Is the average CR4 member in Timbuktu ?

We've all come to that conclusion way back. Don't tell him I said so.

I'll tease you a little.......More to do with finding clusters on a sphere. I may elaborate on that later......

At some stage I shall be disposing of half the globe, though I haven't decided which. You will, in due course, be pancaked. I'd give you an evil smiley, but this computer ain't working :( Prepare yourself for hemispherical distortion. If I'm feeling nice, I'll tell you why.......

I'd say that the final co-ordinates are correct because the Earth sucks. (I'm so down, everything looks like UP TO ME.)

LOL ! Unfortunately, you've got to remain on the surface of the earth for this one.

Since it doesn't affect the real question, I had a little bit of fun in locating all you Americans......not quite where you might expect.

Kris, I think you've gathered the stray needles of CR4 and bundled us into a haystack.. am I right?

PS. in the interest of ambient temperature increase, I shall strive towards the CR4 "average" position from now on.

That's not far off being a good analogy ! You are slightly warmer than anyone else so far...........

could this be possible argument for shifting CR4 HQ to Timbuktu?

Hmmm. Average... Reminds me of the dude that had his head in the oven and his feet in the refrigerator and said: "On the average I feel pretty comfortable."

Ha ! It also illustrates why you can't average longitudes. If only the dude in the kitchen had understood the question here......In a way, he has to decide how his balls feel. Trouble is, he's got to apportion them between oven and fridge. I'm giving too much away.

I could get some meaning full data from your list. Here are top 10 Subscribing nations.

USA 128

India 64

Canada 30

UK 20

Australia 19

New Zealand 11

south Africa 9

Belgium 6

Egypt 6

Pakistan 5

Philippines 5

I'm glad you could read it, although the data has changed slightly since I collected it from here India is now 65, and there are possibly others. No matter, it doesn't affect the overall result significantly. A few anomalies are actually 'beneficial' if I decide to revise the input data. The countries that only have one or two members are equally useful in this respect.

Where would you place the average, and why ? If the answer is good, I may even crunch the numbers for you.

Consider a sphere with a lot of dots on it. How would you attempt to describe the location of the average ?

As you can see from the data you've read, there are 2 main concentrations - USA and India. Would the data be better summarized as two (or more) global focal points, and more importantly - where should they be ? Would you assign other nations to India or the USA ? Perhaps each of the locations would have to shift slightly to incorporate the other countries assigned to them......

The problem is not really about where CR4 members live. That's just a convenient way of introducing a specific problem. I want to see how people from different science/engineering backgrounds tackle it.

'Thinking outside the box' is a much used phrase, and I want to see if anyone here can really do so. Anybody who can see a real-life application would be able to claim that they can think outside the box, as would anybody who can come up with an alternative method to the one I have in mind. Crunching the numbers doesn't matter - it's the method I'd like to hear about.

It's too early to suggest nobody can think outside the box (much as I dislike the yucky phrase, it's reasonably descriptive). I've described the problem in 'Micky Mouse' terms so as to solicit opinion from a diversity of disciplines. The essence of the problem has been described, and I await he or she who can think outside the box'.

Averaging circular values given as vectors is a familiar problem for meteorologists trying to average wind directions. One solution is to convert the measured wind vectors into pairs of cartesian coordinates, averaging the pairs and convering the pair of averages back into a vector. There are other, less intuitive methods.

you could use this accepted trick using both longitude and latitude, weighing by the number of memebers per location.

Some members helpfuly give both latitude and longitude of their location. If not, Google Earth is very helpful.

Still, I can't forget the medical statistician who said that a certain disease hits people in their 40's. His data were a 2 year old child an his 78 year old grandfather... They say that he drowned in a lake with a 1 inch average depth.

Thanks dovy,

It might not be clear from the initial picture given, but I have locations listed in Cartesian form. Locations were snapped off google earth (to the nearest whole degree). A weighting has been applied to the x,y,z values, although that isn't where I'm going with this. It's just a sort of introduction method, to hopefully simplify understanding of what I shall post later.

Meteorology is one area that hadn't crossed my mind (Doh !), so I shall look into it. The suggestion is exactly what I'm interested in - how different disciplines approach similar problems. I suspect that many people use mapping/modelling techniques, unaware that other have tackled similar problems - possibly with a better method. Another related (though not unconnected) area of study might be marine biology - distribution of algae etc. Weather models are highly complex, but I wonder how many conclusions applied to other field of study rely on vague summaries of where some particular phenomena is located. Academia is not geared toward cross-disciplinary knowledge.

