updated: the slides are here.

]]>Everyone new is encouraged to start a thread, in category Chat, introducing themselves and their interests. The subject would be "Introduction: YourName"

]]>Good exposition of concepts, combining informal language with mathematical precision. Calculations get laborious at times.

]]>Category theory keeps on coming to my attention: there is a passage in Zen and the Art of Motorcycle Maintenance where Persig recalls repeatedly seeing references to some important area and compares it to driving along a road and repeatedly seeing signposts to some town. The message you get is that the town, area of study, or whatever, just might be important or at least relevant.

I tend to find that real life gets in the way of continuing study but I really do intend to keep up with this one. A deadline of 25th September is going to help :-)

]]>My interests have been for a long time in the uses of category theory and homotopy theory in providing tools for the description of systems of various types. I used to use categorical models of Petri nets and Discrete Event Systems in my lectures when teaching UG maths courses.

]]>- Keith Conrad, Tensor Products

Whereas in the world of vector spaces, tensors have a clearly visualizable representations, things become more subtle when we generalize to modules over a ring.

He writes:

There isn’t a simple picture of a tensor (even an elementary tensor) analogous to how a vector is an arrow. Some physical manifestations of tensors are in the previous answer, but they won’t help you understand tensor products of modules. Nobody is comfortable with tensor products at first. Two quotes by Cathy O’Neil and Johan de Jong nicely capture the phenomenon of learning about them:

O’Neil: After a few months, though, I realized something. I hadn’t gotten any better at understanding tensor products, but I was getting used to not understanding them. It was pretty amazing. I no longer felt anguished when tensor products came up; I was instead almost amused by their cunning ways.

de Jong: It is the things you can prove that tell you how to think about tensor products. In other words, you let elementary lemmas and examples shape your intuition of the mathematical object in question. There’s nothing else, no magical intuition will magically appear to help you “understand” it.

This is discouraging. Can we do better than this?

There is the construction of the tensor product as the quotient of enormous (free) module by an enormous sub-module, but it doesn't register with my intuition very well.

Regarding this, Conrad says:

From now on forget the explicit construction of M ⊗R N as the quotient of an enormous free module FR(M × N). It will confuse you more than it’s worth to try to think about M ⊗R N in terms of its construction.

He says instead to use the universal mapping property to understand the tensor product. But I don't like the idea of abandoning the definition of something in order to understand it.

Is this a case where it only makes sense to understand things though its morphisms? I hope not, because I like objects as well as arrows :)

]]>Here is a classic reference book:

- Python for Data Analysis, Wes McKinney, O'Reilly Media, 2013.

Here is a recommended primer from the Pandas website:

Here are the main components of the scientific python ecosystem. I am paraphrasing/quoting from McKinney:

NumPy. Short for numerical python, NumPy is the foundational package for scientific computing in Python. It provides a fast and efficient multi-dimensional array object; functions for performing element-wise computations with arrays or mathematical operations between arrays; tools for reading and writing array-based data sets to disk; linear algebra operations, Fourier transform, and random number generation; tools for integrating other languages with Python.

pandas. Pandas provides rich data structures and functions designed to make working with structured data fast, easy and expressive. The primary object in pandas is the DataFrame, a two-dimensional tabular, column-oriented structure with both row and column labels. Pandas combines the high performance array-computing features of NumPy with the flexible data manipulation capabilities of spreadsheets and relational databases.

And, I may add: it is seamlessly integrated with the developed high-level language Python, which contains mechanisms for abstraction, functional programming, object-orientation; extensive platform support libraries for systems programming, web services interfaces, etc., etc.

For users of the R statistical computing language, the DataFrame name will be familiar, as it was named after the similar R data.frame object. They are not the same however, as the functionality provided by the R data frame is essentially a strict subset of that provided by the pandas DataFrame.

matplotlib. The most popular Python library for producing plots and other 2D visualizations. It is maintained by a large team of developers, and is well-suited for creating publication-quality plots.

