $$C \subseteq E$$ such that $$C$$ is homeomorphic to the Cantor set. In particular, $$|E| = \mathfrak{c}$$.

Lemma: Given $$\{E_m\}_{m \in \omega}$$ a set of disjoint Borel sets, and $$\{F_n\}_{n \in \omega}$$ the same, then $$E_i \sim F_i$$ for all $$i \in \omega$$ implies that $$\bigcup_{i \in \omega} E_i \sim \bigcup_{i \in \omega} F_i$$

Theorem 8: Let $$E_1 \subseteq F \subseteq E_0$$ be three Borel subsets of Cantor space. Then $$E_0 \sim E_1$$ implies that $$E_0 \sim F$$.

Proof: Let $$\psi : E_0 \to E_1$$ be the mapping that guarentee the isomorphism. Define recursively, $$E_{n+1} = \psi(E_n)$$, $$A_{n+1} = E_n \setminus F$$, $$B_{n} = F \setminus E_n$$, $$D_{n+1} = E_n \setminus E_{n+1}$$, $$E_{\infty} = \bigcap_{n \in \omega} E_n$$. Via $$\psi^{m-n}$$ and $$\psi^{i-j}$$ , we have $$D_m \sim D_n$$ for all $$m,n \in \omega$$,$$E_i \sim E_j$$ for all $$i,j \in \omega$$. Similarly, it also holds for the class $$\{A_n\}$$ and $$\{B_m\}$$. Thus we have: $$E = E_{\infty} \cup \bigcup_{i \in \omega}A_i \cup \bigcup_{j \in \omega}B_j \sim E_{\infty} \cup \bigcup_{i \in \omega \setminus \{0\}}A_i \cup \bigcup_{j \in \omega}B_j = F$$

Theorem $$9$$:Two Borel sets $$E_1$$ and $$E_2$$ of Polish spaces $$X_1$$ and $$X_2$$ are isomorphic, if they have the same cardinality.

Proof: As shown, WLOG, Let $$X_1 = X_2$$ be the Cantor spaces. The only interesting case is when $$E_1$$ and $$E_2$$ are uncountable. To prove the theorem, it suffice to show $$E_1 \sim E_2 \sim 2^{\omega}$$. Since both $$E_1$$ and $$E_2$$ contains a copy of the Cantor space, by appealing to Theorem $$8$$, we have $$E_1 \sim 2^{\omega}$$ and $$E_2 \sim 2^{\omega}$$. Moreover $$\sim$$ is an equivalence relation, we have $$E_1 \sim E_2$$.

• How does this answer the question? May 1 '13 at 3:15
• @tomasz: I'm sorry I don't follow your comment. There's nothing special about "Lebesgue measure induce on the fat Cantor set". All uncountable Polish spaces are isomorphic. May 1 '13 at 3:21
• The question was about two particular measures, not Borel algebras. Any two uncountable Polish spaces are Borel isomorphic. May 1 '13 at 3:27
• @tomasz: Thank you for pointing that out. To be frank, I didn't read the question carefully and haste to give an answer, which is in effect a supplement on Asaf's comment. Hope you don't mind asking. What can we say about two measures, if we've already known the isomorphism between two Borel algebras. May 1 '13 at 4:01
• As far as I can tell, nothing really; that the Borel algebras are isomorphic is really trivial in this case, because the fat Cantor set is homeomorphic to the usual Cantor set, which is much stronger than mere Borel isomorphism. What the theorem really gives us is that when considering an abstract Borel measure on a Polish space, we can very well assume that the Polish space is any uncountable Polish space we want. But two measures on a Polish space can be rather different (singular), e.g. the Lebesgue measure on reals and the image of the Cantor measure by the usual embedding. May 1 '13 at 8:02


text:
lilypond-devel
[Top][All Lists]

## Re: Let \time in mid-measure work without warning in some cases (issue 1

 From: dak Subject: Re: Let \time in mid-measure work without warning in some cases (issue 143450043 by address@hidden) Date: Thu, 25 Sep 2014 09:46:31 +0000

On 2014/09/25 03:53:35, Dan Eble wrote:

This new patch set works for me.  If you approve of it (take your

time), I'll

start working on documentation.  Thanks.

Well, your code looks simpler than my sketch: I don't see the \partial
1*0 part implemented.  But in my experiments, it does exactly the same
job even in midmeasure.  I just have no idea why and where it manages to
reset measurePosition.

When \time cuts short the previous measure without a \partial, it still
resets the timing and starts a new bar number without placing a bar
line.  That's sort of weirdish, but it has no permanent off-synch
consequences.

I'm not really sure why this works in the case without \partial and an
incomplete measure.  If you can figure it out, adding a code comment at
the place where the measurePosition check/reset used to be might help

If not, still fine.  This covers your use case in a natural manner and
at least in my experiments it did nothing out of line when fed various
strange things.  And it will not require an instruction booklet to use
either.

https://codereview.appspot.com/143450043/


text:
# How can I calculate percent concentration by volume?

Dec 28, 2014

Percent concentration by volume is defined as the volume of the solute divided by the total volume of the solution and multiplied by 100%. This is expressed mathematically as:

v"/"v% = (volume.solute)/(volume.solution) * 100%

Let's take an example to show how this is done. What is the percent concentration by volume of a solution composed of 150.0 mL of rubbing alcohol and 400.0 mL of water.

So, ${V}_{s o l u t e} = {V}_{r u b b i n g . a l c o h o l} = 150.0 m L$ and
${V}_{s o l v e n t} = {V}_{w a t e r} = 400.0 m L$

It is of the utmost importance to know that the volume of the solution is not just the volume of the solvent, but

${V}_{s o l u t i o n} = {V}_{s o l u t e} + {V}_{s o l v e n t} = 150.0 + 400.0 = 550.0 m L$

Therefore, v"/"v% = (150.0mL)/(550.0mL) * 100% = 27.3%


text:
# Exploring Consequences of Simulation Design for Apparent Performance of Statistical Methods. 2: Results from simulations with normally and uniformly distributed sample sizes

This report continues our investigation of effects a simulation design may have on the conclusions on performance of statistical methods. In the context of meta-analysis of log-odds-ratios, we consider five generation mechanisms for control probabilities and log-odds-ratios. Our first report (Kulinskaya et al. 2020) considered constant sample sizes. Here we report on the results for normally and uniformly distributed sample sizes.