More explanation will appear later.........When I do so, I'm hoping that others will trash it by suggesting better methods. I'm holding back to see what suggestions emerge first. The raw data (member numbers, location given, spherical Earth assumption) presented is enough for people to consider the nature of the problem. The average location of CR4 members is not my real interest, it just serves as a convenient set of data.

Thanks for your input

Kris,

You could compare 3 different weight functions:

• number of responses
• number of GAs
• 1/(number of GAs +1)

You may obtain 3 different centres of gravity. I wonder what they would signify?

Is that for the original data set, or just within this thread ?

For the whole set - in order to characterise centres of enthousiasm versus centres of competence versus centres of laziness.

For the restricted set - in order to characterise space-dependent response to vague tasks. But, I'm afraid you won't have a sufficient sample.

Patience, dovy......patience......

Another thought- use polar coordinates instead of cartesian. That way, you hold R constant and analyse the distribution of the two angles for each location...

That is exactly what I meant with keep the location on the globe.

(Latitude and longitude are a polar coordinate system)

As soon as you start working in a carthesian system the center of our activities will move to the middle of the earth.

As soon as you start working in a carthesian system the center of our activities will move to the middle of the earth

Not if you are using them to calculate angular separations........

Kris-

I find it difficult to accept the concept of averaging a surface effect over the volume of the entire sphere. Furthermore, you seem to have ignored those geopolitical areas with no participation at all. And what about areas like Myanmar or Vietnam, where one might encounter participants were it not for government censorship? Should you not provide "negative" scores for these?

I do not have as much spare time as you, so it is difficult for me to develop an appropriate surface distribution model on the sphere. But, then, one also needs to address the issue that the earth is not a perfect sphere (those of you in the Northern and Southern hemispheres reportedly are a lot closer to the center than those of us located nearer the equator).

It would appear that your analysis has oversimplified the situation, resulting in answers of questionable value...Please do not make any major modifications to the geometry of the earth until you have had time to review the issues I have raised...

Let us look at the data (sorry can not look, I don't know from where you ferreted it out.

Do ferrets ferret squirrel ? Then how squireele ferrets out it ?

But the data looks to be statistical _{(ie cooked)}

You know the best thing about statistics ? you can give any information and say it is statistical (we have seen being done so many things ) The important part is it should be practical ie believable.

Now look at the practicality.

0 deg longitude is through greenwich ? so one side is USA the other India and china (almost equi distant) so the population of CR4 balanced on either side

0 deg latitude is unbalanced- upper half is heavily populated, lower half is Aus and NZ with S America.

With sparkstation out nobody to scare away take care of the furry ones.

So the half balance will be between equator and the tropic, So I would have put it some where around 30 deg N or more. (USA centered at 38N and India at 28N China at about US level, and the furry ones at about 50N)

So here there is a bit of non-commonsense statistics .

BTW: I didn't know you were vitalstatistically oriented

0 deg longitude is through greenwich ? so one side is USA the other India and china (almost equi distant) so the population of CR4 balanced on either side

Roughly true, though trying to average longitudes will end in tears . They could all get lumped together in Mali, or maybe 2 poles (not plumbers/waiters/builders) would be better........

The above pancake pictures are not related data. 2 poles are strangely looking like 4.

Surely it does more than 1420rpm?

Are you going to factor in number of posts?
If so I think the average user is somewhere between Etherville and Harlow, which could accout for the number of accidents on the M11.

Excellent work my squirrelly chum, no wonder the KrisDel^TM accounts are in such a mess fine state.
Del

PMSL - If only I'd had ready access to the data for other countries ! Now I think about it, the end result would have been a foregone conclusion. Deciding who's posts we get to claim would be the 'meaty' part.

I would rather like to hunt the cat(average postition) for off topics, Any guess where it could be?

THERE ARE NO "AVERAGE" CR4 MEMBERS!!!!!

The presence of an unknown number of females does somewhat mess up that analysis. Perhaps I should downsize the Texas weighting in order to reflect reality

Nobody has noticed where I put Americans yet

I haven't looked at your actual numbers to discern where you placed us... but I would think if you took Alaska into account, then I would presume that you have all of us in British Columbia, Canada somewhere... or at least Fargo.

