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# Mean Reversion Tests on DARWIN $DWC In a previous post – Quantitative Modeling for Algorithmic Traders – we discussed the importance of Expectation, Variance, Standard Deviation, Covariance and Correlation. In this post we’ll discuss how those concepts can be applied to DARWIN assets. As a practical example, we will employ a series of statistical tests to assess if DARWIN$DWC is a Mean Reverting time series or otherwise.

These will include:

1) Hurst Exponent
3) Half-life of Mean Reversion

In case you missed it, the mean reverting nature of DARWIN $DWC was discussed in our most recent post here. Tests will be conducted on 1-Minute returns from$DWC, results and interpretation being published along the way. As always, please share your comments, feedback and suggestions in the comments at the end.
Note: Different statistical tests don’t always lead to similar outcomes, therefore it’s considered good practice to use at least two when evaluating mean reversion or any other statistical properties.
Before proceeding further, it’s important that we understand what Autocorrelation and Stationarity are.

## Autocorrelation:

Also referred to as Serial Correlation.
It is a measure of the similarity or relationship between a time series and a delayed or “lagged” version of the same time series, over successive periods in time.

## Stationarity:

A time series is considered stationary if its core statistical attributes remain constant over time.
These include mean, variance, standard deviation, autocorrelation, etc.
Stationary series demonstrate high predictability.

If a time series (e.g. DARWIN) can be mathematically transformed to approximately stationary, future Quotes of the time series (or trade entry direction / entries) can be reverse engineered from future points in its forecasted stationary series.
More on this in future blog posts.

## Prior Assumptions:

Prior to conducting these tests on $DWC data, we are expecting to see a reasonable degree of mean reversion for the following reasons: 1. There is visual confirmation (see below) that mean reverting tendency may exist. 2. As$DWC behaves in relation to real time trader sentiment, it is reasonable to assume that it could exhibit cyclical behaviour.

## Mean Reversion Test #1: Hurst Exponent

Mean Reversion in a time series can be assessed in terms of its rate of diffusion from inception.

For a time series X to be considered mean reverting:

##### Rate of Diffusion (X) < Rate of Diffusion of a Geometric Random Walk (GBM)

This rate of diffusion can be measured as the variance of the logarithm of the time series, at a random time interval T:

##### $Var(T) = \left \langle \left | log(t + T) – log(t) \right |^{2} \right \rangle$

If a time series is a GBM, then Var(T) ~ T, as T gets larger:

##### $\left \langle \left | log(t + T) – log(t) \right |^{2} \right \rangle$ ~ T

If a time series is either trending or mean reverting, then:

##### $\left \langle \left | log(t + T) – log(t) \right |^{2} \right \rangle$ ~ $T^{2H}$

.. where H is the Hurst Exponent, a measure of the extent to which the time series trends or mean reverts.

Hurst Exponent Interpretation:
If H > 0.5, the time series is TRENDING
If H < 0.5, the time series is Mean Reverting
If H = 0.5, the time series is a Geometric Random Walk

The DWC’s Hurst Exponent can be easily calculated in R, using the “pracma” library.
Note: For all code examples in this blog post, we have pre-loaded M1 data as “DWC.M1” to save time.

library(pracma)# Print M1 data Hurst Exponent > hurstexp(log(DWC.M1$quote))Simple R/S Hurst estimation: 0.8962816 Corrected R over S Hurst exponent: 0.9945418 Empirical Hurst exponent: 1.001317 Corrected empirical Hurst exponent: 0.9938308 Theoretical Hurst exponent: 0.520278 This first test shows that though this sample of DWC data is not demonstrating mean reverting behaviour (Theoretical Hurst Exponent > 0.5), it is not trending significantly either -> i.e. it is almost behaving like a GBM as per this test’s results (H = 0.520278), reducing the probability of DWC being a non-stationary random walk process. ## Mean Reversion Test #2: Augmented Dickey-Fuller Test If the$DWC time series is not a random walk (non-stationary series), then any Quote in the series will have a proportional relationship with the Quote immediately before it.

### Interpretation of ADF Test Results

Referring back to the autoregressive model earlier:

###### $\Delta x_{t} = \alpha + \beta t + \gamma x_{t-1} + \delta _{1}\Delta x_{t-1} + … + \delta _{p-1}\Delta x_{t-p+1} + \epsilon _{t}$

z.lag.1 = The value of the test-statistic γ (gamma) in the above equation.
tau2 = Critical values corresponding to the null hypothesis (γ = 0)
In order to reject the null hypothesis (γ = 0 – i.e. to reject that DWC is a non-stationary random walk), the value of the test statistic must be smaller than the critical values in tau2 (1%, 5% and 10% confidence intervals).
As z.lag.1 is -3.0566 (smaller than the critical values for the 5% and 10% confidence intervals), the null hypothesis can be rejected at the 90% and 95% confidence intervals, i.e. the probability of DWC being stationary (or not a random walk) is very high.

##### The tests above were also conducted on 30-minute, 1-hour, 2-hour, 4-hour and Daily precision $DWC data. 1. Daily precision lead to the null hypothesis for the presence of a unit root being rejected at the 90% confidence interval. This test will be repeated periodically as more data is accrued over time. 2. 30-minute, 1-hour, 2-hour and 4-hour tests all lead to the null hypothesis for the presence of a unit root being rejected at the 95% confidence interval. ## Mean Reversion Test #3: Half-life of Mean Reversion An alternative to the autoregressive linear model described above, is to consider how long any particular time series takes “to mean revert”. By definition, a change in the next periodic value of a mean-reverting time series is proportional to the difference between the historical mean of the series and the current value. Such time series are referred to as Ornstein-Uhlenbeck processes. The differential of the earlier model leads us to the expected value of x(t): ##### $E(x_{t}) = x_{0}e^{\gamma t} – \frac{\mu }{1 – e^{\gamma t}}$ If DWC is a mean reverting series, and has a negative $\gamma$, then the equation above tells us that DWC prices decay exponentially, with a half-life of $\frac {-log(2)}{\gamma}$. ### This means we now have two tasks ahead of us: 1. Find $\gamma$ and check if it is negative. 2. Calculate the half-life and assess whether it is a practical length of time for traders to consider a mean reverting strategy on DWC. Once again, we can easily conduct both steps in R. ### Step 1: Calculate $\gamma$ and check sign. > M1.data <- as.ts(DWC.M1$quote) > M1.data.lag <- lag(M1.data, -1) > M1.data.delta <- diff(M1.data)> M1.data.frame <- cbind(M1.data, M1.data.lag, M1.data.delta) > M1.data.frame <- M1.data.frame[-1,]
> M1.regression <- lm(M1.data.delta ~ M1.data.lag, data=as.data.frame(M1.data.frame))
> gamma <- summary(M1.regression)$coefficients[2] > print(gamma) [1] -0.0003588994 $\gamma$ is negative (-0.0003588994), so this$DWC 1-minute data sample can be considered mean reverting.

### Step 2: Calculate half-life and assess practicality of mean reversion strategy.

> M1.data.half.life <- -log(2) / gamma> print(paste("Half-life: ", M1.data.half.life, " minutes, or ", M1.data.half.life/60, " Hours", sep="")) [1] "Half-life: 1931.31306610404 minutes, or 32.1885511017341 Hours"
The half-life calculated for this $DWC 1-minute data sample is 32 hours. ##### Another important feature of the calculated half-life, is that it can be used as the period of a moving average employed in a mean reverting trading strategy[1]. If we plot a Simple Moving Average of period 1931 (in minutes, not hours), we get: ## Summary: 1. We conducted three statistical tests to ascertain the degree of mean reversion in$DWC 1-minute data, namely Hurst Exponent, Augmented Dickey-Fuller (ADF) and Half-Life of Mean Reversion.
2. Hurst Exponent did not indicate mean reverting behaviour in the $DWC, but a rather close estimate for possible GBM behaviour. 3. The Augmented Dickey-Fuller test results indicated stationary behaviour at the 95% confidence interval. 4. The Half-life of Mean Reversion test indicated$DWC possesses mean reverting properties.
5. We used the half-life calculated above as the period for a moving average, which when plotted on the chart revealed mean reverting Quote behaviour.

##### References:

[1] Chan, Ernest, 2013. Algorithmic Trading: Winning Strategies and Their Rationale, John Wiley and Sons.

Additional Resource: Measuring Investments’ Risk: Value at Risk (VIDEO)
* please activate CC mode to view subtitles.

Do you have what it takes? – Join the Darwinex Trader Movement!

• ###### Martyn Tinsley
Posted July 22, 2017 at 8:42 am

A very interesting and well written article. Clearly a very complex subject. Luckily due to my own research around cointegration between pairs of instruments and the mean reversion that exists there, I managed to follow most of the message.
However, a very important (and more practical) message that I (and other traders) need to hear if they are to use DWC in real life are things like:
1.How do we know the investment ratio of DWC we should use compared to the other traditional DARWINs in our portfolios. Is is 50% DWC, 50% other? Or is it more complex? If so how do we calculate?
2. Is the strategy around DWC buy and hold? I.e. It always forms a component of an investors portfolio. Or is it a case of buying DWC following the dips in anticipation of traditional DARWINs struggling and then selling when DWC reaches a high?
This current article in invaluable in terms of the background rationale and proof. However I expect many would have struggled with the maths, and therefore would not have benefited from the message. What I think the community would now really benefit from is a more grounded and practical article on its everyday use 🙂 Something that as a minimum should clearly cover my 2 points above, but I’m sure you can also provide many more practical insights and use cases.
Keep up the good work
Martyn

• Post Author
###### The Market Owl
Posted September 13, 2017 at 2:06 pm

Many thanks for the kind comments Martyn – more applied content on its way!