IPython. IPython is the component in the toolset that ties everything together; it provides a robust and productive environment for interactive and exploratory computing.

SciPy. SciPy is a collection of packages addressing a number of different standard problem domains in scientific computing. It includes: scipy.integrate, with numerical integration routines and differential equation solvers; scipy.linalg, with linear algebra and matrix decompostion algorithms; scipy.optimize, with function optimizers and root finding algorithms; scipy.signal, with signal processing tools; scipy.sparse, for sparse matricies and sparse linear system solvers; scipy.stats, with standard continuous and discrete probability distributions, statistical tests, and descriptive statistics; scipy.weave, a tool for using inline C++ code to accelerate array computations.

Together NumPy and SciPy form a reasonably complete computational replacement for much of MATLAB along with some of its add-on toolboxes.

And, I may add: it is free!

]]>**Python data types**

The Python language contains a whole range of standard types, including primitive value types (int, float, etc), lists, tuples, dictionaries (i.e. finite mappings), functions and objects. For tutorials and reference information, see:

**ndarray (NumPy)**

The python module NumPy has an n-dimensional array type. All the elements in an ndarray must be of the same Python type. This is an efficient representation, which gets packed into a contiguous array in memory. This makes it a good format for interfacing with libraries that are external to Python. NumPy provides operators that will apply element-wise operations to entire arrays (vectorization). So, even though the Python interpreter does have performance deficits in comparison with strongly typed compiled languages, by making use of vectorized operators on large data sets, the critical inner loops are being performed in the compiled NumPy library, rather than in the Python interpreter.

**Series and DataFrame (Pandas)**

These two data types (classes in the Pandas module) are built on top of the ndarray data type. They are enrichments of, respectively, the mathematical types Sequence and Relation. A Series is a sequence of values with associated labels, and a DataFrame is a two-dimensional, column-oriented structure with row and column labels.

**Index (Pandas)**

An Index is an object that provides the sequences of labels that are used in the Series and DataFrame objects. An Index may contain multiple levels of hierarchy within it.

This thread will consist of an exposition of the algebra of Series and DataFrames, along with examples of their use.

]]>thanks Daniel

]]>To what extent can this be truly modeled as a random variable, in the technical sense of probability theory? For that we need to have a sample space S consisting of "experimental outcomes," a sigma-algebra of events (subsets) on S, and a probability measure on S; a random variable then has to be a measurable function on S.

So what's the probability space underlying our variable T2000? Would S consist of all "conceivable" histories of the world, and T2000 the function which picks off the temperature at that point in space and time? But this would be a purely fictional construction -- who's to say what's in S and what's not -- and even more artificial would be the assignment of a probability measure to the events in S.

Yet without an underlying probability space, there's no way that we could speak of, say the variance of T2000.

]]>To Save the Planet, Don’t Plant Trees

The article was written by an assistant professor of atmospheric chemistry at Yale.

In the article the author warns of socalled V.O.C.'s, (something I haven't heard of before):

Worse, trees emit reactive volatile gases that contribute to air pollution and are hazardous to human health. These emissions are crucial to trees — to protect themselves from environmental stresses like sweltering heat and bug infestations. In summer, the eastern United States is the world’s major hot spot for volatile organic compounds (V.O.C.s) from trees.

and moreover they write:

Climate scientists have calculated the effect of increasing forest cover on surface temperature. Their conclusion is that planting trees in the tropics would lead to cooling, but in colder regions, it would cause warming.

if I understood the article right then more or less both facts taken together (the carbon cycle and its possible wrong understandings is also mentioned) leads to the recommendation: " Don’t Plant Trees." There are no references with respect to the claims though.

even if, as the author writes:

Planting trees and avoiding deforestation do offer unambiguous benefits to biodiversity and many forms of life. But relying on forestry to slow or reverse global warming is another matter entirely.