But... as for your little experiment... there are two methods depending on what you're looking for... at least from the view of my box.

1. If you talking about finding a common, most prevalent, or average location of entities, for example where is the average location of a data set split between the two geographical locations of Los Angelis and Chicago where twice as many of the data set are in LA... your answer would be Los Angeles because that's where the highest concentration of your test sample is. Sure you could vector it and come up with someplace in Colorado, but none of your sample subjects live there, so how could it be their average location?

Seriously though

2. The other way to look it is geometrically and sense we're talking about a sphere, the best geometric shape to use is a circle, but not using polar coordinates… my though is more simplistic in its approach.

Identify the point location of your data sets and their values. Translate the value into an arc-length based on the radius of your sphere, determine its Cord and set a compass to the Cord length, scribe the circle on the sphere.

The centroid of the area which is shared by all circles is your average location.

If you have to increase the size of your circles in order to get them all to intersect and develop a commonly shared area, you'll have to apply the factor consistently across all data sets to the arc-length value and then recalculate the Cord.

If you find that you have too many circles, then break it down into regions. First plot all the data points from England and Europe and find the centroid of their commonly shared area. Then use the sum of those data sets to determine the size of the joint data set that now originates for your recently calculated centroid.

Below is an example, translated two-dimensionally, finding the average geographical location, based on population density, of Dallas, Memphis, and Chicago. NOTE - This is a concept sketch, actual values not used, neither are these perfect circles, not are they centered… But, in this example, looks like the average is just east of central Missouri.

Anyway… my 2-cents. Not quite sure how to take that concept and translate it to a world map with polar coordinates, but I'd rather have an excuse to scribe circles on a large globe anyway.

JavaHead

The job is yours.

Hi Java,

I like your circles on a map drawing very much, though I need to mull it over a little while.

In case it's not obvious, this is one of my somewhat slower projects. Well, really it's just a bit of fun.

Will get back soonish - appreciate your input.

Certainly a project such as this is one that is done on the side and casually, that's what keeps it fun.

The draw-back to using a two-dimensional map verses a globe though is the natural distortion caused by projecting a spherical surface two-dimensionally. You can only translate three points on a sphere exactly two-dimensionally; 4 or more and you get distortion.

The three-point non-distortion limit was the basis of the Cahill Butterfly Map. But, even with this, everything between the three points gets compressed as it is flattened.

Anyway… if your personal quest is identifying the average locations on a sphere between multiple data sets, and to keep the answer technically correct, then I suppose plotting arc-lengths on a sphere is the way to go.

Otherwise, I guess you could choose to use a standard Winkel tripel projection.

Then modify your circular data values into elliptical data values using Tissot's Indicatrices, which would allow you to work two-dimensionally.

Either way, sounds like a good project. And having a bunch of sphere models lying around will give Del something to play with.

But, if you want to save yourself time, you could just say America is the center of the universe and call it a day.

JavaHead

"Will get back soonish..."

the cheque is in the mail

rofl !

OK, I do have a bit of a record for that one . I'll eventually get back to this and make sure your good self, Java, and others who've made constructive comment get informed.

Just to save you time, Java, and I can't even recall if I got around to mentioning here; The question as posed was a slight subversion. I'm interested in geological mapping of such thing as joint discontinuities etc. Reason for the question being stated the way it was is that I wanted 'fresh eyes' to consider this type of analysis.

Brief summary for you Java (and anyone else still subscribed);

Consider a whole bunch of points marked on a sphere. The points may very likely (though not always) have some degree of clustering). In geotechnics, one might expect a trend toward 1,2, or 3 cluster points (often at 90 degrees to each other).

Not that many years ago, data from a joint survey would be plotted onto a 2D projection. The data points would then be converted to a contour map using fairly primative techniques. An advancement on the most basic method was to use the notion of 'set probability'. which in any other words is akin to giving each point a weighting according to how near it is to other points.

Although the analogy is not exact, because geological representation of joint orientation only requires a single hemisphere (a point is = to the normal to the plane of a joint), the CR4 membership deception still works to an extent.

Somewhere, I have a program that rips the CR4 membership data* and can do various type of averaging on it. To make another, very crude, analogy; We have a sphere, and place weight's on it's surface proportionate to membership and their location. When placed on a flat surface, the sphere will roll to a position indicating the average location.