The Darwinex® brand and the http://www.darwinex.com domain are commercial names used by Tradeslide Trading Tech Limited, a company regulated by the Financial Conduct Authority (FCA) in the United Kingdom with FRN 586466, with company registration number 08061368 and registered office in Acre House, 11-15 William Road, London NW1 3ER, UK. and by Sapiens Markets EU Sociedad de Valores SA, a company regulated by the Comisión Nacional del Mercado de Valores (CNMV) in Spain under the number 311, with CIF A10537348 and registered office in Calle de Recoletos, 19, Bajo, 28001 Madrid, Spain.

CFDs are complex instruments and come with a high risk of losing money rapidly due to leverage. 64% of retail investor accounts lose money when trading CFDs with this provider. You should consider whether you understand how CFDs work and whether you can afford to take the high risk of losing your money.


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It is well known that Wilson loop is a quite hard observable to compute. In the case in which the QFT is dual to a gravitation theory in AdS space, we can use holography to compute the Wilson loop, which is given by

$$W(\mathcal{C}) = <\mathcal{P}\,e^{\oint_\mathcal{C} A_\mu dx^\mu}>$$
where $\mathcal{P}$ is the path-ordered product.

The contour $\mathcal{C}$ describes the worldline of the quark-antiquark pair on the field theory side which is created at an initial time and then annihilated.

The Wilson loop on the contour measures the potential energy of the pair and signals if the theory is confining.
A very massive quark in the field theory side can be seen as a open string in the bulk with one end-point attached to N D3 branes and the other one to another D3 brane which is sent to infinity (the boundary of AdS) to make the mass very large (infinity), see the next figure.

However, in many reviews, it's mentioned that the contour $\mathcal{C}$ can be seen as the boundary of a minimal surface worldsheet of a string. As in the following figure, is this string attaching to the quark and anti-quark? If yes, is it an open string?

I am confused whether the string is attaching both the quark and the anti-quark or the quarks themselves are the strings. If they are the strings, how do you close the worldsheet?

+ 1 like - 0 dislike

Indeed, on the gravity side the Wilson loop is computed by evaluating an area of the minimal surface ending on the boundary at the Wilson loop.

On the gauge theory side there are no strings, because it is just SYM theory, so the quarks and anti-quarks are point particles.

answered Jul 7, 2017 by (904 points)

Sorry but this does not answer my question...

@apt45 Hm... On the gauge side of the duality there are no strings, because it is a theory of point particles -- SYM theory. On the gravity side, the object that corresponds to the Wilson loop of the SYM is the surface of the worldsheet of the open string ending on the boundary. Is it OK now?

I know you don't have strings on the gauge theory side. In the gravity side instead, you can attach a string to the quarks living at the boundary and make this worldsheet

@apt45 There are no quarks living on the boundary on the gravity side. There is just an open strings going to conformal boundary of AdS, which doesn't support any dynamical degrees of freedom.

+ 1 like - 0 dislike

I think the string is attaching to the D-brane and the two end points of the open string are quark and anti-quark, respectively. In the figure, the world-sheet is still a world-sheet of open string (take the vertical line as the time direction, say). It looks different from normal world-sheets for open strings because now the trace of the end points is a loop. Note that, the world-sheet for a closed loop can not be attached to a D-brane.

answered Jul 7, 2017 by (195 points)

This is also strange, because in the gravity side we have type II B superstring on $AdS_5 \times S_5$ which is a theory of only closed strings... don't?

@apt45 Yes, in the gravity side, we have type 2B superstring on $AdS_5\times S_5$ which is a theory of only closed strings. But in your post, we are considering the gauge theory side, aren't we? In the both sides of AdS/CFT, we always have a D3-brane in 10 dimensional space-time. When the D3-brane is considered in the perturbative regime, it is described by the fields on the D3-brane which are induced by the massless excitations of open strings. However, when this exactly the same system is in the non-perturbative regime, the D3-brane is described as a solitonic object of superstring/suprgravity theory which inducing the tyep 2B superstring theory in the near-horizon region. That is, the duality is built on one and the same system but within different regimes. In any regime, we have the bulk and the D3-brane simultaneously. The conjecture now is, in any regime, the system can be euqally be described by either of these two theories which are actually the same theory. Hence, when the t' Hooft coupling is strong, we do not know how to solve the gauge theory but we shall still have the picture of open strings ending on the brane.

@apt45 Type IIB string theory is not a theory of only closed strings (as it is known since 1995). It is a string theory with chiral supersymmetries (which differs it from Type IIA) with identical left- and right-moving SCFTs (which differs it from Heterotic strings). There are odd-dimensional branes in the theory (in particular, the 3-brane which is a "base" of canonical example of AdS/CFT), on which open strings can end.

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[This article was first published on Data Imaginist, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

I’m happy to announce that farver 2.0 has landed on CRAN. This is a big release comprising of a rewrite of much of the internals along with a range of new functions and improvements. Read on to find out what this is all about.

## The case for farver

The first version of farver really came out of necessity as I identified a major performance bottleneck in gganimate related to converting colours into Lab colour space and back when tweening them. This was a result of grDevices::convertColor() not being meant for use with millions of colour values. I build farver in order to address this very specific need, which in turn made Brodie Gaslam look into speeding up the grDevices function. The bottom line is that, while farver is still the fastest at converting between colour spaces, grDevices is now so fast that I probably wouldn’t have bothered to build farver in the first place had it been like this all along. I find this a prime example of fruitful open source competition and couldn’t be happier that Brodie took it upon him.

So why a new shiny version? As part of removing compiled code from scales, we decided to adopt farver for colour interpolation, and the code could use a brush-up. I’ve become much more trained in writing compiled code, and further there were some shortcomings in the original implementation that needed to be addressed if scales (and thus ggplot2) should depend on it. Further, I usually write on larger frameworks and there is a certain joy in picking a niche area that you care about and go ridiculously overboard in tooling without worrying about if it benefits any other than yourself (ambient is another example of such indulgence).

## The new old

The former version of farver was quite limited in functionality. It had two functions: convert_colour() and compare_colour() that did colour space conversion and colour distance calculations respectively. No outward changes has been made to these functions, but internally a lot has happened. The old versions had no input validation, so passing in colours with NA, NaN, Inf, and -Inf would give you some truly weird results back. Further, the input and output was not capped to the range of the given colour space, so you could in theory end up with negative RGB values if you converted from a colour space with a larger gamut than sRGB. Both of these issues has been rectified in the new version. Any non-finite value in any channel will result in NA in all channels in the output (for conversion) or an NA distance (for comparison).

library(farver)
colours <- cbind(r = c(0, NA, 255), g = c(55, 165, 20), b = c(-Inf, 120, 200))
colours
##        r   g    b
## [1,]   0  55 -Inf
## [2,]  NA 165  120
## [3,] 255  20  200
convert_colour(colours, from = 'rgb', to = 'yxy')
##            y1        x        y2
## [1,]       NA       NA        NA
## [2,]       NA       NA        NA
## [3,] 25.93626 0.385264 0.1924651

Further, input is now capped to the channel range (if any) before conversion, and output is capped again before returning the result. The later means that convert_colour() is only symmetric (ignoring rounding errors) if the colours are within gamut in both colour spaces.

# Equal output because values are capped between 0 and 255
colours <- cbind(r = c(1000, 255), g = 55, b = 120)
convert_colour(colours, 'rgb', 'lab')
##             l        a        b
## [1,] 57.41976 76.10097 12.44826
## [2,] 57.41976 76.10097 12.44826

Lastly, a new colour space has been added: CIELch(uv) (in farver hcl) has been added as a cousin of CIELch(ab) (lch). Both are polar transformations, but the former is based on luv values and the latter on lab. Both colour spaces are used interchangeably (though not equivalent), and as the grDevices::hcl() function is based on the luv space it made sense to provide an equivalent in farver.

## The new new

The new functionality mainly revolves around the encoding of colour in text strings. In many programming languages colour can be encoded into strings as #RRGGBB where each channel is given in hexadecimal digits. This is also how colours are passed around in R mostly (R also has a list of recognized colour names that can be given as aliases instead of the hex string - see grDevices::colour() for a list). The encoding is convenient as it allows colours to be encoded into vectors, and thus into data frame columns or arrays, but means that if you need to perform operations on it you’d have to first decode the string into channels, potentially convert it into the required colour space, do the manipulation, convert back to sRGB, and encode it into strings. Encoding and decoding has been supported in grDevices with rgb() and col2rgb() respectively, both of which are pretty fast. col2rgb() has a quirk in that the output has the channels in the rows instead of the columns, contrary to how decoded colours are presented everywhere else:

grDevices::col2rgb(c('#56fec2', 'red'))
##       [,1] [,2]
## red     86  255
## green  254    0
## blue   194    0

farver sports two new functions that, besides providing consistency in the output format also eliminates some steps in the workflow described above:

# Decode strings with decode_colour
colours <- decode_colour(c('#56fec2', 'red'))
colours
##        r   g   b
## [1,]  86 254 194
## [2,] 255   0   0
# Encode with encode_colour
encode_colour(colours)
## [1] "#56FEC2" "#FF0000"

Besides the basic use shown above, both function allows input/output from other colour spaces than sRGB. That means that if you need to manipulate some colour in Lab space, you can simply decode directly into that, do the manipulation and encode directly back. The functionality is baked into the compiled code, meaning that a lot of memory allocation is spared, making this substantially faster than a grDevices-based workflow:

library(ggplot2)