If you look at that pretty foto traumawald by Christian Miersch, who had recently commented here on Azimuth, then it seems indeed to be an important question wether science is able to determine the right measures to adress climate change.

For me this article brought however up some question, which I've been tossing around for quite a while, which is the question of the role of certain thermodynamic quantities like entropy and chemical energy in the question of global warming. That is a dark surface absorbs a lot of infrared (thats what I figure is behind the assertion: planting trees in the colder regions would lead to cooling etc. that is the net albedo change in reversing grasslands and other soils into forests seems to be different in differetn climates, where I am not sure wether I understood all the resonings behind this)) but one question is also: what's happening with the absorbed infrared. That is black body radiation is only a fist approximation and it might be worthwhile to think about effects like conversion into chemical energy etc. Like if I would look at this example of upconversion then the upconverted light of a dark looking leave would differently contribute to the overall radiation and in particular to the infrared balance, which plays an important role in the green house effect. In that context I am also asking myself how big are the cooling effects of human efforts in killing biodiversity and building rigid structures like streets and houses. That is exageratedly speaking: if earth would be covered with concrete then this could be seen as lowering the overall earth entropy, and if this would be the case some of the sun's energy would have needed to go into that entropy lowering and not into heat. I was hesitating to ask this question, because I always had some unease with certain thermodynamical laws (visible e.g. here.), but I am not sure how much of this under-understanding is due to missing out some literature or forgetting learned content.

]]>The most baffling aspect of our physical world to me is the handedness which is imposed on most living beings! No explanation for it at all, and how it came about even larger mystery.

Handedness appears in Maxwell's Equations (cross product) which is even more exciting to note.

Even oddest:

Let V be a finite-dimensional vector product algebra, then d=Sum (e_i.e_i) e_i some orthonormal basis of V satisfies d(d-1)(d-3)(d-7)=0

Vector product algebra dimensions

For Real d, then spaces supporting such vector products are limited to dimensions 0,1,3 and 7.

]]>Actually you gave me a fun idea, Graham! In a sandpile when the sand is at the critical angle of repose, as steep as possible, small landslides occur... and at least in theoretical models, these landslides are roughly scale-invariant: there are small ones and big ones and bigger ones, with the frequency of a landslide of size $x$ being $\propto x^{-p}$ for some power $p$. Under some conditions sand naturally organizes itself into dunes that are near the critical angle of repose: this is called self-organized criticality. The idea is that this system naturally has a second-order phase transition as some sort of attractor.

Maybe Pacific warm water that's just about ready to slosh back east is a bit like a sandpile at its critical angle of repose! If so, there might be a second-order phase transition here.

I feel this idea is a overly naive, but it might have some merit, or lead to some better ideas.

I was thinking about this idea and searched the forum to see if it came up before. The discussions of sloshing driving El Nino sound a lot like self organized criticality to me. That suggests one possible approach.

Bialek, Nemenman & co as well as Sejnowski & Saremi have papers on measuring criticality in complex natural signals, particularly images and neural data by treating the pixel/signal intensities as the order parameter. This approach could be applied to the various gridded data sets like the NOAA surface tempreature, pressure, humidity ... data sets. It sounds like the they should be in a near critical state most of the time, and El Nino's should correspond to departures from criticality.

I have seen a paper claiming that epilepsy attacks are departures from criticality I also think one that claims it for stock market crashes, but everything eventually get claimed to cause those.

Link strength sounds like a partial indicator of criticality. Looking for criticality on the full data could be more promising.

I found the following older paper:

- J S Andrade Jr, I Wainer, J M Filho, J E Moreira. Self-organized criticality in the El Nino southern oscillation Physica A: Statistical Mechanics and its Applications 215, 331--338 (1995). (The versions linked on Scholar are paywalled, but it is possible to find free versions on the internet.)