The above paragraph is only an initial part, because as mentioned, in a real situation it's no use to simply average all the data. What's more useful is a 3D contour map. I have the number crunching ability to do this, though lack the graphical skills to produce an image that can be rotated in order to view 'hot spots' of population concentration.

The analysis of joint data is not as simple as punching raw data in abnd getting a result the says "there are 4 main clusters at location <whatever>. It needs visualizing with graphics. If I were to visit the USA and only had info such as, "well, the Rocky mountains are pretty big, and the Blue Ridge Mountains of Virgina can be a bugger, and heck, watch out for them Dakato Badlands", I'd be a bit pissed off. I'd want a contour map in order to make not just a numerical but qualitative assesment as to where I walked.

Forgive any lapse in explanation, it's been a while since I considered this question. It does, however, remain on my stove. The fact that a lot of cool software is available now is beside the point. One of the interesting issues in technology is that people can become tunnel visioned. A new recruit to a company will often lack the confidence to say, "hang on, that's all wrong - I have an idea". In much the same way, experts in different fields can find themselves striving to solve what is essentially the same problem. They are simply not aware of techniques evolved by other in a seemingly unrelated area.

The whole thread is slightly mischievous of me, but hopefully the reasons for my deceptive method of posing the question are now clearer. It was somewhere about #1 (sorry if it wasn't you exactly #1) that I though 'to heck with this, you got no imagination' (comment being a rather snide "you got much time?"). It's not $ related, just a bit of mild curiosity (about both the fundamental problem and other engineers to see beyond the ubiquitous 'box').

Chris, and several others have made very constructive input (on thread and by PM). Thanks for your time and input too, Java. The project is, as said, a bit of curiosity on my part. It's certainly going to get resolved long before the BBT one ever does !

* I somewhat dread going back - apart from locating the most recent version, I have to sort my sloppy coding to see what methodology I'd used in handling the data .

ASNF ? Forget "Men of Honour", the 'S' is for squirrel . Excuse me for a few days whilst I go shoot a few elephants

To begin with, I have not thought this completly through but here it is.

Record the longitude, as nunbers, not degrees, of each member. After 180, continue on numeriaclly to 359. If there are more than one member in a city, record for each member. Average the sum to get the average longitude. Set up a similar system for latidudes but start it with zero at one of the poles instead of the equator, though that may not matter as long as you continue past 90 and go to 360.

Hi Jag,

Since you ******'s seem determined to make me see this thru, I' guess I'm goin ti have to

Bear with me, it may take some weeks (shaddup, Chrisq), and I'll get back with more explanation. Averaging positions on a sphere is no big deal. My slight embarrasment is that I have neither the software of knowledge to transport data form a 'number crunching' program to a decent visual form.

I can see that I'm going to have to post some full and comprehensive answer to the original question soon. Around about Summer 2001 might be a good time to check .

Cheers for coming back, Jag, it's appreciated. Consider yourself on the list

Oh, I already consider myself on lists.

Yes, check it again- the KrisDel Time Machine never misses. You just knew I was going to say that.......I wonder why

"It's an ill wind that gets up nobody's backside"

Dad's Army

I mean it in the nicest possible way, Chris, but you are a complete and utter git ! Who on earth is responsible for making you privy to Dad's Army ? That's a Brit show, dammit ! You do, I hope, know that you lay yourself open to quips about Due South, Terrence and Phillip. and any sentence ending in 'eah ?'. Even in these remote isles, we are tutored in how to taunt Canadians.

South Park aside, Brits love Canadians (North Americans in general). In Glencoe is one of the most serene places - story goes that some dude built it for his native American wife. He built a 'lochan' there and surrounded it with the kind of trees one would see in Canada. It is exquisite.Very early in the morning, the place with it's maple'd shores and still water is a fine site to see. Damn good place to find you're car nicked as well, but if ever in the area it's a a 'must see'

huh? git?

now I'm really confused...

you introduced Dad's army.. as to the 'ill wind' I thought it was hilarious and thought you would find it so also... but perhaps Ive... well I've no idea what I've done wrong.

but sorry anyway.

Chris

Oh, Chris my dear chum - It was said humorously ! Chucking an insulting word is often done in jest over here. Sorry I left you baffled - I'm going to have to be more careful

Don't apologise to him, it'll only encourage him. Throw some imitation nuts at him and see how he likes that....

Never said a word, Kris. Nuffink to do wiv me, 'onest.