# Create some random colour strings
colour_strings <- sample(grDevices::colours(), 5000, replace = TRUE)

# Get Lab values from a string
timing <- bench::mark(
farver = decode_colour(colour_strings, to = 'lab'),
grDevices = convertColor(t(col2rgb(colour_strings)), 'sRGB', 'Lab', scale.in = 255),
check = FALSE,
min_iterations = 100
)
plot(timing, type = 'ridge') +
theme_minimal() +
labs(x = NULL, y = NULL)

Can we do better than this? If the purpose is simply to manipulate a single channel in a colour encoded as a string, we may forego the encoding and decoding completely and do it all in compiled code. farver provides a family of functions for doing channel manipulation in string encoded colours. The channels can be any channel in any colour space supported by farver, and the decoding, manipulation and encoding is done in one pass. If you have a lot of colours and need to increase e.g. darkness, this can save a lot of memory allocation:

# a lot of colours
colour_strings <- sample(grDevices::colours(), 500000, replace = TRUE)

darken <- function(colour, by) {
colour <- t(col2rgb(colour))
colour <- convertColor(colour, from = 'sRGB', 'Lab', scale.in = 255)
colour[, 'L'] <- colour[, 'L'] * by
colour <- convertColor(colour, from = 'Lab', to = 'sRGB')
rgb(colour)
}
timing <- bench::mark(
farver = multiply_channel(colour_strings, channel = 'l', value = 1.2, space = 'lab'),
grDevices = darken(colour_strings, 1.2),
check = FALSE,
min_iterations = 100
)
plot(timing, type = 'ridge') +
theme_minimal() +
labs(x = NULL, y = NULL)

## The bottom line

The new release of farver provides invisible improvements to the existing functions and a range of new functionality for working efficiently with string encoded colours. You will be using it indirectly following the next release of scales if you are plotting with ggplot2, but you shouldn’t be able to tell. If you somehow ends up having to manipulate millions of colours, then farver is still the king of the hill by a large margin when it comes to performance, but I personally believe that it also provides a much cleaner API than any of the alternatives.


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# Question #7229d

May 7, 2017

The original pressure was $250 \cdot m m \cdot H g$...........

#### Explanation:

Old Boyle's Law holds that for a given quantity of gas, at a given temperature, $P$ varies inversely according to $V$

$i . e . P \propto \frac{1}{V}$, and thus $P V = k$, where $k$ is some constant of proportionality; and thus the defining relationship when we solve for $k$:

${P}_{1} {V}_{1} = {P}_{2} {V}_{2}$

And thus ${P}_{1} = \frac{{P}_{2} {V}_{2}}{V} _ 1 = \frac{500 \cdot m m \cdot H g \times 0.25 \cdot L}{0.50 \cdot L}$

$= 250 \cdot m m \cdot H g$.

How many atmospheres of gas pressure does this represent? Is the drop in pressure consistent with the details of the experiment?


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# Calculators and complex numbers (Part 7)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number $z = a+bi$ is

$z = r(\cos \theta + i \sin \theta)$

where $r = |z| = \sqrt{a^2 + b^2}$ and $\tan \theta = b/a$, with $\theta$ in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

There’s a shorthand notation for the right-hand side ($r e^{i \theta}$) that I’ll justify later in this series.

This post builds off the previous two posts by completing the proof of De Moivre’s Theorem.

Theorem. If $n$ is an integer, then $\left[ r (\cos \theta + i \sin \theta) \right]^n = r^n (\cos n \theta + i \sin n \theta)$.

The proof has two parts:

1. For $n \ge 0$: proof by induction.
2. For $n < 0$: let $n = -m$, and then use part 1.

In the previous post, I presented how I describe the proof of step 1 to students. Today, I’ll discuss how I present step 2.

My personal opinion is that the proof of step 2 goes easiest when a numerical example is done first. Let $n = -5$. Students can usually volunteer the successive steps of this special case:

$\left[ r (\cos \theta + i \sin \theta) \right]^{-5} = \displaystyle \frac{1}{ \left[ r (\cos \theta + i \sin \theta) \right]^{5} }$

$= \displaystyle \frac{1}{r^5 (\cos 5 \theta + i \sin 5 \theta)}$

At this point, students usually want to multiply by the conjugate of the denominator. There’s nothing wrong with doing that, of course, but it’s more elegant to write the numerator in trigonometric form:

$= \displaystyle \frac{1(\cos 0 + i \sin 0)}{r^5 (\cos 5 \theta + i \sin 5 \theta)}$

$= r^{-5} (\cos [0-5\theta] + i \sin[0-5\theta] )$

$= r^{-5} (\cos [-5\theta] + i \sin[ - 5\theta])$,

which matches what we would have expected. Students are now prepared for the proof, which I try to place alongside the above computation for $n = -5$.

Proof for $n < 0$. Let $n = -m$. I tell students to imagine that $n$ is $-5$, so that $m$ is equal to (students volunteer) $5$. In other words, $n$ is negative and $m$ is positive. Then

$\left[ r (\cos \theta + i \sin \theta) \right]^{n} = \displaystyle \frac{1}{ \left[ r (\cos \theta + i \sin \theta) \right]^{-n} }$

I again remind everyone that $-n = m$ is positive, and so (like a good MIT freshman) the previous work applies:

$= \displaystyle \frac{1}{ \left[ r (\cos \theta + i \sin \theta) \right]^m }$

The remaining steps mirror the calculation above:

$= \displaystyle \frac{1}{r^m (\cos m \theta + i \sin m \theta)}$

$= \displaystyle \frac{1(\cos 0 + i \sin 0)}{r^m (\cos m \theta + i \sin m \theta)}$

$= r^{-m} (\cos [0-m\theta] + i \sin[0-m\theta] )$

$= r^{-m} (\cos [-m\theta] + i \sin[ - m \theta])$

$= r^{n} (\cos [n\theta] + i \sin[ n \theta])$,

thus ending the proof.

For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.


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## December 13, 2009

### Can Five-Year-Olds Compute Coproducts?

#### Posted by John Baez

Charlie Clingen points out the following article:

The abstract explains what the authors are up to:

Transitive inference, class inclusion and a variety of other inferential abilities have strikingly similar developmental profiles—all are acquired around the age of five. Yet, little is known about the reasons for this correspondence. Category theory was invented as a formal means of establishing commonalities between various mathematical structures. We use category theory to show that transitive inference and class inclusion involve dual mathematical structures, called product and coproduct. Other inferential tasks with similar developmental profiles, including matrix completion, cardinality, dimensional changed card sorting, balance-scale (weight-distance integration), and Theory of Mind also involve these structures. By contrast, (co)products are not involved in the behaviours exhibited by younger children on these tasks, or simplified versions that are within their ability. These results point to a fundamental cognitive principle under development during childhood that is the capacity to compute (co)products in the categorical sense.

A lot of the paper is taken up with explaining coproducts, which suggests a variant of the old Groucho Marx joke:

Coproducts are so simple, any five-year-old child can understand them. Quick, someone find me a five-year-old child!

But mathematicians who already understand coproducts might find this the most interesting part:

Children acquire various reasoning skills over remarkably similar periods of development. Transitive Inference and Class Inclusion are two behaviours among a suite of inferential abilities that have strikingly similar developmental profiles—all are acquired around the age of five years. For example, older children can infer that if John is taller than Mary, and Mary is taller than Sue, then John is taller than Sue. This form of reasoning is called Transitive Inference. Older children also understand that a grocery store will contain more fruit than apples. That is, the number of items belonging to the superclass is greater than the number of items in any one of its subclasses. This form of reasoning is called Class Inclusion. These two types of inference appear to have little in common. Transitive Inference typically involves physical relationships between objects, while Class Inclusion involves abstract relative sizes of object classes. Nonetheless, explicit tests of these and other inferences for a range of age groups revealed that success was attained from about the median age of five years.

Since Piaget, decades of research have revealed important clues regarding the development of inference, yet little is known about the reasons underlying these correspondences. A common theme in two recent proposals is the computing of relational information. In regard to Relational Complexity theory, the correspondence between commonly acquired cognitive behaviours is based on the maximum arity of relations that must be processed (e.g., tasks acquired after age five involve ternary relations, i.e., relations between three items). In regard to Cognitive Complexity and Control theory, the correspondence is based on the common depth of relation hierarchies. Although a relational approach to cognitive behaviour has a formal basis in relational algebra, certain assumptions must be made about the units of analysis. For tasks as diverse in procedure and content as Transitive Inference and Class Inclusion, it is difficult to see how the analysis of one task leads naturally to the other. For Relational Complexity theory, Transitive Inference is considered to involve the integration of two binary relations between task elements into an ordered triple, or ternary relation; whereas Class Inclusion is regarded as the integration of three binary relations between three sets of elements (one complement and two containments) into a ternary relation. For Cognitive Complexity and Control theory, Transitive Inference involves relations over items; whereas Class Inclusion involves relations over sets of items.

This theoretical difficulty is symptomatic of the general problem in cognitive science where the basic components of cognition are unknown. In the absence of such detailed knowledge, cognitive modelers have been forced to assume a particular representational format (e.g., symbolic, or subsymbolic). This approach, however, does not lend itself to the current problem, because the elements of Transitive Inference and Class Inclusion tasks (i.e., objects and classes of objects) do not share a common basis. Understandably, then, these sorts of behaviours have tended to be studied in detailed isolation, narrowing the scope for identifying general principles.