## Authors

• 8 publications
• 8 publications
• 8 publications
• ### Exploring Consequences of Simulation Design for Apparent Performance of Statistical Methods. 1: Results from simulations with constant sample sizes

Contemporary statistical publications rely on simulation to evaluate per...
06/30/2020 ∙ by Elena Kulinskaya, et al. ∙ 0

• ### Odds Ratios are far from "portable": A call to use realistic models for effect variation in meta-analysis

Objective: Recently Doi et al. argued that risk ratios should be replace...
06/04/2021 ∙ by Mengli Xiao, et al. ∙ 0

• ### Completely uniformly distributed sequences based on de Bruijn sequences

We study a construction published by Donald Knuth in 1965 yielding a com...
09/24/2019 ∙ by Emilio Almansi, et al. ∙ 0

• ### Statistical Reliability of 10 Years of Cyber Security User Studies (Extended Version)

Background. In recent years, cyber security security user studies have b...
10/05/2020 ∙ by Thomas Groß, et al. ∙ 0

• ### Improved Summation from Shuffling

A protocol by Ishai et al. (FOCS 2006) showing how to implement distribu...
09/24/2019 ∙ by Borja Balle, et al. ∙ 0

• ### Detecting the skewness of data from the sample size and the five-number summary

For clinical studies with continuous outcomes, when the data are potenti...
10/12/2020 ∙ by Jiandong Shi, et al. ∙ 0

• ### A note on the modeling of the effects of experimental time in psycholinguistic experiments

Thul et al. (2020) called attention to problems that arise when chronome...
05/28/2021 ∙ by R. Harald Baayen, et al. ∙ 0

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 1 Introduction

Our interest lies in effects that simulation design choices may have on conclusions on the comparative merits of various methods, taking as an example meta-analysis of odds ratios. The basic data from studies involve binomial variables, for and or (for the Control or Treatment arm); those data underlie the odds-ratios for the meta-analysis.

A design specifies the number of studies, ; the sample sizes, ; the nuisance parameters (control-arm probabilities,

, or, equivalently, their logits,

); the overall log-odds-ratio,

; and the between-study variance,

. For each situation the simulation uses replications, where is typically large, say 10,000.

For simplicity, we consider equal arm-level sample sizes, . The control probabilities or their logits can be constant or generated from some distribution. Normal and uniform distributions are the typical choices. As in our previous report arXiv_LOR_simulation_equal_sample_sizes, we consider five possible generation mechanisms for control-arm probabilities and log-odds-ratios under the random-effects model of meta-analysis.

We consider two fixed-intercept random-effects models (FIM1 and FIM2) and two random-intercept random-effects models (RIM1 and RIM2), as in bakbergenuly2018GLMM. These models are equivalent to Models 2 and 4 (for FIM) and Models 3 and 5 (for RIM), respectively, of jackson2018comparison. Briefly, the FIMs include fixed control-arm effects (log-odds of the control-arm probabilities), and the RIMs replace these fixed effects with random effects. We also consider a model with uniformly distributed control-arm probabilities (URIM1).

Studies also vary in how they specify the sample sizes . In our previous report (arXiv_LOR_simulation_equal_sample_sizes) we set in all replications. Here we investigate the use of normal and uniform distributions to generate a new set of in each replication.

## 2 Generation of sample sizes

Several authors Cheng2016; bakbergenuly2018GLMM

use constant study-level sample sizes, either equal or unequal, in all replications. More often, however, authors generate sample sizes from a uniform or normal distribution.

jackson2018comparison use (mostly with ) sample sizes from discrete . Langan_2018_RSM_1316 use either constant and equal sample sizes within and across studies, or sample sizes from and ; sidik2007 use ; and AboZaid2013 use and . viechtbauer2007confidence generates study-level sample sizes () from ( is the variance) with . In an extensive simulation study for sparse data, kuss2015statistical uses FIM1 and the corresponding model with , along with a large number of fitting methods; he generates both the number of studies and their sample sizes

from log-normal distributions: LN(0.65, 1.2) and LN(3.05, 0.97) for

and LN(4.615, 1.1) for sample sizes.

In general, if mutually independent random variables

have a common distribution , and is independent of the , the sum has a compound distribution compound

. A binomial distribution with a random number of trials is a compound Bernoulli distribution. The first two moments of such a distribution are

and . This variance is larger than the variance of the distribution. Therefore, random generation of sample sizes produces an overdispersed Binomial (compound Bernoulli) distribution for the control arm, and may also inflate, though in a more complicated way, the variance in the treatment arm.

In particular, when , the compound Bernoulli distribution has variance . And when , .

## 3 Variances of estimated log-odds-ratios for random sample sizes

The (conditional, given and

) variance of the estimated log-odds-ratio

, derived by the delta method, is

 v2i=Var(^θi)=1niTpiT(1−piT)+1niCpiC(1−piC), (3.1)

estimated by substituting for . (We follow the particular method’s procedure for calculating .)

Under the binomial-normal random-effects model (REM), the true study-level effects, , follow a normal distribution: .

To calculate the variance of when sample sizes are random, we use the law of total variance:

 Var(^θi)=E(Var(^θi|ni))+Var(E(^θi|ni)).

The second term is , and the first term is obtained by substituting and in an expression for the variance of under fixed sample sizes.

For a random sample size , using the delta method,

 E(N−1)=(E(N))−1(1+[CV(N)]2), (3.2)

where CV

is the coefficient of variation (i.e., the ratio of the standard deviation of

to its mean). Therefore, to order , random generation of sample sizes inflates the variance of if and only if the coefficient of variation of the distribution of sample sizes is of order . In the simulations of viechtbauer2007confidence, where , , so the variance is not inflated. In contrast, generating sample sizes from would result in and would inflate variance. (Use of such a combination of mean and variance, however, is unlikely to produce realistic sets of sample sizes, and the probability of generating a negative sample size exceeds 2%.)

The variance of a uniform distribution on an interval of width centered at is , and its CV is . Therefore, is of order 1 whenever the width of the interval is of the same order as its center. Hence, variance is inflated in simulations by jackson2018comparison, Langan_2018_RSM_1316, sidik2007, and AboZaid2013, who all use wide intervals for .

## 4 Design of the simulations for randomly distributed sample sizes

Our simulations keep the arm-level sample sizes equal and the control-arm probabilities and the log-odds-ratios independent. Table 1 shows the components of the simulations for normally and uniformly distributed sample sizes: parameters, data-generation mechanisms, and estimation targets. Our first report arXiv_LOR_simulation_equal_sample_sizes provides more details. We included the DerSimonian-Laird (DL), restricted maximum-likelihood (REML), Mandel-Paule (MP), and Kulinskaya-Dollinger (KD) estimators of with corresponding inverse-variance-weighted estimators of

and confidence intervals with critical values from the normal distribution.