Following the citing and related papers on Scholar shows that scaling and criticality in climate is a lively cottage industry in its own right. One way this might be usable in El Nino prediction is to try and estimate the amout of energy built up in the system. This should primarily be a function of the water temperature differential across the the El Nino region and the sea level differential. If El Nino is really an SOC system the probability and likely size of the next event should be a function of the built up energy.

If El Nino is SOC, then there ought be mini/micro El Ninos happening on all space and time scales. These would be small backflows eastward from the warm pools. Is there a data set from which these would be detected easily? I know Paul has been looking into the details of the sloshing behavior. What data did you use to create your visualizations?

I should probably actually read the paper first :)

]]>The second mechanism is a weakening of these westerly atmospheric flows leading to a change in the thermocline profile to produce El Nino conditions.

It seems obvious to ask where this weakening comes from: at latitudes to the west of the warm water pool, to the east or both. Fortunately the laws of causality exclude the possibility that changes come from somewhere else.

Naively this seems to boil down to asking if easterlies are generated in the Indian Ocean pushing the Walker circulation eastward or whether some drop in central Pacific pressure pulls the Walker circulation eastwards?

Are significant amounts of water evaporated from the Pacific warm water pool and then dumped on Australia as is common in strong La Ninas? This would need to be distinguished from the contribution to Australian precipitation from evaporation in the Indian Ocean. This seems to me to be needed for any simple volumetric mass balance description of ENSO.

What is the effect of the Indian Ocean dipole on ENSO?

The influence of the Indian Ocean Dipole (IOD) on the interannual atmospheric pressure variability of the Indo-Pacific sector is investigated. Statistical correlation between the IOD index and the global sea level pressure anomalies demonstrates that loadings of opposite polarity occupy the western and the eastern parts of the Indian Ocean. The area of positive correlation coefficient in the eastern part even extends to the Australian region, and the IOD index has a peak correlation coefficient of about 0.4 with the Darwin pressure index, i.e. the western pole of the Southern Oscillation, when the former leads the latter by one month. The correlation analysis with seasonally stratified data further confirms the lead role of the IOD. The IOD-Darwin relation has undergone interdecadal changes; in the last 50 years the correlation is highest during the most recent decade of 1990–99, and weakest during 1980–89.

Saji et al. (1999) found that the coupled ocean-atmosphere phenomenon evolves with an east-west dipole in the SST anomaly, and named it the Indian Ocean Dipole. The Dipole Mode Index (DMI ) is thus defined as the SST anomaly difference between the eastern and the western tropical Indian Ocean (see insets in Fig. 3a for the regions used to compute the DMI ). The changes in the SST during the IOD events are found to be associated with the changes in the surface wind of the central equatorial Indian Ocean. In fact, winds reverse direction from westerlies to easterlies during the peak phase of the positive IOD events when SST is cool in the east and warm in the west. The effect of the wind is even more significant at the thermocline depths through the oceanic adjustment process (Rao et al. 2002); the ther ...

The correlation coeffient between the pressure index and the SST index time series is 0.65 (0.74 for June-November) when the latter leads the former by one month.

Fig.3 shows cross correlation coefficients above 2.5 with 99% CL for:

- WP-DMI with a -4.5 month WP lag : 0.3
- Darwin-DMI with a -1 month Darwin lag : 0.4

and coefficients below -0.25 with 99% CL for:

- IOSPL-POSLP with a -2 month IOSPL lag : -0.275
- SOI-DMI with a -2.5 month SOI lag. : -0.4

where

WP : western Pacific DMI : dipole mode index IOSPL : Indian ocean sea level pressure POSPL : Pacific ocean sea level pressure SOI : southern oscillation index ...

During positive IOD events sea level pressure anomalies in the Indonesia-Australia region are positive and those in the western Indian Ocean region are negative.

The Nino3 index has a broad 3-6 year spectral peak whereas the spectral peak of the IOD index is around 2 years.