Category theory was born out of a desire to establish formal commonalities between various mathematical structures, and has since been applied to the analysis of computational structures in computer science. The seminal insight was a shift from objects as the primary focus of analysis to their transformations. Contrast, for instance, sets defined in terms of (the properties of) the objects they contain—Set Theory—against sets defined in terms of the morphisms that map to or from them—Category Theory. This insight motivates our categorical approach to the analysis of inference, and our way around the current impasse. In cognitive science, several authors have used category theory for a conceptual analysis of space and time, though we know of only one other application that has modeled empirical data. Since our application of category theory to cognitive behaviour is novel, we first introduce the basic category theory constructs needed for our subsequent analysis of Transitive Inference, Class Inclusion, and other paradigms. The analysis begins with a brief introduction of the sort of data our approach is intended to explain, which primarily concerns contrasts between younger and older children relative to age five, and correlations across paradigms. Finally, we extend our categorical approach to more complex levels of inference. Our main point is that, despite the apparent lack of resemblance, all these tasks are formally connected via the categorical (co)product, to be defined below. The significance of this result is that it opens the door to an entirely new (empirical) approach to identifying general principles, particularly in regard to the development of inferential abilities, that are less likely to be revealed by standard modeling methods.

How convincingly do the authors argue that products and coproducts are the unifying theme of the reasoning skills that develop around the age of 5? I’m not sure… they tackle this later in the paper, but I haven’t read it carefully yet.

Posted at December 13, 2009 5:27 AM UTC

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### Re: Can Five-Year-Olds Compute Coproducts?

The basic point seems to me to be that the more difficult tasks (the ones that children older than about 5 years are able to perform while children younger than about 5 years are not) involve analysing two pieces of data at once, while the simpler tasks (that even younger children are able to perform) involve analysing only one piece of data at a time (and other pieces are irrelevant even if present). Of course, this is just the sort of thing that may be modelled categorially with products.

Why use category theory at all? One answer may be to get other researchers to focus on the relevant points. While the others are worrying about the level of complexity of each piece of data (such whether it refers to individuals or to sets of individuals), the authors want to consider the relationships between the data.

From the introduction:

This theoretical difficulty is symptomatic of the general problem in cognitive science where the basic components of cognition are unknown. In the absence of such detailed knowledge, cognitive modelers have been forced to assume a particular representational format (e.g., symbolic [6], or subsymbolic [7]). This approach, however, does not lend itself to the current problem, because the elements of Transitive Inference and Class Inclusion tasks (i.e., objects and classes of objects) do not share a common basis. Understandably, then, these sorts of behaviours have tended to be studied in detailed isolation, narrowing the scope for identifying general principles.

Posted by: Toby Bartels on December 13, 2009 6:52 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

What a coincidence. Just over the past few days Todd and I were discussing Piaget and cognitive development , keeping in mind Piaget’s consideration of category theory (however superficial) in his theory of genetic epistemology, lo and behold, you brought up this paper! This should make some interesting reading.

For what its worth (and perhaps others may have thought along these lines), I have always suspected that there is a good reason why CT is able to unify so many diverse areas of mathematics, that reason being we are perhaps genetically endowed with cognitive structures that help us reason categorically with varying degrees of success. To me, a promising line of research in cognitive neuroscience would be one that verifies (perhaps elucidates) such structures. Of course, such structures are not meant to be located in specific parts of the brain; rather, they must exist as dynamic representations in the mind/brain.

Posted by: Vishal Lama on December 13, 2009 11:21 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

There are a range of related studies brought up in this thread. Also some nice papers in The Logical foundations of cognition by John Macnamara and Gonzalo E. Reyes.

Posted by: David Corfield on December 13, 2009 11:48 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Had a look a some of the papers in “The Logical foundations of cognition” since the prospect that category theory could be of use in psychology made me curious.
But I have to admit that I’m confused by the use of some of the math nomenclature. Is it just me?
If you take a look at the introduction of chapter 5 “Category Theory as a Conceptual Tool in the Study of Cognition” you’ll find:

“2. Category theory provides means to circumscribe and study
what is universal in mathematics and other scientific disciplines.”

Here “universal” is meant to denote something that several scientific disciplines have in common - somewhere else “universal” is explicitly used to denote some structure that all natural languages have in commen, yet it is also introduced to denote what is called a “universal object resp. property” in category theory. This is off putting.

Do they do this on purpose? Has my math education spoiled me?

Posted by: Tim vB on December 14, 2009 1:45 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Tim wrote:

Has my math education spoiled me?

Yes. You are no longer comfortable with ordinary language.

I think the sentence you quote is using ‘universal’ in the ordinary sense, not the technical one. We can’t expect people in cognitive psychology to avoid using all words that have been borrowed by mathematicians and given technical meanings. Of course this can get a bit confusing when applying category theory to cognitive psychology. But I think we should be forgiving, because these applications are potentially important — and the subject may advance more quickly if some mathematicians contribute. Of course one thing mathematicians are good at doing is detecting and rejecting ambiguity and baloney, and this is one way to contribute. But if an effort seems sincere, and has interesting ideas behind it, it seems unfortunate to focus on the fact that the authors don’t write exactly the way mathematicians would.

Are there interesting ideas in the chapter you read?

Posted by: John Baez on December 14, 2009 3:52 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I’ll report back the moment I understand something…
Some moments after my post it occured to me too that a psychologist might even strive to use “universal” both in the ordinary sense and in the technical sense (one the same page!) in order to MOTIVATE the abstract term to his intended audience.

As a mathematician I instinctively avoid that (just think about “The function of an operator is to function as a function of functions, while a function that operates on operators is a function with operators functioning as argument.”)

But I’ll try to adapt :-)

Posted by: Tim vB on December 14, 2009 4:47 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I’ll try to explain my understanding why category theory is of interest to cognitive science at all:
Cognitive science as understood here tries to describe processes of learning that are universial in the sense that they do not depend on time, culture or other environmental influences. (The concurrent use of universal in this sense and in the sense of category theory - vexing to me on the first reading - seems to be deliberate and is intended to point out the analogy of the two concepts).

If you try to understand how a child learns to count, you need a theoretical model of counting. You may use set theory, use finite sets (or rather the isomorphism classes of finite sets, isomorph = having equal numbers of elements) to model numbers and set-theoretic constructions to model arithmetic operations.
Here’s the problem: To model addition, you need a concept of disjoint union of finite sets. This concept is not “universial”, because there is no “natural = canonical way” to define it.

If you buy yourself a category, you get a “universial” definition (in the sense of canonical) of disjoint union of sets for free, you substitute it with the definition of the product of two objects (which is a universal object in the sense of category theory).

Thus I think I’m not mistaken to state that the motivation to get interested in category theory is very close to the reasons that mathematicians are thinking about it.

(I’ll stop here, but there are also nice papers about children learning nouns from adults, by being shown objects, “look, there is a dog!” It’s fun to compare these to your personal experience, if you happen to have children. My nephew had already aquired the use of the noun “dog” at a very young age, when we met a dog and it’s owner during a walk. The owner pointed at her dog and said to my nephew “look, there is a wuff wuff” - which he instantly mimicked - his mother then told him “stop the bullshit, it’s a dog, for god’s sake!”).

Posted by: Tim vB on December 15, 2009 2:30 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

To model addition, you need a concept of disjoint union of finite sets. This concept is not “universial”, because there is no “natural = canonical way” to define it.

Wait, hold up.. I missed something huge here, because last time I checked disjoint union of finite sets was the coproduct in the category of finite sets.

Posted by: John Armstrong on December 15, 2009 4:11 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

First, the meta-question: My problem with these papers is that they mix trivial mathematics with subtle points about their vision of cognition science, which makes it hard to know if you understand what they are going for or not. (The math is trivial from my point of view, but, considering that the authors aren’t mathematicians but psychologists - with the exception of Lavwere - I hold them in high regard for this).
So, yes, “am I missing something huge here?” is my constant companion, too.

The point that Frangois Magnan and Gonzalo E. Reyes try to illustrate in chapter 5 with this example is IMHO this: They define the disjoint union of two sets A and B as union of A x {0} and B x {1}. See, the choice of 0 and 1 is arbitrary. The definition of a (co)product however does only use the very basic parts that a category is made of (objects, morphisms, composition of morphisms, existence of identity morphisms and associativity of composition). No arbitrary choices enter the scene after you have settled for the concept of category.

They do have other examples that justify the use of categories, I tried to pick the easiest one.

Posted by: Tim vB on December 15, 2009 4:40 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

They define the disjoint union of two sets A and B as union of A x {0} and B x {1}. See, the choice of 0 and 1 is arbitrary. […] They do have other examples that justify the use of categories, I tried to pick the easiest one.

How about product then? (For multiplication of numbers.) Even after the arbitrary choice of 0 and 1, you've still got an arbitrary choice of how to define the cartesian product. (Another arbitrary choice: whether to use A x {0} or {0} x A.)

Posted by: Toby Bartels on December 15, 2009 7:00 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

(Another arbitrary choice: whether to use A x {0} or {0} x A.)

(Actually, this is balanced by a corresponding choice in the definition of cartesian product; we shouldn't count it twice.)

It seems that what they really need here is structural set theory instead of material set theory.

Posted by: Toby Bartels on December 15, 2009 7:07 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Well yes, sure, you pick some definition of disjoint sets. Within set theory you also pick some definition of numerals (Church, von Neumann, etc, etc) as we’ve discussed many times in structuralism conversations here.