Bakbergenuly2020 studied those inverse-variance-weighted estimators in detail. We also included the SSW point estimator of , whose weights depend only on the studies’ arm-level sample sizes, and a corresponding confidence interval, which uses as the midpoint, in the estimate of its variance, and critical values from the distribution on degrees of freedom. Among the estimators, FIM2 and RIM2 denote the estimators in the corresponding GLMMs.

We generated the arm-level sample sizes, , from a normal or a uniform distribution centered at 40, 100, 250, and 1000.

In generating sample sizes from a normal distribution, we want negative sample sizes to have reasonably small probability. For our choice of this probability is . Unfortunately, we were still getting a small number of values below zero out of thousands of simulated values, so we additionally truncate the values generated from a normal distribution at 10. Truncation happens with probability .

To make uniform distributions of sample sizes comparable to the normal distributions, we centered them at the same value, , and equated their variances. If a normal distribution has variance , a uniform distribution with the same variance has interval width . We set , resulting in and a squared CV of . Therefore, by Equation (3.2), our simulations with random inflate variances and covariances by in comparison with simulations with constant . Wider intervals of would inflate variances more, but in generating sample sizes from a corresponding normal distribution, we wanted negative sample sizes to have reasonably small probability. For our choice of this probability is .

## 5 Summary of the results

Our simulations explored two main components of design: the data-generation mechanism and the distribution of study-level sample sizes. Results of our simulations with normally distributed sample sizes are provided in Appendix A, and those with uniformly distributed sample sizes in Appendix B.

The five data-generation mechanisms (FIM1, FIM2, RIM1, RIM2, and URIM1) often produced different results for at least one of the measures of performance (bias of estimators of , bias of estimators of , and coverage of confidence intervals for ). In the most frequent pattern FIM2 and RIM2 yield similar results, and FIM1, RIM1, and URIM1 also yield results that are similar but different from those of FIM2 and RIM2. In some situations URIM1 stands apart.

We also expected the coverage of to suffer because random sample sizes increase the variance of generated log-odds-ratios. However, generation of sample sizes from normal and uniform distributions had essentially no impact, as can be seen by comparing the results from this report with those from our report arXiv_LOR_simulation_equal_sample_sizes on the simulations with constant sample sizes. The explanation may lie in our choice of variance (not large enough) for the normal and uniform distributions of the sample sizes, causing an increase of just 10% in the variance of LORs, or in the rather low coverage, even under constant sample sizes, resulting from considerable biases of estimators of .


text:
# Symbols of circuit elements

• Engineering

## Homework Statement

I wanted to ask why the symbols of capacitor , resistor and inductor is such which they have..?? Why are we denoting capacitor by 2 parallel plates why not 2 zig-zag plates..and same for inductor??

## The Attempt at a Solution

As much I got , resistor has such symbol as it acts as a barrir to flow of electrons..

Last edited by a moderator:

Mark44
Mentor

## Homework Statement

I wanted to ask why the symbols of capacitor , resistor and inductor is such which they have..?? Why are we denoting capacitor by 2 parallel plates why not 2 zig-zag plates..and same for inductor??

## The Attempt at a Solution

As much I got , resistor has such symbol as it acts as a barrir to flow of electrons..
The simplest capacitor would be two parallel plates at a fixed distance from each other, so the || symbol makes more sense than two zigzag shapes. An inductor is essentially a coil of wire, so the symbol for an inductor captures that idea. I'm not sure why they chose a sawtooth symbol for resistors, although its rough shape does suggest an obstacle.

ranju
CWatters


text:
Continue to Site

# Common mode rejection ratio with opamps?

#### cupoftea

Hi,
If we have an opamp with 0v and +5V supply, then its common mode voltage is +2.5V, and so the Common mode rejection ratio will be max for signals near 2.5V?
If we have an opamp with +5V and -5V supply, then its common mode voltage is 0V , and common mode rejection ratio will be greater for signals near 0V?
If its an InAmp with 0V and +5V supply, then the CM rejection ratio is the same no matter where the signal is between 0V and +5V?

No.

#### cupoftea

Thanks,
Though would you agree that the attached is an instrumentation amplifier with a gain of 1?
Its common mode rejection ratio will be as good as an Inst Amp because there is no amplification to a common mode signal at its inputs.?

Also, its difficult to find an OTS In Amp with a gain of 1, so the attached, which could be done with a cheap quad OpAmp, looks like a game changer?
LTspice and jpeg attached

#### Attachments

• Instrumentation Amplifier with gain = 1.jpg
111.5 KB · Views: 36
• Instrumentation Amp.zip
1 KB · Views: 18

cupoftea

### cupoftea

Points: 2

#### dick_freebird

DC CMRR generally comes from the forward gain and a
stable Vio (front end gain and balance), through the
feedback network. The same is true of PSRR. Of
course poor design choices can degrade things, from
there.

As inputs or outputs get close to a supply rail there is
likely to be some degradation to AVOL, Vio or both. If
that happens then CMRR will degrade as well.

Now many op amp topologies will be fine at one end
of the range and bad at the other - look for hints in the
datasheet regarding input headroom, at where the
output load-driving attributes are measured, etc. It's
a good bet that outboard of those ranges, occurs
"stuff" that the vendor isn't real proud of. Like the
"good" ol' LM124 with its Vcc-(2Vbe+Vce(sat)) upper
bound in input voltage for meeting main specs - no
way around needing to keep the front end lit. Though
more modern "rail-rail-I/O" architectures are better,
you can still see (if you peel off a couple layers of the
datasheet onion) that gain is depressed within a volt
or so of the rails. They probably neglect to mention the
other "fellow traveler" attributes which suffer the same.

### cupoftea

Points: 2

#### KlausST

##### Super Moderator
Staff member
Hi,

The descriptions - especially about common mode voltage - of post#1 and post#3 don't match at all.

Often INAs are made for higher gain.
And the higher gain stage is in the first stage and it designed to amplify common mode with gain of 1 and differential modd with the desired gain.

Your circuit of post#3 thus is a "special case" of instrumentation amplifier. It does not benefit from the classical INA design.
One can say it is a classical differential amplifier circuit with buffered inputs.

I can not find any "game changing".

So I did a search on "instrumentation amplifier datasheet" and checked the first 5 hits:
* INA115 ... can be operated the same way as circuit of post#3
* INA114 ... too
* AD8223 ... is something "special" can work with gain>=5

What "game changing" do you see?