]]>Thus the inherent periodicity associated with the IOD events, which is different from that of the Pacific ENSO events, may provide covariability between the IOD and the pressure fluctuation at Darwin, i.e., one pole for the Southern Oscillation. The correlation analysis in the present study clearly supports this hypothesis; the IOD index shows a significant correlation with the sea level pressure anomaly at Darwin. The correlation analysis further shows the lead role of IOD in determining the evolution of such a correlation with the Darwin and central-west Pacific indices.

Though the IOD correlation with the pressure variations in the eastern Pacific is insignificant, the former has a significant correlation with the SOI, and the pressure difference index of the tropical Pacific. It can be explained by the fact that pressure variations in the western Pacific will set up anomalous winds that force the oceanic Kelvin waves initiating changes in the eastern Pacific. The above novel relationship, however, undergoes decadal modulation.

A significant reduction of the IOD impact on the Darwin pressure is observed for the recent decade; 1980 through 1989. Understanding the physical mechanism that determines such decadal modulations of the ocean-atmosphere coupled system in the Indo-Pacific sector is underway using sophisticated coupled ocean atmosphere models.

```
1) free
2) allow me to build my own graph of pages. That is, I want to be able to have hyperlinks from anywhere to anywhere.
```

I spent a few hours looking around without any luck. Every host required that I go through a web site builder that allowed no user control over the structure of the resulting site. Years ago I used yolasite, but their site builder has changed and seems to be no longer of any use to me. Even then I found the site builder a hindrance and I'd would rather have uploaded my own HTML. Any help?

]]>The vector fields will from the rate equation, applied to various two-species Petri nets.

I'm looking for any and all suggestions for how to go about this. On the one hand, I like the idea of using a text-file interface between the output of the computational component, and the input of the rendering component. Generate data, store it in a declarative file, and then display it. This has the advantage of decoupling the choice of computation and display languages.

Some kind of standalone plotting system, like gnuplot. Or use R. R must have nice packages (right?) that will take a functional description of the vector field, and apply nice numerical methods, to compute and display the flow lines, etc. Maybe this outweighs the software engineering wish to decouple the computation language from the rendering system.

Thanks in advance for your thoughts.

]]>You can save 20 acres of rainforest (which stores about 200 tons of carbon per acre) for a mere 10 dollars via this link:

http://www.rainforesttrust.org/acres-for-50cents/

And until the initial fundraising goal of $646,000, your contributions will be quadrupled because of a 3:1 matching challenge by other supporters.

After giving, you can share with others via these social networking tools:

http://www.rainforesttrust.org/social-media/

If you wish to give by check, please make the check payable to Rainforest Trust (with Sierra del Divisor in the memo or "for" area of the check) and please mail the check to: Rainforest Trust, 25 Horner St., Warrenton, VA 20186.

The Rainforest Trust is a 4 star rated charity (by Charity Navigator) and over 90 percent of all donates go to saving specific rainforest acres (by land purchase or establishment of preserves and parks). 100 percent of donations to the Sierra del Divisor project go to saving acres of rainforest (less a small transaction fee charged by Network for Good for credit card payments).

As a method to reduce carbon dioxide emissions, saving an acre of rainforest for 50 cents is over 1,000 times more cost effective than buying a hybrid or electric car (rather than a conventional gas-powered car) or installing solar panels on your home, each of which cost well over $1,000 and each individually do less to prevent carbon dioxide emissions than saving one acre of rainforest. This 1,000+ ratio holds true in comparison to other alternatives as well.

Here are some interesting bits of information that you may find informative, interesting and/or motivating:

I have much skin in the game: I have funded projects to save over 750,000 acres of rainforest this year, and have committed to fund another 250,000 acres (I will fund the majority of this in 2014).

Much due diligence behind this project: During 2013, I spent many weeks of full time work searching for the most cost effective ways to save rainforest (visiting the Amazon twice and working with many conservation charities) to develop the most cost effective projects to save rainforest. This project is the result, and others will follow. Note that I gobbled up (via donations) all other cost effective projects I could find during 2013, making it necessary to push charities to develop larger projects, such as this one.