So it seems that if their objection is the one you cite, then they shouldn’t come anywhere near addition-as-disjoint-union before it crops up. They should note that counting numbers themselves are “non-canonical”.

Posted by: John Armstrong on December 15, 2009 7:40 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I'm guessing that they haven't swallowed the material Kool-Aid so much as to believe that everything is a set. (Really, I think that only mathematicians are ever that far gone.) So if I'm right, then they don't care about any method of representing numbers as sets as such.

However, they may still think of numbers as being cardinal numbers of sets (themselves sets of, say, blocks that children manipulate). And then once they get sets into the picture, then they worry about defining operations on them.

Posted by: Toby Bartels on December 15, 2009 8:09 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Toby wrote:

I’m guessing that they haven’t swallowed the material Kool-Aid so much as to believe that everything is a set. (Really, I think that only mathematicians are ever that far gone.)

I’m not sure which ‘they’ you mean.

If you mean the authors of Category theory as a conceptual tool in the study of cognition, namely Frangois Magnan and Gonzalo E. Reyes, it’s worth noting that Gonzalo Reyes is a mathematician.

But, it’s also worth noting that he’s an excellent category theorist! He wrote a great paper called ‘The history of categorical logic 1963-1977’ with Jean-Pierre Marquis, which unfortunately Marquis was unable to put on the arXiv. And he’s also written lots of papers on synthetic differential geometry — i.e., doing differential geometry in topoi that allow for infinitesimals.

So, he has certainly not swallowed the Kool-Aid of material set theory.

But what’s more interesting to me, ultimately, is what the 5-year-olds think about all this!

Posted by: John Baez on December 15, 2009 8:41 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I’m not sure which ‘they’ you mean.

I was thinking of the authors of the paper cited in the original post, but I guess that I should have been talking about Magnan & Reyes. And of course Reyes is an expert in category-theoretic logic, so no fears about him.

Posted by: Toby Bartels on December 15, 2009 9:22 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Well, I don’t have any five year olds around here and now, but my own experience with children who are virtually grown was that they could very easily deal with abstractions. I never attempted to teach them category theory — my bad. But I certainly taught my boys stuff that their peers did not know. They never feared negative numbers, became quasi-comfortable with modular arithmetic, and learned various algebraic facts were true via arithmetic experimentations.

I believe that the biggest flaw in education is to assume that young people cannot grapple with abstraction at an early age. The whole world is an category of abstractions to children. The problem with teaching children the neat mathematical stuff that is there in the world is that there are far too few educated adults to re-enforce the ideas.

Posted by: Scott Carter on December 16, 2009 1:19 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

What would be the question to ask a five year old to find out what (s)he thinks about coproducts?

If you try to find out what abstraction the human mind uses when it learns counting, transitive inference etc. you should look into dyscalculia.

There is much to learn from children who fail to learn certain abstractions. Here are two examples:

• Some children don’t understand the order relation of natural numbers (the way we do), i.e. why 1 < 2? As far as I now, we don’t yet understand the reason - but from my personal experience it could be something like “2apples are smaller than 1 pumpkin, so sometimes 2 is smaller than 1”.

• Some children don’t understand equality (the way we do): A = A is obviously false, because one of the A’s is on the left of the equality sign, one is on the right, they are obviously not the same. Or this: A = a is right, because both sides sound the same.

Posted by: Tim vB on December 16, 2009 9:37 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I do have a five-year old around so if anyone does come up with a good question to test whether or not five-year olds can compute coproducts, I’d be happy to try them out.

If you want comparative data, then I can also try them out on a four-year old, and if I ask the questions now and then wait a couple of weeks and ask them again I can get data for a six-year old, too.

(I guess there’s not a lot of point in asking the two-year old.)

For comparison, this particular five-year old is fine with limits: a common game in our house is for the children to cling on tightly to something (usually an adult’s leg) and declare “I’m a limit!”[1].

More seriously, he did make the leap from the statements “I’m the youngest in my class” and “I’m the eldest at home” to figuring out a group in which he was both the youngest and eldest. That’s a limit, right?

[1] s/(?<=m)i/pe/

Posted by: Andrew Stacey on December 16, 2009 11:13 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

aha! no 2 things can be equal
because then there wouldn’t be 2 of them

Posted by: jim stasheff on December 16, 2009 1:11 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

The notion of equality is subtle, which I learned from a child that had serious problems to understand it, which was quite an experience because the notion is so selfevident even to most little children!

I was reminded of this because one topic of the book “Category Theory as a Conceptual Tool in the Study of Cognition” is “reference”: If you point to a dog and tell a child “this is a dog”, how does it know what “dog” means? After all, it could be the proper name of that specific individual (like “Freddy”), it could refer to a part of the dog (“it is grey” or “look, the fur”) etc.

If you say A = A is true, you are using a list of abstract properties to identify both objects, but which are they? Let’s have a little test:

Is A = A true? (yes, but why do I think that?)

• a = A ? (no, but why not?)
• A = A ?
• A = A ?
• A (imagine this letter to be red, is there a way to change the color of text here?) = A ?

In order to learn the count noun “dog”, a child has to be able to understand the list of properties that identify an object as a “dog”, and I think this list is not constant, but depends on the context. “dog” may not be a good example of this (this is the one the authors use throughout the book), but try to list the properties that identify an object as a “table”.

(Remark: If you know little children you may have noticed that this learning process sometimes does not work and e.g. a child, for some time, calls every dog “Freddie”).

Posted by: Tim vB on December 16, 2009 3:05 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Maybe you already advanced the discussion far beyond the point the authors tried to make, the paper I talked about is an introduction to the subject and tries to illustrate and motivate the concept of a category to an audience that is unfamiliar with it.

This excerpt from page 61 could help to clarify this:

“The set-theoretical operations underlying the arithmetical ones are so natural that they are taken for granted when counting, which creates the impression that they are not involved in this process. We believe that these operations constitute universal “blueprints” of the mind and
we may ask what distinguishes them from others. Set theory does not give us too much help with this problem. Indeed, from a purely set-theoretical point of view, nothing seems to distinguish these operations from countless others that may be defined starting from two sets. To
make things worse, the definition of disjoint union is rather artificial. The main point is to transform the original sets so as to make them disjoint and this could be achieved in a variety of ways. How can one decide which, if any, is the right operation? It is at this point that category theory comes to our rescue with a new idea: instead of seeing a cartesian product as a set, let us view it as
a structure consisting of a set together with the relations between this set and the original ones. This structure satisfies a universal property which essentially characterizes it.”

Posted by: Tim vB on December 16, 2009 8:53 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

As I recall, Levi-Strauss states in no uncertain terms in several places that the sense of ‘structure’ that he uses is pretty much the same as in mathematics. I think Piaget’s introductory book on structuralism has group theory in it somewhere, for example. At a superficial level, structuralists were looking for (approximate) isomorphisms between various concrete structures (e.g. families). Perhaps one could say they tried to model a concrete collection of objects as a single category, with varying degrees of success.

Since category theory, on the other hand, is interested in functors between different kinds of structures, one might have suspected it to be a more suitably flexible framework for making the kind of comparisons anthropologists or cognitive scientists are interested in. So I had assumed that someone had already done something of this sort, convincingly or otherwise.

Posted by: Minhyong Kim on December 13, 2009 12:53 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Do you know the paper by Morava – On the canonical formula of C. Levi-Strauss:

The anthropologist Claude Levi-Strauss has formulated a theory of the structure of myths using a formalism borrowed from mathematics, which has been difficult to interpret, and is somewhat controversial. Nevertheless, Levi-Strauss’s old school chum Andre Weil took his work seriously, and in this note I propose an interpretation of Levi-Strauss’s ‘canonical formula’ in terms of an anti-automorphism of the quaternion group of order eight.

Posted by: David Corfield on December 14, 2009 5:37 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Re the title…

I am 38 years old and cannot compute coproducts :)

PS: Greetings from HK :)

Posted by: Eric on December 15, 2009 11:25 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Yes you can!

Posted by: Dan Piponi on December 16, 2009 1:27 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Who let Bob the Builder in here?

Posted by: Andrew Stacey on December 18, 2009 9:20 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

You’ve computed coproducts at least since the age of 5, Eric. For example:

Problem: Suppose Fred starts with an empty bucket. He puts an onion in bucket. Then he puts in a tomato. What’s in the bucket?

The hard part is not computing coproducts. The hard part is learning what ‘coproduct’ means.

PS - Lisa just got back from Hong Kong today! I really miss that place. Luckily I’ll get to visit it when we’re in Singapore next year.

Posted by: John Baez on December 16, 2009 3:03 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Suppose Fred starts with an empty bucket. He puts an onion in bucket. Then he puts in a tomato. What’s in the bucket?

That is a very basic example of having two think about two things at once. I can imagine that there is some age at which a child will answer simply ‘a tomato’. But surely that age is younger than four years?

Posted by: Toby Bartels on December 16, 2009 3:39 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I’ve been trying to analyse why I don’t like this example. I think it’s the bucket.

For me, the key step about colimits would be creating the colimit out of “nothing”: building it up from the pieces rather than starting with a container and putting all the pieces safely inside it and then working purely within that container. Of course, there always is a container but if it’s big enough then it’s not obvious.

So perhaps Lego is a better example! What’s the colimit of a collection of bricks? The Millennium Falcon, of course.

A slightly more serious example is the question: “I have 10 kroner, you have 5 kroner, how much money is there?”. This might still be too simplistic, but it springs to mind because of Calvin’s answer (that’s Calvin as in Calvin and Hobbes) which was (paraphrased): “10 kroner: that’s _my_ 5 kroner there and you’re not having any of it.”.