Klaus

cupoftea

### cupoftea

Points: 2

#### cupoftea

Thanks,
The "game changing" is in that INA115 and co cost $5+ whereas LM124 (etc) Quad opamp is <$1

And the higher gain stage is in the first stage and it designed to amplify common mode with gain of 1 and differential modd with the desired gain.
Thanks, and since CMRR is (DM gain/CM gain), then if the first stage only has a gain of one, then the InAmp has very poor common mode rejection?

#### KlausST

##### Super Moderator
Staff member
Hi,

You really compare a LM124 with an INA115?
Vos: 2mV vs 0.05mV
VOSDRIFT: 7uV/°C vs 0.25uV/°C
IB: 20nA vs 2nA
CMRR: poor 70dB (and where are the resistors? Dana's post) vs 115dB including laser trimmed resistors

Just the cost: What is the cost for LM124 plus 4 laser trimmed resistors to get a CMRR of 115dB?
(Indeed I'd say it is impossible to get close to the CMRR (only!) performance of the INA115)

My opinion: one could say the invention of an INA115 was "game changing".

I can not find a single parameter where 3(out of 4 parts of) LM124 beat the INA115.
Not even power supply 3× 1.5mA vs 3mA .... nor speed (2 x 1.3MHz in series) vs 1 MHz

Klaus
--- Updated ---

then the InAmp has very poor common mode rejection?
Even then the INA has 45dB better CMRR. This is about a factor of 30,000.

It's the other way round: if you have a higher input gain stage, then the INA has additional benefit.

Klaus

Last edited:
cupoftea

### cupoftea

Points: 2

#### cupoftea

It's the other way round: if you have a higher input gain stage, then the INA has additional benefit
Thanks, yes, but we only want a gain of 1.
We just want to pass a feedback signal between PWM controllers for whom ground varies slightly.
We cant just pass the error voltage with respect to ground, as there are ground drops between the PWM controllers....(due to current flows)

Last edited:

#### KlausST

##### Super Moderator
Staff member
Hi,

A "current" feedback from a shunt?

--> Which solution to choose depends on many things, like frequency, and source impedance... and so on.

We have no idea how your source looks like nor how the output signal should look like.

A "gain of 1" application isn´t the usual target for an INA.

Klaus

#### cupoftea

It's the other way round: if you have a higher input gain stage, then the INA has additional benefit
Thanks, yes i think its best for a "classical differential amplifier circuit with buffered inputs", as you kindly called it.


text:
## Friday, 1 July 2011

### Equations for a falling body near the surface of the Earth

Ball falling freely under gravity. See text for description.
Under an assumption of constant gravity, Newton's law of universal gravitation simplifies to F = mg, where m is the mass of the body and g is a constant vector with an average magnitude of 9.81 m/s2. The acceleration due to gravity is equal to this g. An initially stationary object which is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. The image on the right, spanning half a second, was captured with a stroboscopic flash at 20 flashes per second. During the first 120 of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 220 it has dropped at total of 4 units; by 320, 9 units and so on.
Under the same constant gravity assumptions, the potential energy, Ep, of a body at height h is given by Ep = mgh (or Ep = Wh, with W meaning weight). This expression is valid only over small distances h from the surface of the Earth. Similarly, the expression $h = \tfrac{v^2}{2g}$ for the maximum height reached by a vertically projected body with velocity v is useful for small heights and small initial velocities only.

### Gravity and astronomy

The discovery and application of Newton's law of gravity accounts for the detailed information we have about the planets in our solar system, the mass of the Sun, the distance to stars, quasars and even the theory of dark matter. Although we have not traveled to all the planets nor to the Sun, we know their masses. These masses are obtained by applying the laws of gravity to the measured characteristics of the orbit. In space an object maintains its orbit because of the force of gravity acting upon it. Planets orbit stars, stars orbit Galactic Centers, galaxies orbit a center of mass in clusters, and clusters orbit in superclusters. The force of gravity is proportional to the mass of an object and inversely proportional to the square of the distance between the objects.

### Gravitational radiation

In general relativity, gravitational radiation is generated in situations where the curvature of spacetime is oscillating, such as is the case with co-orbiting objects. The gravitational radiation emitted by the Solar System is far too small to measure. However, gravitational radiation has been indirectly observed as an energy loss over time in binary pulsar systems such as PSR B1913+16. It is believed that neutron star mergers and black hole formation may create detectable amounts of gravitational radiation. Gravitational radiation observatories such as LIGO have been created to study the problem. No confirmed detections have been made of this hypothetical radiation, but as the science behind LIGO is refined and as the instruments themselves are endowed with greater sensitivity over the next decade, this may change.


text:
# Superbase

In chemistry, a superbase is an extremely strong base. It is a compound that has a high affinity for protons.

Hydroxide ion is the strongest base possible in water (aqueous) solutions, but superbases are much stronger than water-based bases. Such bases are useful in organic synthesis and are fundamental to physical organic chemistry. Superbases have been described and used since the 1850s.[1] Reactions involving superbases often require special techniques since they are destroyed by water, carbon dioxide, and oxygen in the air. Inert atmosphere techniques and low temperatures minimize these side reactions.

## Definitions

IUPAC defines superbases simply as a "compound having a very high basicity, such as lithium diisopropylamide."[2] Caubère defines superbases qualitatively but more precisely: "The term superbases should only be applied to bases resulting from a mixing of two (or more) bases leading to new basic species possessing inherent new properties. The term superbase does not mean a base is thermodynamically and/or kinetically stronger than another, instead it means that a basic reagent is created by combining the characteristics of several different bases."[3]

Superbases have also been defined in terms of numbers (semi-quantitatively) as any species with a higher absolute proton affinity (APA = 245.3 kcal/mol) and intrinsic gas phase basicity (GB = 239 kcal/mol) than Alder's canonical proton sponge (1,8-bis-(dimethylamino)-naphthalene).[4]

The "strength" of bases is measured by pKb, the "dissociation constant" of each base. Historically, the equilibrium constant Kb for a base has been defined as the association constant for protonation of the base, B, to form the conjugate acid, HB+.

B + H2O HB+ + OH

So, the equilibrium of this reaction is:

$The base association constant K b equals the concentration of the protonated form H B +, times the concentration of the hydroxyl anion O H minus, all divided by the concentration of the base B.$

and pKb is defined as:

pKb = − log10 Kb. The range for aqueous solutions (pH) is 1 to 14. But superbases are outside this range.