I am a fiduciary with respect to this project: I recently joined the Rainforest Trust board.

While I am new to this group, I am very serious about this issue. Since retiring as a venture capitalist last year, I have become aware of the full potential threat of global warming through much study and research, and have become obsessed with pursuing the most cost-effective means to mitigate global warming. You can read a bit more about my background here:

http://www.rainforesttrust.org/about/our-board-of-directors/

]]>- Mark Newman,
*Networks: An Introduction*, Oxford U. Press, 2010.

I would like to create a Wiki page with reading notes, but not sure about a good way to name it. Calling it "Networks: An Introduction" would make it sound like an introduction. One way to go would be to add a prefix like "Notes - Networks: An Introduction," like the way we have Blog as a prefix. Any suggestions welcome here.

]]>Let's start with the simplest case of a Petri net, which has just one species $U$, and one transition $Z$. (I guess you could have a net with zero species that is completely vacant.) Suppose that $Z$ has $m$ input connections to $U$, and $n$ output connections to $U$. Let $\alpha$ be the rate constant for $Z$.

Let $u(t)$ be the continuous amount stored at $U$ at time $t$.

Then the firing rate for $Z$ at time $t$ is $\alpha u(t)^m$.

The general, time-invariant "direction vector" for $Z$ is $(n - m)$ — this is how many tokens would get added for each firing of the transition, back in the discrete model. Let's call this $DirVec(Z)$.

Then the rate equation states that:

$$u'(t) = rate(u(t)) \cdot DirVec(Z) = \alpha \cdot u(t)^m \cdot (n - m)$$

So the general form of the equation that is raised is:

$u'(t) = \beta \, u(t)^m$

When $m = 0$, the general solution is $u(t) =$ affine function of $t$.

When $m = 1$, the general solution is $u(t) =$ exponential function of $t$.

When $m = 2$, a solution is $u(t) = - 1 /(t + const)$. I found this one on the web. Is this the general form of the solution for $m = 2$?

I find this to be a curious sequence of functions: linear, exponential, reciprocal of a polynomial.

What's next?

Does anyone know a general solution, that covers all cases of $m$?

See, here is the pull of the formula-based approach.

]]>Are there already examples of where this is done on the Wiki, and how do you use the category system towards this end?

What I'm picturing is a Wiki page for each Book that is to be worked through. Since I'm getting the Sudbery book, I would start a page on that. Then post selected problems, along with whatever solutions I could work out. The page would be developed by the group, and discussions would take place in an associated Forum thread.

Would this go under an "Experiments " page. And do we have a category for such an animal. If there's no convention already, I suggest using a prefix other than Experiments, it could be "Problem Sets," which is different from the idea of an Experiments page.

In theory we could also work out problems in the setting of a blog, which has the advantage of larger readership and more participants. There could be a blog "header" article that announces e.g. that I will be going through the Sudbery book, and will be posting some problems and solutions in location ABC. A big drawback of this is that our blog setup does not provide Wiki capabilities. Though it could contain postings that describe individual problems, and have a URL that points to the page on the Azimuth wiki.

Another technical issue is that the blog could then generate too much traffic for the general readers of the blog. According to the scheme I just described (which is growing on me), I would make a post for every problem that I want to work on. That could be a lot of little posts, which would be annoying, and would wipe out the limited history buffer that the Wordpress blog is maintaining.

To reduce posts, there could just be the header blog article, that points to the Wiki page. But then there would be no notification mechanism, for people who want to follow the progress of the problem sets that are being worked out.

Maybe a companion blog for the working out of problem sets?

]]>- John Baez and Jacob Biamonte,
*Notes on Quantum Techniques for Stochastic Mechanics*.

Re: Section 15, Dirichlet operators and electrical circuits.