But I’m still not completely happy with this example so I shan’t test it out on my 5-year old. A better one would be where there’s a non-trivial intersection between the two sets. Maybe, “I’ve got all the pieces to build the plane and all the pieces to build the rocket but not all the pieces to build both at the same time.”

Posted by: Andrew Stacey on December 18, 2009 9:28 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Hehe :)

Yeah, I know, I’ve computed things that are “coproducts”, but to recognize it as a coproduct requires that you know what a category is and what a colimit is, etc. Looking at colimit diagrams still makes my brain hurt (although they are pretty).

PS: I don’t know if you heard (I mentioned it here and there), but I am in HK permanently. I moved here last month. Finally! :) So definitely let me know next time you’re in the neighborhood.

Posted by: Eric on December 16, 2009 3:43 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Eric wrote:

Yeah, I know, I’ve computed things that are “coproducts”, but to recognize it as a coproduct requires that you know what a category is and what a colimit is, etc.

Right. This is related to the distinction raised by Rajesh Kasturirangan.

Looking at colimit diagrams still makes my brain hurt (although they are pretty).

I’m convinced this ‘brain ache’ feeling is related to the growth of new neurons, which is now known to occur during learning. If so, it’s a good thing, not to be avoided — just like the ‘muscle ache’ feeling you get when you exercise.

I don’t know if you heard (I mentioned it here and there), but I am in HK permanently. I moved here last month. Finally! :) So definitely let me know next time you’re in the neighborhood.

You didn’t tell me, but you linked to your blog, and you explained it there. Congratulations!

Like you, I’m fleeing the sinking United States for Asia… but unfortunately, only for a year. It’s looking 99.9% certain that I’ll be spending a year at the Centre for Quantum Technologies in Singapore. And my friend Jiang-Hua Lu, who works on Poisson geometry, Lie groups, and Hopf algebras, has invited me to visit the University of Hong Kong, so I’ll almost certainly see you.

Posted by: John Baez on December 16, 2009 6:13 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

At the risk of sounding pedantic, I think it is worth distinguishing between:

Hypothesis A: Children compute coproducts (starting at age X or so)

Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

In cognitive science the distinction between hypothesis A and hypothesis B is quite important - what Chomsky has called the difference between competence and performance. For example, if A is true, we would be entitled to look for brain mechanisms that implement coproducts, while if B is the better hypothesis, it would be premature if not false to look for brain mechanisms.

In my admittedly biased view, good cognitive science has mostly happened when research has proceeded along directions that resemble hypothesis-B, resulting in competence theories. Unfortunately, the authors of this paper have not clarified their views on the matter, though it seems as if they have a hypothesis-A like view.

What I find interesting about the tasks that they test in children is that all of them involve a comparison between one kind of structure represented in one mental state and another structure represented in another mental state (such as the appearance/reality false belief test in the theory of mind segment). A strong case could be made that some category theory like formalism is needed to perform cross structural comparisons, especially if the comparison abilities are systematic and productive. However, arguing for category theory on grounds of systematicity is a competence argument, not a performance argument and the authors lose out on an opportunity as a result of their confusion between the two categories.

To give a linguistic analogy, English speakers are able to perform wh-movements (replace an assertion by a question, for example). Generative grammarians have tried to describe children’s knowledge of wh-movements using constraints on tree structures that represent the grammatical structure of a sentence. However, Chomsky and other generative grammarians never claimed that children actually compute those trees in their heads or that those tree operations are implemented somewhere in our brains.

Another analogy: we can clearly predict the motion of planets using differential equations, but no one believes that planets compute the differential calculus. (I should say ‘almost no one’, for there are people like Ed Fredkin, Norm Morgolus and Steve Wolfram who believe the universe IS a computer).

I find it strange that views that are demonstrably false and/or problematic in physics are so easily accepted when it comes to cognitive science and the philosophy of mind.

Posted by: RajeshKasturirangan on December 16, 2009 3:24 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

At the risk of sounding pedantic, I think it is worth distinguishing between: Hypothesis A: Children compute coproducts (starting at age X or so) Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

The Phillips/Wilson/Halford paper does not mention this distinction - or I missed it, too. But John Macnamara and Gonzalo E. Reyes do it in their paper “Foundational Issues in the Learning of Proper Names, Count Nouns and Mass Nouns”. Maybe it goes without saying from the point of view of most scientists in the field?

Excerpt from page 146:

“Since representations of the learner’s knowledge are at the core of our theory, we must clarify our notion of representation. A representation need not resemble what it represents in the way a map or a photograph do. For example, in dynamics the trajectory of a projectile is represented by a system of equations, which enable one to calculate height, velocity and direction at each point in the trajectory. There is no suggestion that the projectile itself is computing any of these values. Similarly, there is no suggestion in our representations that children express the knowledge in precisely the form we offer. Rather, we claim that they must represent the same information in some manner. In particular, we do not expect that children can tell other people the information they have learned. We do, however, expect them to reveal their knowledge indirectly, in their utterances and actions.”

RajeshKasturirangan said:

I find it strange that views that are demonstrably false and/or problematic in physics are so easily accepted when it comes to cognitive science and the philosophy of mind.

I grant others the benefit of the doubt, after all the methods used to aquire knowledge that work for mathematicians and for physicists are not universally applicable :-)

Posted by: Tim vB on December 16, 2009 4:27 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

You quote the paper saying:

For example, in dynamics the trajectory of a projectile is represented by a system of equations, which enable one to calculate height, velocity and direction at each point in the trajectory. There is no suggestion that the projectile itself is computing any of these values. Similarly, there is no suggestion in our representations that children express the knowledge in precisely the form we offer.

It can be put even more clearly by anyone who has tried to tutor an NCAA star basketball player in calculus.

Yes, we can express the problem of specifying a launch angle and force that will make the ball go through the hoop in terms of a calculus problem. And yes, he makes some absurd percentage of his free throws. But he most certainly isn’t thinking of it in terms of calculus problems, or he’d do a lot better on these exams!

Posted by: John Armstrong on December 16, 2009 7:08 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I had an interesting chat the other day with an ex-colleague now at the engineering department in Cambridge. He contrasted two ways to have a machine move a cart along a track so as to balance a rod upright which is attached to it. Either you work out the equations of motion for the rod and use control theory, or else you use machine learning techniques and some learning experience with feedback of how well its doing.

It appears that the latter method works very well with little ‘prior knowledge’ encoded in the algorithm. This also worked for balancing a jointed rod, but as yet only in a simulation.

There’s an interesting challenge now to see which technique works to allow a robot to ride a unicycle. It would not surprise me if the machine learning method worked best - clearly we don’t know the equations of motion for a bicycle when we learn to ride.

Posted by: David Corfield on December 17, 2009 9:35 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

If we compare different machine learning algorithms to humans, maybe chess could be of interest: What do chess masters think when contemplating a specific situation and is it possible to get computers to do the same?

The wikipedia article (chess psychology) does not mention this, unfortunatly, here is a book about what I mean:

The book I read is available in German only, it would seem:

Reinhard-Munzert (amazon)

One point of interest: Munzert describes chess masters that record their stream of conciousness, which revealed that they don’t dig deeper than the average player, but instinctivly/unconciously pick the best moves and spend their time thinking about those. The average player wastes his time thinking about other moves.

For some time developers of chess programs tried to understand this and to implement it - via heuristic algorithms - in computer programs. These programs lost to the brutal force approach (compute every possibility until you run out of time) somewhere during the 1990ies, I don’t know if anyone still tries to advance in this direction.

Posted by: Tim vB on December 17, 2009 10:54 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Rodney Brooks style robotics is based on the idea that knowing-how (to ride a bicycle, for example) doesn’t require detailed descriptions of the underlying dynamics. In contrast, the equations of motion that capture the laws of classical mechanics are a form of knowing-that.

To reverse the robot riding a bicycle challenge, I would be very curious to find out what happens when you train the best machine learning algorithms on the kind of data that Kepler and Copernicus used to study planetary motion. Are they as good as the classical Newtonian solutions? Can they handle 3-body problems?

In his classic paper “Intelligence Without Representation” Brooks gives good arguments as to why we do not need internalized equations of motion to solve robotic navigation etc. Of course, Brooks style subsumption architecture doesnt run on machine learning algortithms, but Tommy Poggio made a similar argument on the basis of his work on Support Vector Machines (see Poggio’s “How the Brain Works”).

Other cognitive scientists - J. J. Gibson, Francisco Varela etc - have also made a similar argument. There is now a pretty thriving field in cognitive science called enactive cognition that is trying to shift the focus of cognitive epistemology from knowing-that to knowing-how.

Posted by: RajeshKasturirangan on December 17, 2009 2:37 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Yes, in highschool I was very frustrated that I could calculate how I should throw the ball, but could not do it.

But aren’t we working towards another necessary refinement? We already distinguish if humans use categories (Hypothesis A) or if categories are a useful tool to describe how humans manage their knowledge (Hypothesis B).
If you look at Hypothesis A you can further distinguish if humans know that they use categories (or calculus) or if they don’t.
Your star basketball player could have a robotic arm with a microcontroller that has a program performing (approximatly) calculus, he neither needs to know that nor to understand how it works :-)

The passage I quoted seems to state that the authors don’t claim that Hypothesis A is true, nor that it can be tested by asking children how they manage their knowledge, because even if Hypothesis A is true, you cannot expect them to be able to know and explain it.