## Classes of superbases

There are three main classes of superbases: organic, organometallic and inorganic.

### Organic

Organic superbases are almost always neutral, nitrogen-containing species. Despite enormous proton affinity, chemists like organosuperbases for their high reactivity tempered by low nucleophilicity and relatively mild conditions of use. Increasingly important in organic synthesis, these include the phosphazenes, amidines and guanidines. Other organic compounds also meet the physicochemical or structural definitions of 'superbase'. Proton chelators like the aromatic proton sponges and the bispidines are also superbases. Multicyclic polyamines, like DABCO might also be loosely included in this category.[5]

### Organometallic

Organometallic compounds of reactive metals are often superbases, including organolithium and organomagnesium (Grignard reagent) compounds. Another type of organic superbase has a reactive metal exchanged for a hydrogen on a heteroatom, such as oxygen (unstabilized alkoxides) or nitrogen (metal amides such as lithium diisopropylamide). A desirable property in many cases is low nucleophilicity, i.e. a non-nucleophilic base. Unhindered alkyllithiums, for example, cannot be used with electrophiles such as carbonyl groups, because they attack the electrophiles as nucleophiles.

In organic synthesis, the Schlosser base (or Lochmann-Schlosser base), the combination of n-butyllithium and potassium tert-butoxide, is a commonly used superbase. n-Butyllithium and potassium tert-butoxide form a mixed aggregate of greater reactivity than either reagent alone and with distinctly different properties in comparison to tert-butylpotassium.[6]

### Inorganic

Inorganic superbases are typically salt-like compounds with small, highly charged anions, e.g. lithium nitride. Alkali lanthanides and alkali metal hydrides potassium hydride and sodium hydride are superbases. Such species are insoluble in all solvents owing to the strong cation-anion interactions. But the surfaces of these materials are highly reactive and slurries are useful in synthesis.

## References

1. "BBC - h2g2 - History of Chemistry - Acids and Bases". Retrieved 2009-08-30.
2. "IUPAC Gold Book - superacid". Retrieved 2009-08-30.
3. Caubère, P. (1993) Unimetal super bases Chemical Reviews, 93, 2317-2334. doi:10.1021/cr00022a012
4. Raczynska, E. D., Decouzon, M., Gal, J.-F. et al.(1998) Superbases and superacids in the gas phase. Trends in Organic Chemistry, 7, 95-103.
5. Superbases for Organic Synthesis Ed. Ishikawa, T., John Wiley and Sons, Ltd.: West Sussex, UK. 2009.
6. Schlosser, M. (1988). Superbases for organic synthesis, Pure and Applied Chemistry, 60(11), 1627-1634. doi:10.1351/pac198860111627


text:
# Publishing house / Banach Center Publications / All volumes

## On some class of hypersurfaces with three distinct principal curvatures

### Volume 69 / 2005

Banach Center Publications 69 (2005), 145-156 MSC: Primary 53B20, 53B25; Secondary 53C25. DOI: 10.4064/bc69-0-9

#### Abstract

We investigate hypersurfaces $M$ in spaces of constant curvature with some special minimal polynomial of the second fundamental tensor $H$ of third degree. We present a curvature characterization of pseudosymmetry type for such hypersurfaces. We also prove that if such a hypersurface is a manifold with pseudosymmetric Weyl tensor then it must be pseudosymmetric.

#### Authors

• Katarzyna SawiczInstitute of Econometrics and Computer Science
Technical University of Cz/estochowa
Armii Krajowej 19B, 42-200 Cz/estochowa, Poland
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.


text:
# ) Describe what assets value would be if Liabilities are $12,000 and Owners' Equity is$50,000 by...

) Describe what assets value would be if Liabilities are $12,000 and Owners' Equity is$50,000 by showing the Accounting Equation (10 points), and (2) provide an example of two asset accounts that could contain the value


text:
0
Research Papers

# Axial Transonic Rotor and Stage Behavior Near the Stability Limit

[+] Author and Article Information
Anthony J. Gannon

Department of MAE, Naval Postgraduate School, 700 Dyer Road, RM 245, Monterey, CA 92943ajgannon@nps.edu

Garth V. Hobson

Department of MAE, Naval Postgraduate School, 700 Dyer Road, RM 245, Monterey, CA 92943gvhobson@nps.edu

William L. Davis

Department of MAE, Naval Postgraduate School, 700 Dyer Road, RM 245, Monterey, CA 92943william.l.davis@uscg.mil

J. Turbomach 134(1), 011009 (May 26, 2011) (8 pages) doi:10.1115/1.4003225 History: Received September 01, 2010; Revised September 07, 2010; Published May 26, 2011; Online May 26, 2011

## Abstract

Transient casing pressure data from a transonic rotor and rotor-stator stage measured using high-speed pressure probes embedded in the casewall over the rotor tips are analyzed. Using long data sets sampled at a high frequency, low-frequency (less than once-per-revolution) nonaxisymmetric flow phenomena were detected while operating at steady-state conditions near stall. Both the rotor and stage cases are investigated, and the difference in behavior of a rotor with and without a stator blade row is investigated. Data for both cases over the speed range 70–100% of design and from choke to near the stability limit (stall or surge) are presented. The root mean square power of the low-frequency signal as well as its fraction of the total pressure signal is presented. It was thought that the behavior of these signals as stall was approached could lead to some method of detecting the proximity of stall. For the rotor-only configuration, the strength of these nonaxisymmetric phenomena increased as stall was approached for all speed-lines. However, for the stage configuration, more representative of an operational machine, these were of a lower magnitude and did not exhibit a clearly increasing trend as stall was approached. This would seem to indicate that the stator suppressed these signals somewhat. It is also shown that these nonaxisymmetric phenomena led to a significant variation of the mean relative inlet flow angle into the rotor blade. During stable operation near to stall at 100% speed for the rotor-only case, a 1.9 deg variation of this angle was measured. This compared with a 5.6 deg variation over the entire speed-line. Further, it was observed that while the rotor and stage cases had different stability limits, their peak relative inlet flow angles near stall were similar for both along most speed-lines.