Well, at some point today there were 42,484 registered CR4 accounts.

Of them, 20,559 are Browsers, meaning they have never posted a message.

3,968 are Active Contributors (10 postings plus) or greater.

1,926 are Associates (25 postings plus) or greater

1,083 are Commentators (55 postings plus) or greater

673 are Power-Users (100 postings plus) or greater

198 are Gurus (500 postings plus)

Only 108 members have 1,000 or more postings

and only 3 members have 10,000 or more postings (who knew fuzzies could type so much)

So depending on what sampling [technique] you choose, the center of the CR4 universe is probably in the UK or wherever "the bothy, 7 chains down the line from Dodman's Lane level crossing." is.

Regardless, I'll be coming over for tea… or at least a drink at the Bath and Cr4wl because me thinks once you start plotting points a vortex might open and you'll discover that the average location of all the technical minds on this forum is in the 4th dimension otherwise knows as the BBT.

rofl - yeah, BBT had some serious lurking last night as another momentous number approached . My best ever sighting was nobody online, until the admin team popped up one by one in recovery mode after a 'few' visitors from Taiwan crashed the place. Wish I'd screen captured the moment it was just me and the admins - life would have been so simple if I could have drawn a line and marked it 1/5th the way from London to Troy .

Chris: 'few' is Brit silliness in understating something

Darn it, I was sure I'd be back in time for the BBT race - instead I was driving back along B roads to the Pie Town.

I'm sure you spend more time on here than all the admin combined....

The plotting of local densities is separate from the finding of the mean/average/centre of gravity, a bit like burying tulip bulbs and acorns together. What am I saying, you're a bloddy squiggle! So really you're trying to solve two problems for the price of one! Cheapskate!

Your point about tunnel vision is well made. I've been at an EMC conference this week and in one of the lectures, a method for working something out (can't remember the details, I've slept since then - yes, in the lecture) was presented a recent important step forward. I looked at it and thought "It's bloddy obvious, been in use for years...for acoustic and vibration interpretation" and then thought, but I bet you sparkies never bother to talk to us clankies about how we solve such equations!

Oh, and yes, I am still subscribed!

Ooooph....me, Cheap-skate? I wasn't bloody invited !

Yes and no to the 2 for the price of 1 notion. It's really about finding if there is more than 1 significant concentration of data points. In all likelyhood the situation I'm interested in would have 2 or 3 such places. In that situation, the issue of set allocation arises for the more remote data points.

Lumping all the data into one would certainly give a defined 'average' location, but with a wide spread of data. Two positions defined as maxima of concentration might be more meaningful (with data points asigned, in part, to one or the other). Three positions, etc etc. Oh, my, I'm at risk of becoming deviant

Mention of Pie-Town (sounds like Sidney) has reminded me it's about time for coffee and some bickies. I hear Cam has decided you can have your cake and eat if you go to a meeting that lasts more than 4 hours.

Mention of Pie-Town (sounds like Sidney)

mmmmm, mushy peeas

They are the work of the devil and all who peddle them should be burnt at the stake (using electrodes of course. Ooops, wrong thread)

Will get back soonish

Some dictionaries have a problem with this word (probably an American "invention"), but of course.........you do not speak the Queen's English, do you?????..........hahahahaha

One dictionary said..........actually it did not say anything.........but it said

"In two shakes of a lambs tail"

Perhaps one could say "I have soonished," when you get back to us.........or...........I am soonishing" when you have not quite finished, but you are working on it.

Good onya mate

1:26 !

Yes, there are some very "good" examples of the the "Queen's English" in a few of those examples............at least the Brits can make good comedies!

All above average? Or just outside the norm?

Apropos of nothing, most of my recent statistical work has been actually automating the identification of the clusters you started with so the question is sort of backwards to my thinking of late.

Were you planning a party with all of us as invitees and you wanted to minimize travel, a more suitable measure might be the geographical mean or median - weighting for population - and so a fairly standard logistics application of your data.

The mechanicals of imposing the data on a globe is another interesting aspect of the problem.....the average of one member on each pole could be anywhere on the entire globe.....

Hmmmm

But I sympathetically submit you have too much time on your hands.

No tea and cucumber sandwiches . Can you elaborate on what method you'd use to derive some meaningful summary of the data ? Treating it as dots of <whatever> on a sphere is fine. Number and location of dots as detailed. Is there a standard methodology you know of to identify and quantify cluster(s) on a sphere ? All the locations must be taken into account.