Posted by: Tim vB on December 17, 2009 9:48 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Rajesh Kasturirangan wrote:

At the risk of sounding pedantic, I think it is worth distinguishing between:

Hypothesis A: Children compute coproducts (starting at age X or so)

Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

I agree that this is a crucial distinction! Of course when it comes to 5-year-olds, I only think Hypothesis B is interesting. My title ‘Can Five-Year-Olds Compute Coproducts?’ was merely meant to be eye-catching, amusing and terse.

I assume that any cognitive psychologist who hasn’t been asleep for many decades is aware of this distinction. Like Tim vB, I am willing to grant them the benefit of the doubt instead of demanding that they demonstrate their awareness of it.

Posted by: John Baez on December 18, 2009 4:29 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

As a mathematician turned cognitive scientist, my experience is that psychologists are aware of the distinction between hypotheses A and B but they often slip when doing their work and writing their papers as can be seen in the PLOS paper that started this discussion, and unlike the Macnamara-Reyes paper Tim vB cited (which I am certain is because of Macnamara’s familiarity with Chomskian linguistics; Macnamara is a particularly insightful cognitive psychologist).

Chomsky made the distinction between competence and performance precisely because he wanted to address the concerns of psychologists and others who were critiquing his work on Hyp A grounds when the claims were being made on Hyp B grounds.

Posted by: Rajesh Kasturirangan on December 18, 2009 5:56 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

My colleague Chris Athorne sent me the following by email, which I’m reposting here with his permission.

I still maintain that I taught Thomas how to “count” before he could speak by giving him a set of cards with numbers of dots marked on them and showing him that he should sort them into correspondence classes, which he was able to do. Cards of the same … er, Card … were not geometrically congruent so he did not have any patterns to offer clues. I told the child psychologist this and she gave me a look over her glasses as though I’d just claimed there were fairies at the bottom of my garden. Which, of course, there are but I thought it prudent not to go there!

Posted by: Tom Leinster on December 17, 2009 8:00 PM | Permalink | Reply to this

### Your left fusiform gyrus Mileage May Vary; Re: Can Five-Year-Olds Compute Coproducts?

I believe Tom Leinster. My father was dubious about my claim that my son, younger than 18 months, knew the alphabet flawlessly. When my (Harvard Cum Laude) father visited, he tested my son with alphabet blocks. “Show me the “X.”
“What is the name of this one?”

In less than a year, my son was beating my Dad at chess.

I kept notes of his language acquisition. He jumped in one month from 3-word sentences (i.e. “It fall down.”) to this one, which stunned my father when my boy said so, still much younger than 2:

“Mommy, I want you to cut the tag from the back of Daddy’s pajama shirt.”

What we are born able to do, or predetermined to be able to acquire easily, varies widely. I observe this in Mathematics, in Reading (via left fusiform gyrus – which is necessary for normal, rapid understanding of the meaning of written text as well as correct word spelling – per Tsapkini et al. The orthography-specific functions of the left fusiform gyrus: Evidence of modality and category specificity. Cortex, 2009), and in Learning generally.

Education: Learning Styles Debunked
Journal References:

1. Rohrer et al. Increasing Retention Without Increasing Study Time. Current Directions in Psychological Science, 2007; 16 (4): 183 DOI: 10.1111/j.1467-8721.2007.00500.x
2. McDaniel et al. The Read-Recite-Review Study Strategy: Effective and Portable. Psychological Science, 2009; 20 (4): 516 DOI: 10.1111/j.1467-9280.2009.02325.x
3. Roediger et al. Test-Enhanced Learning. Taking Memory Tests Improves Long-Term Retention. Psychological Science, 2006; 17 (3): 249 DOI: 10.1111/j.1467-9280.2006.01693.x
4. Hal Pashler et al. Learning Styles: Concepts and Evidence. Psychological Science in the Public Interest, (in press)

Association for Psychological Science (2009, December 17). Education: Learning styles debunked. ScienceDaily. Retrieved December 17, 2009, from http://www.sciencedaily.com /releases/2009/12/091216162356.htm

Posted by: Jonathan Vos Post on December 17, 2009 10:46 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Why was the child psychologist sceptical?

As a child psychologist she was surely aware that humans differ considerably in their development during childhood, and that an abstract concept like counting does not rely on language (see the introductory remarks about nonverbal thinking in this essay about human intelligence: Norimasa Kobayashi: An Essay on Human Intelligence ).

Posted by: Tim vB on December 18, 2009 12:45 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I don’t know about psychologists, but from my personal experiences with physicians, although they may recognize on an intellectual level that such variation exists, many seem to react skeptically (or assume evidence of some problem) when actually presented with individuals who differ significantly from the median. I’d guess many psychologists are similar.

Posted by: Mark Meckes on December 20, 2009 12:52 AM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Well, yes, I know this from my personal experience, too…but what was the reason in this case?

We know that even animals can count.
And of course we know of many Wunderkinder that were able to do amazing tasks at young ages. (I do not say that being able to count before you are able to speak proves that you are a Wunderkind).

One of my favorite stories is about Samuel Reshevsky who could beat world class players at chess when he was only a few years old. Asked how he was able to do it he used to say “I don’t now, it just happens, like breathing”.
As a child he toured the world. When he visited Vienna, people called Milan Vidmar, then one of the world’s top players, because no one else could hold a candle to him (this anecdote is mentioned briefly here: game agains Milan Vidmar). First, Vidmar did not take him seriously and lost, then gave his best and won, which caused Sammy to burst into tears. Vidmar later recalled that he never regretted to win a chess game more than on this occasion.

Posted by: Tim vB on December 20, 2009 3:28 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Conc. ‘abstract’ thinking and language, I find archeological hints about language-free precursors of modern humans interesting. On tool using skills , here is an interesting article.

Posted by: Thomas on December 20, 2009 12:39 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I think we have to distinguish between subitizing, and counting, the former is hard-wired into our brains, while the latter is acquired at a later age. Thus lot of animals can be demonstrated to be counting, but instead they are just using their brains hard-wired pattern recognition algorithm, while not understanding the underlying correspondence between numbers, and quantities.

Posted by: Németh Lukács on December 26, 2009 1:12 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

I think we have to distinguish between subitizing, and counting

Ah, so there may be a language barrier; what psychologists mean by ‘counting’ is more specific that what mathematicians mean by that term.

But if Tom used cards with large enough Card, then the psychologists should still accept it as counting.

Posted by: Toby Bartels on December 26, 2009 2:14 PM | Permalink | Reply to this

### Re: Can Five-Year-Olds Compute Coproducts?

Interesting! I did not know that. Follow up question would be if there is a clear way to distinguish the two, but one way that does not seem very promising would be to search for an upper bound of items to count for the subitizing ability to still work.

Why not?

Well, it would seem that even ants can “count” (i.e. subitize):

Do ants count their steps to find their way home?

Posted by: Tim vB on January 2, 2010 12:55 PM | Permalink | Reply to this

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# If $g(x) + x \sin g(x) = x^2,$ find $g'(0).$

## $\frac{d x}{d y}=\frac{-2 x^{4} y+x^{3}-6 x y^{2}}{4 x^{3} y^{2}-3 x^{2} y+2 y^{3}}$

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if this problem were given any question on and years to find their function. G when exes. Zira, let's first find the value of that function when x zero Solis for G zero. So we're gonna play zero in terms of eggs. So we have g 00 was zero times sine GOP zero as people to zero this term is zero. So from this BC, that G o P zero is zero. And let's tag narrative All think of the question. You have G primal X plus breaking news for a cool for a second term of left hand side, we have a sign your ex waas um X times consign g o eggs times deeper X is equal to x. No less plays you're in. You have g p zero plus sign G off zero plus zero times. Co sign g +00 times zero is zero. This storm is zero. We have signed you up zero but be known at jobs series itself, his ears off signing zero sign of serious. You're so from this we see that GI problems here put there a physical zero. So Jeep ERM of zero is the U

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# Scientific Notation

## Writing and reading scientific notation

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Scientific Notation

In science, measurements are often extremely small or extremely large.  It is inefficient to write the many zeroes in very small numbers like 0.00000000000000523.  Usually, the order of magnitude and the first few digits of the number are what people are interested in.  How should you represent these extreme numbers?

#### Guidance

Scientific notation is a means of representing very large and very small numbers in a more efficient way.  The general form of scientific notation is \begin{align*}a \cdot 10^b\end{align*}

The \begin{align*}a\end{align*}  is a number between 1 and 10 and most often includes a decimal.  The integer \begin{align*}b\end{align*}  is called the order of magnitude and is a measure of the general size of the number.  If \begin{align*}b\end{align*}  is negative then the number is small and if \begin{align*}b\end{align*}  is positive then the number is large.

\begin{align*}1, 240, 000 &= 1.24 \cdot 10^6\\ 0.0000354 &= 3.54 \cdot 10^{-5}\end{align*}

Note that when switching to and from scientific notation the sign of \begin{align*}b\end{align*}  indicates which direction and how many places to move the decimal point.

Multiplying and dividing numbers that are in scientific notation is just an exercise in exponent rules:

\begin{align*}(a \cdot 10^x) \cdot (b \cdot 10^y) &= a \cdot b \cdot 10^{x+y}\\ (a \cdot 10^x) \div (b \cdot 10^y) &= \frac{a}{b} \cdot 10^{x-y}\end{align*}

Addition and subtraction require the numbers to have identical order of magnitudes.

\begin{align*}1.2 \cdot 10^6-5.5 \cdot 10^5=12 \cdot 10^5-5.5 \cdot 10^5=6.5 \cdot 10^5\end{align*}

Example A

An electron’s mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg.