<>

## Figures

Figure 1

Transonic test rotor-stator stage

Figure 2

Rotor-only and stage stagnation pressure ratios

Figure 3

(a) Rotor-only and (b) stage isentropic efficiencies

Figure 4

Rotor-only and stage rotor-tip relative inlet Mach numbers

Figure 5

Kulite high-speed pressure probe positions

Figure 6

Rotor-only 100% near stall wall pressure contours

Figure 7

Rotor-only 100% speed spectral analysis

Figure 8

Filter performance and band-pass regions

Figure 9

Unfiltered and filtered pressure signal

Figure 10

Filtered pressure signal

Figure 11

Axial Mach number variation

Figure 12

(a) Rotor-only and (b) stage axial Mach number peak variation

Figure 13

(a) Rotor-only and (b) stage comparison of filtered signal magnitude and distribution

Figure 14

(a) Rotor-only and (b) stage low-frequency rms powers

Figure 15

(a) Rotor-only and (b) stage total rms power

Figure 16

(a) Rotor-only and (b) stage low-frequency/total rms nondimensional

Figure 17

Varying relative inlet flow angle

Figure 18

(a) Rotor-only and (b) stage mean-to-max relative inlet angle variation

Figure 19

(a) Rotor-only and (b) stage maximum relative inlet angle

## Discussions

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text:
Question

2) An office building has 10 floors of the same height. How many times as far from the ground is the ninth floor as the third floor?

it will be 3 times as far.
Step-by-step explanation:
9a/3a
a=5
(9×5)/(3×5)
45/15=3
this work with any number for a.


text:
# Specifying vertical alignment in a table, column-by-column

I am creating a very simple table with three columns and one row. The alignment and content of each column is as follows:

• Col. 1 - (H) center, (V) middle, single character
• Col. 2 - (H) left, (V) top, multiple lines w/ breaks
• Col. 3 - (H) left, (V) top, wrapping text string

It seems like a simple task. I've looked through countless questions, here and elsewhere, and I can't seem to find a way to do this. Here are a couple things I've tried:

Attempt 1

The only thing this needs is a way to vertically-center the first column. Modifying this code would be my first choice, because it's very simple.

\documentclass{article}
\usepackage{lipsum}
\usepackage{array}

\begin{document}

\begin{tabular}{|>{\centering\arraybackslash}p{0.5in}|p{2in}|p{2in}|}
\hline
A &
Line 1 \newline Line 2 \newline Line 3 \newline Line 4 &
\lipsum*[75] \\
\hline
\end{tabular}

\end{document}

Attempt 2

Same as above, except the last column is inside a \parbox. This makes the first column vertically-centered as desired, but the adjusted baseline causes the second column to start halfway down the cell.

\documentclass{article}
\usepackage{lipsum}
\usepackage{array}

\begin{document}

\begin{tabular}{|>{\centering\arraybackslash}p{0.5in}|p{2in}|p{2in}|}
\hline
A &
Line 1 \newline Line 2 \newline Line 3 \newline Line 4 &
\parbox{2in}{\lipsum*[75]} \\
\hline
\end{tabular}

\end{document}

Attempt 3

Creating a table with several rows and \multirow works to align the first column, but it won't allow the text to wrap in the third.

\documentclass{article}
\usepackage{lipsum}
\usepackage{array}
\usepackage{multirow}

\begin{document}

\begin{tabular}{|>{\centering\arraybackslash}p{0.5in}|p{2in}|p{2in}|}
\hline
\multirow{4}{*}{A} & Line 1 & \multirow{4}{*}{\lipsum*[75]} \\
& Line 2 & \\
& Line 3 & \\
& Line 4 & \\
\hline
\end{tabular}

\end{document}

Any suggestions? I'm looking for a simple method with limited use of extra packages. This code will be compiled and run server-side on a fairly large scale, so the fewer dependencies, the better.

• Welcome to TeX.SX! – jub0bs Jan 23 '14 at 20:04
• Thanks! I've been lurking for a while and have gleaned some fantastic information here. Hopefully I'll be able to offer some help sometime, too! – higrafey Jan 24 '14 at 18:30

\documentclass{article}
\usepackage{lipsum}
\usepackage{array}

\begin{document}

\setlength\extrarowheight{2pt}

\begin{tabular}{|>{\centering\arraybackslash}m{0.5in}|c|}
\hline
A &
\begin{tabular}{@{}p{2in}|p{2in}@{}}
Line 1 \newline Line 2 \newline Line 3 \newline Line 4 &
\lipsum*[75]
\end{tabular} \\
\hline
\end{tabular}

\end{document}

• That's a really clever solution. Thank you! I played around with the nested tabular beforehand, but never got it working right. Is there any reason I wouldn't want to use this method for other tables with additional columns, specifying the vertical alignment as I go? – higrafey Jan 24 '14 at 18:33

You might want to try cals package:

\documentclass{article}
\usepackage{lipsum}
\usepackage{cals}

\begin{document}

\begin{calstable}
\colwidths{{0.5in}{1.5in}{2.5in}}

% Add frame around the table
\makeatletter \def\cals@framecs@width{.4pt}\def\cals@framers@width{.4pt}

\brow
\alignC\cell{\vfil A}\alignL
\cell{Line 1 \newline Line 2 \newline Line 3 \newline Line 4}
\cell{\lipsum*[75]}
\erow
\end{calstable}

\end{document}

• The cals package looks pretty neat. I hadn't heard of it before, but I think it might come in handy. I like how the alignment can be specified on a cell-by-cell basis. Thanks for showing it to me! – higrafey Jan 28 '14 at 14:29


text:
# Decoding DART

DART, the Double Asteroid Redirection Test, is a NASA mission that launched last Wednesday from Vandenberg. The goal of this mission is to crash the spacecraft into the small asteroid Dimorphos, allowing us to measure the small change in the orbit of the asteroid caused by the impact.

From the communications perspective, this spacecraft is the first to use a Spiral Radial Line Slot Array (RLSA) as high-gain antenna. Details about the antenna design can be seen in this paper. The paper shows that antenna polarization is LHCP. Most DSN communications use RHCP, although there are a few notable exceptions (for instance Emirates Mars Mission), and the DSN stations are equipped to handle both polarizations. I’m not sure if DART is indeed using LHCP or if this is just a matter of the convention in the definition of the polarization used in the paper (there are actually two opposite conventions to define the sense of circular polarization).

A few hours after launch, as the spacecraft passed over Europe, Miguel CT1BYM and Iban EB3FRN recorded the X-band telemetry signal from DART at 8421.79 MHz. This post is a first analysis of the signal.

The modulation used by DART in these recordings is 600 kbaud PCM/PM/NRZ, which means that the carrier is directly phase-modulated with the baseband NRZ data. Turbo rate coding with r=1/6 and 8920 information bits is used, so the data rate is approximately 100 kbps. Initially in the mission, a lower rate of 2 kbps was used, since the smaller antennas such as the 4.5 metre in New Norcia weren’t able to receive reliably the higher rate signal.