Gloss: This shows how a Hamiltonian framework can be applied to a network of resistors. Let the nodes in the circuit be x1,...,xn. Between xi and xi is a resistor, with conductance (= reciprocal of resistance) cij. Form the symmetric matrix H which has cij in each non-diagonal entry, i.e., Hij = Hji = cij, and let Hii be minus the sum of all the other values in the ith row (or ith column, same thing here). By construction, H is both self-adjoint and infinitesimal stochastic. Such a matrix is called a Dirichlet operator. Because H is self-adjoint, it is a valid quantum mechanical operator, and because it is infinitesmal stochastic, it is a valid stochastic mechanical operator. So it is in an overlapping territory between the two theories.

Let V be a vector in R^n, representing the voltage at each of the points x1,...,xn. The book shows that <V, H(V)> equals the power consumed by circuit!

Here I will add a few points, to this charming topic that has been introduced.

(1) H(V) itself has a physical interpretation, which is the vector I of *currents* that is induced by the voltage vector across the resistor network. The signs are oriented so Ij is the net flow of current *into* the node xj.

It just requires a simple calculation to show this. To get some practice, let's do it for an example with three nodes x1, x2, x3, where there is a 1 ohm resistor connecting each pair of points.

Then H = ((-2, 1, 1), (1, -2, 1), (1, 1, -2)).

Let the voltage vector be (a,b,c).

Then:

H(a,b,c) = (-2a + b + c, a - 2b + c, a + b - 2c) = (b-a + c-a, a-b + c-b, a-c + b-c) = (inflow into a, inflow into b, inflow into c) = current I

(2) As a conclusion, it follows that <V, H(V)> = <V, I> = power consumed by the network.

(3) To give a full physical interpretation of the Hamiltonian dynamic on this network of resistors, we should connect a capacitor between each node and a ground point. Give them all the same unit capacitance. This is what will make the current H(V)(j) into node j actually produce a rising voltage at V(j) -- this condition is needed to fulfill the equation dV / dt = H(V).

(4) As was pointed out in the text, any constant voltage vector K = (a,a,a,...) will satisfy H(K) = 0. Interpretation: There is no current when all voltages are equal.

(5) Intuitively, we see that any given initial state of the network, given by a voltage vector V, the network will asymptotically charge/discharge to an equilibrium state where, for each connected component of the graph, all voltages are the same.

For each connected component C, the final voltage is easily calculated, by dividing the total charge in C in the network by the number of nodes in C:

FinalVoltage = Sum(V(j)) / n = mean(V(C)), for all j in C.

(6) An eigenvector is a voltage vector V that heads in a "straight line" towards the equilibrium vector. A non-eigenvector does some "turning" as it heads towards the equilibrium vector.

]]>It would be cool to get some of these guys to work on equations relevant to Azimuth.

]]>Now, I wonder how big the variance in death figures over a 13 week period is? Since I don’t have the time to chase up data on deaths over a 13 week period for the past 10 years, I don’t know. But there is a way to get a lower bound on it. Conveniently, they provide a table of deaths in the individual weeks 12-25. For all the weeks in 2011 and 2010 combined, the standard deviation of those weekly figures is 556 deaths. Multiplying by the square root of 13, we get an estimate of the standard deviation of the sum of 13 weeks. And the answer is: 2006 deaths.

I just read this on the blog and I'm replying here.

This is not a good way to estimate the standard deviation of the weekly figures. The paper is about differences between various 14 week periods. If you combine the data for two of them, including the allegedly unusual one, your estimate for the standard deviation will likely too high, since it will be increased by differences between periods. Even if you use the periods separately, and ignore the period after Mar 2011, there may be seasonal effects within the periods which boost the estimated standard deviation. (There certainly are such effects around Xmas.)

Like Roko Mijic, I can't be bothered to download more data. It is not difficult to do, but it is too like my day job. Using the data from the paper, I think the best way to estimate the standard deviation, allowing for seasonal effects, is to take the two 'before' periods (Dec 2009 - Mar 2010 and Dec 2010 - Mar 2011), subtract corresponding weeks, and estimate the standard deviation of the differences, and then divide by sqrt(2). I get 332 for the sd of the weekly figures.