Write this number in scientific notation.

Solution: \begin{align*}9.109 \ 3822 \cdot 10^{-31}\end{align*}

Example B

The Earth’s circumference is approximately 40,000,000 meters.  What is the radius of the earth in scientific notation?

Solution:  The relationship between circumference and radius is \begin{align*}C=2 \pi r\end{align*}

\begin{align*}4.0 \cdot 10^7 &= 2 \pi r\\ r &= \frac{4.0}{2 \pi} \cdot 10^7 \approx 0.6366 \ldots \cdot 10^7=6.366 \cdot 10^6\end{align*}

Note that the number of significant digits required depends on the context.

Example C

Simplify the following expression.

\begin{align*}x=(4.56 \cdot 10^7) \cdot (2.89 \cdot 10^8) \div (7.15 \cdot 10^{-15})+216\end{align*}

Solution: \begin{align*}x=4.56 \cdot 2.89 \cdot 7.15 \cdot 10^0+216=310.22556\end{align*}

Concept Problem Revisited

In order to represent an extremely large or small number you should count the number of moves necessary for the decimal point to be directly after the first non-zero digit.  This count will be the order of magnitude and will be used as the exponent of 10 as a means of representing how large or small the number is.

#### Vocabulary

Order of magnitude is formally the exponent in scientific notation.  Informally it refers to size.  Two objects or numbers are of the same order of magnitude are relatively similar sizes.  A marble and a planet are not of the same order of magnitude, but Earth and Venus are.

Scientific notation is a means of representing a number as a product of a number between 1 and 10 and a power of 10.

#### Guided Practice

1. Order the following numbers from least to greatest.

\begin{align*}5.411 \cdot 10^{-3} \quad 7.837 \cdot 10^{-4} \quad 9.999 \cdot 10^3 \quad 9.5983 \cdot 10^{-7} \quad 8.0984 \cdot 10^3\end{align*}

2. Compute the following number and use scientific notation.

\begin{align*}2,000,000^3 \cdot 3,000^4\end{align*}

3. Simplify the following expression.

\begin{align*}(4.713 \cdot 10^7)+(8.985 \cdot 10^5)-(4.987 \cdot 10^2) \cdot (7.3 \cdot 10^{-6}) \div (6.74 \cdot 10^{-9})\end{align*}

1. First consider the order of magnitude of each number.  Small numbers have negative exponents.  If two numbers have the same order of magnitude, then compare the actual digits.

\begin{align*}9.5983 \cdot 10^{-7} < 7.837 \cdot 10^{-4} < 5.411 \cdot 10^{-3} < 8.0984 \cdot 10^3 < 9.999 \cdot 10^3\end{align*}

2. First convert each number to scientific notation individually, then process the exponent and multiplication.

\begin{align*}2,000,000^3 \cdot 3,000^4 &= (2 \cdot 10^6)^3 \cdot (3 \cdot 10^3)^4\\ &= 8 \cdot 10^{18} \cdot 81 \cdot 10^{12}\\ &= 648 \cdot 10^{30}\\ &= 6.48 \cdot 10^{32}\end{align*}

3. Resolve in order of standard order of operations

\begin{align*}(4.713 \cdot 10^7)+(8.985 \cdot 10^5)-(4.987 \cdot 10^2) \cdot (7.3 \cdot 10^{-6}) \div (6.74 \cdot 10^{-9})\end{align*}

\begin{align*}&= (4.713 \cdot 10^7) + (8.985 \cdot 10^5)-(5.40135 \cdot 10^5)\\ &= (471.3 \cdot 10^5)+(8.985 \cdot 10^5)-(5.40135 \cdot 10^5)\\ &= 474.8836499 \cdot 10^5\\ &= 4.748836499 \cdot 10^7\end{align*}

#### Practice

Write the following numbers in scientific notation.

1. 152,780

2. 0.00003256

3. 56, 320

4. 0.0821

5. 1, 000, 000, 000, 000, 000, 000, 000

6. 7.32

7. If the federal budget is \$1.5 trillion, how much does it cost each individual, on average, if there are 300,000,000 people?

8. The Library of Congress has about 60,000,000 items.  How could you express this number in scientific notation?

9. The sun develops \begin{align*}5 \times 10^{23}\end{align*}  horsepower per second.  How much horsepower is developed in a day? In a year with 365 days?

10. A light-year is about 5,869,713,600 miles.  A spacecraft travels \begin{align*}8.23 \times 10^4\end{align*}  miles per hour.  How long will it take the spacecraft to travel a light–year?

11. Compute the following number and use scientific notation: \begin{align*}324, 000 \cdot 30,000^3\end{align*} .

12. Compute the following number and use scientific notation: \begin{align*}14, 300 \cdot 20, 200^2\end{align*} .

Simplify the following expressions.

13. \begin{align*}(3.29 \cdot 10^4)-(3.295 \cdot 10^5)+(1.25 \cdot 10^2) \cdot (3.97 \cdot 10^{15}) \cdot (5.8 \cdot 10^{-6})\end{align*}

14. \begin{align*}(1.95 \cdot 10^2)+(6.798 \cdot 10^6)+(2.896 \cdot 10^3) \cdot (5.6 \cdot 10^{-3}) \div (2.89 \cdot 10^4)\end{align*}

15. \begin{align*}(2.158 \cdot 10^7) \cdot (1.679 \cdot 10^6) - (9.98 \cdot 10^4) \cdot (3.4 \cdot 10^{-2})\end{align*}

### Vocabulary Language: English

order of magnitude

order of magnitude

Formally, the order of magnitude is the exponent in scientific notation. Informally it refers to size. Two objects or numbers are of the same order of magnitude are relatively similar sizes.
Scientific Notation

Scientific Notation

Scientific notation is a means of representing a number as a product of a number that is at least 1 but less than 10 and a power of 10.
Standard Form

Standard Form

As opposed to scientific notation, standard form means writing numbers in the usual way with all of the zeros accounted for in the value.


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# The electromagnetic field tensor in curvilinear coordinate systems

I've been researching all things tensor so I can start studying general relativity. One of the main reasons to use tensors is because of their ability to describe physical laws in various coordinate systems. During my research, I found out about the electromagnetic field tensor, $F^{\mu\nu}$, and how it can be used to sum up Maxwell's equations in two equations, these: $$\partial_{\mu}F^{\mu\nu}=\mu_0J^{\nu}$$ $$\partial_{[{\gamma}}F_{\mu\nu]}=0$$ Wikipedia states the form of $F^{\mu\nu}$as this: $$F^{\mu\nu}=\begin{pmatrix}0 & -E_x/c & -E_x/c &-E_x/c\\E_x/c & 0 & -B_z & B_y\\E_y/c & B_z &0 & -B_x\\E_x/c &-B_y & B_x &0\end{pmatrix}$$ However, this definition is in Cartestian coordinates, which defeats the purpose of tensor equations being able to define in all coordinate systems. So, my question is how can I generally define the electromagnetic field tensor for all curvilinear coordinate systems?

NOTE: I have been here, https://www.physicsforums.com/threads/electromagnetic-field-tensor-in-curvilinear-coordinates.618984/ , where this same question was asked. However, I do not understand the way they explained their method, or really the method itself. Also, this is different from the question Electromagnetic tensor in cylindrical coordinates from scratch, because I am asking for a way to find it in all curvilinear coordinate systems, not just cylindical.

• Wouldn't it just be applying the coordinate transformation twice (once for each index)? For example, $F^{\mu\nu}=\Lambda^\mu_{\alpha}\Lambda^\nu_\beta F^{\alpha\beta}$? – Kyle Kanos Feb 21 '17 at 11:22

The explicit form for $F^{\mu\nu}$ is just one option to express $F^{\mu\nu}$ in cartesian coordinates using the EM field itself.

The definition over the four potential is coordinate independent and covariant: $$F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu.$$ For an overview the link provided by Leonardo Francisco Cavenaghi in his answer is a good starting point.

For a more detailed but still compact discussion of Maxwells equations and electromagnetism in flat and curved spacetime I can recommend this paper W. C. dos Santos, 2016, Introduction to Einstein-Maxwell equations and the Rainich conditions.

The appropriate definition uses the concept of manifolds and bundles. The electromagnetic tensor is in fact a $2-$form and in particular, the curvature of a particular bundle. You can find more informations here: https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism

The field-strength tensor $F$ can be described in an entirely coordinate-independent manner using the language of differential forms.

The gauge group of electromagnetism, $U(1)$, is one-dimensional and all structure constants vanish, that is, $f^\gamma_{\alpha\beta} = 0$. The $U(1)$ bundle over four-dimensional Minkowski space $M$ is $\mathbb R^4 \times U(1)$ and is trivial; this can also be seen by the fact $M$ is contractible to a point.

The gauge potential is simply $A = A_\mu dx^\mu$ and $F = dA = 2\partial_{[\mu}A_{\nu]} dx^\mu \wedge dx^\nu$ known as the field-strength corresponds simply to the curvature of the $U(1)$-valued connection or gauge potential.

We can make the identification (up to possibly factors of $i$) that $E_i = F_{i0}$ and $B_i = \frac12 \epsilon_{ijk}F_{jk}$ which makes no explicit reference to the coordinate system we must use.

As an aside, Maxwell's equations in vacuum remarkably reduce to $dF = d\star F = 0$; unpacking these gives you the equivalent of the Bianchi identity and the fact that $F$ is a closed form due to it being an exact form. These two conditions reduce to the four Maxwell equations.