The GNU Radio flowgraph I’m using to decode the signal can be seen in the figure below. It is very similar to other flowgraphs I’ve been showing in recent posts, such as the one I’m using for Lucy. The flowgraph can be found here.

The GUI of the flowgraph can be seen below running on one of the recordings done by Iban. The signal is rather weak, and the data modulation is below the noise floor. However, the correlations with the 192-bit ASM show up clearly, and the Turbo decoder is able to decode correctly some of the frames. A 16-bit CRC is used to filter out only correctly decoded frames.

At this SNR I think that the clock recovery loop is not doing a good job. One of the challenging aspects of low rate Turbo coding is that r=1/6 Turbo coding can work down to approximately -0.25 dB Eb/N0. At that level, the Es/N0 is -8 dB, so the clock recovery loop will have a hard time. I think that an open loop approach at clock recovery, using the ASM correlations will work better. However I think there aren’t yet ready made tools to do this in GNU Radio.

The recording that Miguel made had somewhat better SNR. Unfortunately, his recording was losing blocks of samples from the SDR, so no frames can be decoded. Iban made two recordings of 5 minutes and 10 minutes starting at 08:24:50 and 08:34:42 UTC on 2021-11-24. I could decode a total of 438 frames from these recordings. The decoded frames can be found here. Since frames take approximately 90 ms to transmit, this corresponds to 39.23 seconds worth of decoded data, which is 4.36% of the total number of frames in the recordings.

The frames are CCSDS TM Space Data Link frames. Handling these has been a nice change from all the latest spacecraft I’ve been working with, which use the AOS Space Data Link protocol instead. There are two virtual channels in use: virtual channel 0, which contains Only Idle Data, and virtual channel 7, which contains the telemetry. The frames have a TM Secondary Header and an Operational Control Field.

The figure below shows the master channel frame count in the received frames. This has been unwrapped with the help of the timestamps in the secondary header (described below). The large jump corresponds to the gap between the two recordings.

We see that the frame loss is not at all uniform. There are certain moments where many more frames are lost.

The TM Secondary Header has a size of 8 bytes. The first byte is the transfer frame secondary header ID field. The second byte is always 0x34. I don’t know what this field encodes. The last 6 bytes contain a 48-bit timestamp as the number of units of $$2^{-16}$$ seconds elapsed since 2010-01-01 00:00:00 UTC. Here we can see the evolution of the timestamps and the master channel frame count. Even though the second recording lasts 10 minutes, the signal level drops significantly after the first few minutes, so there are no decoded frames afterwards.

The Operational Control Field contains a Communications Link Control Word as described in the TC Space Data Link Protocol Blue Book. Its contents do not change throughout these recordings. They are listed below. They show that the spacecraft’s uplink receiver is correctly locked.

control_word_type = False
clcw_version_number = 0
status_field = 0
cop_in_effect = 1
virtual_channel_identification = 5
rsvd_spare = 0
no_rf_avail = False
no_bit_lock = False
lock_out = False
wait = False
retransmit = False
farm_b_counter = 2
rsvd_spare2 = 0
report_value = 0

Virtual channel 0 contains Only Idle Frames. The transfer frame data field of these is filled with an 8-bit counter. This counter wraps 4 times during each frame, and is reset at the beginning of each frame. The figure below shows a raster plot of the frames in this virtual channel. We can see the counter increasing and wrapping from left to right.

Virtual channel 7 contains the telemetry data. The raster plot of its frames is shown here.

The first header pointer field in the TM Primary Headers is used to indicate the beginning of the first packet within the transfer frame data field of each frame. Interestingly, the packets are not CCSDS Space Packets, since when trying to interpret as Space Packet Primary Headers the beginnings of these frames, the values we obtain don’t make sense.

It seems that the beginning of each frame contains a counter or timestamp. For the first packets, the first 4 bytes are always 0x1660bbd3, but then this value keeps increasing in later packets. I don’t know how the length of each packet can be determined, since there is no obvious field at the start of the packet indicating it.

I have used a heuristic algorithm to extract the packets. I maintain a 4 byte counter that starts at 0x1660bbd3. For each frame, I start at the position indicated by the first header pointer and then search after this point for either the current value of the counter or an increased version of it (and there is a limit to how much it can increase). If one of these is found, the new increased value of this counter is committed for the next packets, and we try to search the next packet in this frame. If none is found, there are no further packets we can extract from this frame and we continue with the next frame.

This algorithm misses packets that are fragmented across several frames, but since we are missing lots of frames, it wouldn’t be possible to recover most of these anyway. The rest of the packets are successfully recovered without a previous knowledge of their lengths.

We see that there are several different lengths that can appear. Classifying the packets by their length, we note that the packets with the same length have similar contents, so probably they are the same packet type.

The most common packets have length 64. These are shown in the raster map below.

I haven’t spotted anything that looks very interesting. The data for each of the packet sizes doesn’t vary much with time. The plots for all the packets can be found in the Jupyter notebook.

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text:
# Explosion test in 2-D

The two dimensional Euler equations are solved on a square domain 2x2 in the xy-plane. The initial conditions consists of circular discontinuity of radius 0.4 centered at (1,1). The values inside and outside the circle are given in the table below.

 Variable Inside Outside Density 1.0 0.125 x velcity 0.0 0.0 y velocity 0.0 0.0 Pressure 1.0 0.1

A typical mesh size that can be used consists of 100x100 cells. It is good to use cell-averaged data as initial conditions. Otherwise the initial data consists of a staircase configuration leading to small amplitude waves at initial times.

The solution exhibits a circular shock wave traveling away from the center, a circular contact surface traveling in the same direction and a circular rarefaction traveling towards the origin (1,1). As time evolves, a complex wave pattern emerges. The circular shock wave travels outwards and becomes weaker. The contact surface follows the shock and also becomes weaker; at some point in time the contact comes to rest and then travels inwards. The rarefaction traveling towards the center reflects, as a rarefaction, and over-expands the flow so as to create an inward traveling shock wave; this circular shock wave implodes into the origin, reflects and travels outwards colliding with the contact surface.

Since the solution has circular symmetry, the 2-D Euler equations can be reduced to 1-D Euler equations in the radial coordinate r,

$\frac{\partial U}{\partial t} + \frac{\partial F}{\partial r} = S(U)$

where

$U=\begin{bmatrix} \rho \\ \rho u\\ E \end{bmatrix}, \quad F = \begin{bmatrix} \rho u \\ p + \rho u^2 \\ (E + p)u \end{bmatrix}, \quad S = - \frac{1}{r} \begin{bmatrix} \rho u \\ \rho u^2 \\ (E + p)u \end{bmatrix}$

This 1-D problem can be solved using a large number of grid points. The 1-D solution can then be used to assess the accuracy of the 2-D solution.


text:
#### Howdy, Stranger!