Next, I simulated data from a normal with mean 11000 and sd 332, and found out how often the method in the paper produced a result which is more unusual than the value of the statistic they calculated in Appendix Table 4. This happens about 20% of the time for a two-sided result or 10% for a one-sided result.

Here is my R code for you to check.

```
x2010spring <- c(11010, 11097, 11075, 10712, 10940, 10549, 10637, 10389,
10491, 10352, 9894, 10781, 10178, 10290)
x2011spring <- c(12137, 11739, 12052, 10928, 10743, 10826, 11251, 11300,
11132, 10839, 9538, 10770, 10981, 10779)
x2010winter <- c(10323, 7942, 8288, 11557, 11299, 10110, 10832,
10524, 9877, 9802, 10198, 10586, 10699, 9969)
x2011winter <- c(10702, 8339, 8194, 11804, 10775, 10689, 10420,
10295, 10700, 10952, 10762, 10779, 10639, 10274)
par(mfrow=c(2,1))
maxd <- max(x2010spring, x2011spring, x2010winter, x2011winter)
mind <- min(x2010spring, x2011spring, x2010winter, x2011winter)
plot(c(x2010winter,x2010spring), ylim=c(mind, maxd))
plot(c(x2011winter,x2011spring), ylim=c(mind, maxd))
esd <- sd(x2010winter-x2011winter)/sqrt(2)
n <- 0
for (i in 1:10000) {
w0 <- sum(rnorm(14, mean=11000, sd=esd))
w1 <- sum(rnorm(14, mean=11000, sd=esd))
s0 <- sum(rnorm(14, mean=11000, sd=esd))
s1 <- sum(rnorm(14, mean=11000, sd=esd))
O <- s1/s0
E <- w1/w0
mean1 <- sqrt(s1)^-1 * O
mean2 <- sqrt(s0)^-1 * E
X <- (O-E)/sqrt(mean1^2+mean2^2)
if (X > 5.7) { n <- n+1 }
cat (X, "\n")
}
n
```

]]>Probably most here already know this, but its nice to see someone evangelizing these concepts. Does anyone know the people doing this research?

]]>Let me say about water drainage issues, the topic which is now considered often in central Europe. The level of underground water in urban and suburban areas is dropping steadily in last several decades. There are several reasons connected to human presence. First of all, large area of the land is covered with parking lots, roofs and so on. Water falls on a roof, road or similar area and goes directly into the channels or canalization systems, it can not get into the ground which is covered. Thus only the water falling on grassland, parks and so on gets absorbed into the ground. The water under ground of course gets partly spread through natural underground streams, what does not concern much surface water, but that means that the underground streams, has lower supply. Another problem is that if the rain is big, all the water from covered areas gets into the channels soon and this raises flooding treats. If the water is first getting into ground like forest or meadows area, then it releases water mor slowly and the wave of released water is diminished, so the treat of flooding is smaller.

Many solutions are considered. The areas of parking lots, pathways, roads and so on should not be continuous, but should be interspersed with areas where water can get into the grassland, grass strips, if possible, equipped with efficient drainage structure at places, so that water gets absorbed into the ground fast. This makes flooding buffer and also helps maintain natural level of underground water what is important for example for survival of trees in surrounding mini-green areas, for health of gardens, and as a supply of water for many purposes (if not hi quality, local water is often used in urban area for gardening, industrial use, local construction and so on). This also concerns the survival of local small lakes.

Maybe we should write about such ideas in Azimuth. And if somebody has some contacts for my cousin, let me know. It would be ideal if the scientists in some research institute concerned with urbanism and resource efficiency needs contract collaboration with an experienced person from architecture and civil engineering, I am sure such exist somewhere but the right profile is probably not easy to find and match.

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