The real world application I have in mind is nothing new, I just want to see how people from different backgrounds might approach it. There are probably people working in different fields, using completely different approaches to handle what is essentially the same problem. I don't necessarily mean this particular one, but in general. Most people probably leave University or wherever with a fairly blinkered view point. Inter-disciplinary study might help grease the wheels of innovation. As someone once said to me, we start off knowing ***-all about a lot, and end up knowing a lot about ****-all (at least to the outside world). Advances in knowledge make it harder for colleges to find time to teach students beyond their chosen field. Good engineers/scientist become so through experience and awareness of the bigger picture. We all get tunnel-vision, so occasionally pausing to see what others do is no bad thing. Part of the reason why CR4 works so well is that members have an interest in all spheres of science and engineering.

Strewth, I'm rambling ! It's just a fun way of seeing how folk approach finding the 'average' location of places on a sphere. I added the number of members to cut down on the number crunching - it doesn't affect the general principle of the problem.

It's taken me longer to paste the question in, and look at the replies, than it did to do the initial analysis. I'll keep popping back to see how it's brewing

Some other creative individuals with too much time on their hands!

Thanks for initiative make a human network , I agree with you and initial activity.

You can get an average of CR4 members but as stated prior, None of us are average, [or we would not be here].

I stand corrected we have a poster who is average:

The infamous guest who in the same thread can be genius, an blithering idiot, slow, interesting, non responding, flaming angry(thinks guest is a badge to hide behind) and all in all average.

So how do you average in the infamous guest.

Kris project is the one in correct direction.

I remember long back read an Asimov story - where the computer selects a man or a woman above 18 and decides that he is average and hence representing the whole CR4 human world. And then the total President or Director or whatever the name of the post is will be selected just by one vote.

Such a great project. I hope this will be a great thorn cog in tomorrows wheel.

Being trained in simple decent statistics I have to inform you that you somehow abused the data you had.

You should never have replaced the planar location with 3D locations.

Within a band of -400m to + 9000m we all live on the surface ot the earth. As long as you don't take the height above sea level of our capitals just don't do anything with it.

Just work with the Longitude and Latitude and assume someone living west of Greenwich living more than 180° east of it.

Just make the average of the two figures and the result is again a location on the sphere.

Now we can discuss wheter our capital is a good reference for our average location, for my little country I agree (Brussels is only 40km from here), but using Washington DC as the location for our average US friends is perhaps a few miles of the reality. (a good portion of our subscribers reside in CA which is a few hours drive from DC)

It would be great if we could add files to the threads and each do our own Minitap analysis.

Just work with the Longitude and Latitude and assume someone living west of Greenwich living more than 180° east of it.

Just make the average of the two figures and the result is again a location on the sphere

hmmm...Suppose you have only the equator to deal with. A population at 45^o East, and a population at 315^o East (ie 45^o West). Your proposition would average them out to a location of 180^o. Not quite true, methinks. You cannot use a reference grid on the sphere surface that has a zero start point and 360 (or whatever) at the other end. x,y,z is the way to go, though it's better to define direction cosines to each axis. More on that later.

My centre of gravity approach was intended as an analogy for people to visualize the problem. The result would inevitably end up somewhere pretty absurd (perhaps I should say 'hellish'), though it's a reasonable first method. If the populations were considered as point weights on the inside of a spherical surface, the sphere would roll so that Mali was lowest. In a somewhat dubious way, it represents the location that most of the members (that I used) are closest to.

Hight above sea level is not considered at all, neither is the oblateness of Earth. I'm simply using CR4's membership of country data as input for analysis of data points on the surface of a sphere. To cut down the data for anyone mad enough to contest my final results, I've made generalizations. The capital city is use for each country, although I felt it more amusing to use Troy, NY, for our American chums (stops the Texans from screaming for my blood to loudly)

I'll remedy any gross mistakes in my tabulation of city locations, though any change in numbers who sport their national flag is insignificant to the overall problem.

Fear not, I intend to impose a large amount of international mixing in the final analysis. Whether America moves towards Europe, or the other way around, remains to be seen. Tempting as it is, I shall ignore the fact that we have Obama visiting early next month.

I shall leave a very ambiguous clue as to the nature of the real problem. (Enjoy, Del !)