It looks like you're new here. If you want to get involved, click one of these buttons!

Options

# Nitrification inhibitor

## Comments

• Options
1.
edited May 2014

Interesting. As I keep running around telling everyone, right now humans turn more atmospheric nitrogen into reactive forms than all natural processes combined. But how can you put nitrification inhibitors in fertilizers and still have them function as fertilizers? Obviously I'm missing something.

Comment Source:Interesting. As I keep running around telling everyone, right now humans turn more atmospheric nitrogen into reactive forms than all natural processes combined. But how can you put nitrification inhibitors in fertilizers and still have them function as fertilizers? Obviously I'm missing something.
• Options
2.
edited May 2014

The idea is that more nitrogen goes from the fertilizer into the plants, instead of into N2O in the air, or nitrates in water. I edited the page in the hope of making this clearer.

Comment Source:The idea is that more nitrogen goes from the fertilizer into the plants, instead of into N2O in the air, or nitrates in water. I edited the page in the hope of making this clearer.
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text:
# Thin triangles vs Slim triangles in hyperbolic spaces

What is the difference between thin triangles and slim triangles in $\delta$ hyperbolic spaces? Google search seems to consider thin and slim as synonyms and shows the same results for the two.

• They are slightly different concepts, but turn out to encode the same notions of Gromov hyperbolicity, in the sense that: every geodesic triangle is $\delta$-thin iff every geodesic triangle is $\varepsilon$-slim, possibly with different constants. A great reference is Bridson, Haefliger, Metric spaces of non-positive curvature; have a look at def 1.1, 1.16 and Prop 1.17 in Chapter III.H – Lor Nov 19 '14 at 20:49


text:
# Set Position of multiple bodies

I have a character composed of five bodies which are tied together by a lot of joints. On of them is the overall chassis, to which all forces and impulses are applied to move the whole Character.

All in all that works very fine, except one thing: I need to set the Position of the Character so that it get Beamed from one place to the other in one single frame.

Unfortunately I cannot get this to work. I tried the following code, without any success…

playerbodies.forEach(function (bd) {
bd.SetLinearVelocity(new b2.Vec2());
var t = bd.GetTransform();
t.p.x -= 10;
bd.SetTransform(t, bd.GetAngle());
});

How can I make that happen?

• What do you mean by "without any success"? Apr 22 '14 at 19:33
• total crash, or the body move to somewhere and is left with strange movements… Apr 22 '14 at 19:46
• Please don't edit your solution into the question; rather, post it as an answer, since it is an answer to your quesiton. (Also, don't edit [Solved] into your title - this is not a forum.) Apr 23 '14 at 12:51
• Google "box2d teleport". I am sure you will find your solution. Oct 21 '15 at 8:28

As suggested in the comments… The answer is quiet simple and the right code is:

1| playerbodies.forEach(function (bd) {
2|     bd.SetLinearVelocity(new b2.Vec2());
3|     var t = bd.GetTransform();
4|     t.p.x -= 10;
5|     bd.SetTransform(t.p, bd.GetAngle());
6| });

whereby line 6 is corrected with t.p.

Since Javascript is not typesafe this mistake totally broke the simulation of the bodies, instead of throwing an error.


text:
# Superposition principles of electrostatic force

The topic which we are going to talk about today is one of the important topics of electrostatic. Today we are going to talk about the principle of superposition of electrostatic force. This principle is very important and widely used in acoustic, wave optics, and current electricity and engineering.

So it becomes very important for us to understand this principle and learn about this. For understanding the basic concept of superposition principles. Let’s consider a situation.

## Superposition principle

Think of a charge $q_1$ in the open space, which is surrounded by a system of n stationary charges at different distances. The charges which surround the charge $q_1$ are $q_2, q_3 …..$ and $q_n$. We know that these electrical charges exert electrostatic force on each other, which can be measured by the coulomb’s Law. But what is the electrostatic force on the charge $q_1$ due to all charges? This is the situation where the superposition principle comes into play.

Superposition principles – According to this principle, force on any charge is the vector sum of all the forces on that charge due to the presence of other charges at a time. The main feature of this principle is that the individual charge remains unaffected due to the presence of other charges.

The charge which we have thought of in the open space i.e the charge $q_1$, will experiences force from the other charges i.e $q_2, q_3 …$, and $q_n$.

The force experienced by the charge $q_1$ due to other charges $q_2, q_3…$ and $q_n$ is given as – $F_{12}, F_{13} …. \text{and}\; F_{1n} \qquad…(1)$

According to the superposition principle, the total force experienced by the charge $q_1$ is given as – $$F = F_{12} + F_{13} + …. + F_{1n}$$ If the distance between the charge $q_1$ and $q_2$ is denoted by $r_{12}$ and $r_{21}$ is the unit vector from the charge $q_2$ and $q_1$, then $$F_{12} = \frac{q_1 q_2}{4\pi\epsilon_0 r_{12}^2}r_{21}$$

Similarly, the force on the charge $q_1$ due to other charges is – \begin{align}F_{13}& = \frac{q_1 q_3}{4\pi\epsilon_0 r_{13}^2}r_{31}\\ F_{1n} &= \frac{q_1 q_n}{4\pi\epsilon_0 r_{1n}^2}r_{n1}\end{align} Putting these force in the equation (1) , then we get –

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### Home > MC1 > Chapter 7 > Lesson 7.2.4 > Problem7-83

7-83.

Consider the diagram of three pieces of lumber shown below.

$\underline{{4}\text{ meters}} \quad \underline{\ \ \ {x}\ \ \ } \quad \underline{\ \ \ {x}\ \ \ }$

1. If $x = 6$ meters, how long are all three pieces of lumber laid end to end?

Since $x = 6$ meters, the three pieces of lumber now look like this:

$\underline{{4}\text{ meters}} \quad \underline{{6}\text{ meters}} \quad \underline{{6}\text{ meters}}$

Can you find the total length of the lumber pieces?

1. If the total length of all three pieces laid end to end is $20$ meters, what is the value of $x$?

Using the diagram above, can you write an expression to represent the pieces of lumber?

If $4 + x + x = 20,$ why must $x + x = 16$?

Can you find the value of $x$?