Statistical Mechanics Revisited (2)

Issues of order and disorder occupy the top rungs of that branch of physics we call statistical mechanics. In this discipline, we apply principles from classical mechanics as well as microscopic thermodynamics to arrive at properties of complex systems and their evolution. In so doing, we can expose the conditions under which spontaneous order can arise from chaos, as well as ascertain systems in which equilibrium applies.

One example is the so-called Markov process which is at the heart of an important model used in statistical mechanics. As an example of this (Ehrenfest) model, one can consider a number of marbles, N, distributed among containers C1 and C2. At precise periods, say every 2 seconds, one marble is randomly selected from C1 (with X initial marbles at time to) and moved to C2 which must have N – X marbles by inference. It can be shown that at some arbitrary future time:

t(o) < t  < t(o) + tN/2


There will be either X + 1 or X –1 marbles, with transition probabilities: X/N corresponding to X -> X - 1, and 1 – X/N for X -> X + 1. Eventually, assuming the initial number of marbles large, equalization (‘equilibrium’) will occur with N/2 marbles in each container. The sketch of the approach to equilibrium is shown in Fig. 1.

We expect that as the process continues, oscillations will stabilize about the line X = N/2, so effectively the transition probabilities equalize. The important point of this process is that each such transition is independent of the system’s prior history. Say, there are just 16 marbles in all initially with 11 in C1 and 5 in C2 at t(o). Then successive distributions might appear like this, leading toward equilibrium (equalization of C1-C2 distributions):

C1: (11) -> (10) -> (9) ->(7) -> (9) -> (7) -> (9)

C2: (5) -> (6) -> (7) -> (9) -> (7) ->(9) ->(7)

Notice how the numbers relative to each container ultimately settle into small deviations from the equilibrium value (N/2 = 8). Of course, the probabilities reflect this, and can be represented on a separate graph for probability density plotted against X. This is shown in Fig. 2. In effect, this is the standard ‘Gaussian’ distribution. Mathematically, one can obtain it starting with the basic diffusion equation:

    ¶ r / ¶ t    = D*DIV 2  r 

And applying suitable boundary conditions, then using a Fourier transform to solve. This indicates all Markov processes can reduce to a form of diffusion.  

The probability density peak at N/2 discloses this to be the most probable state. The reader should again bear in mind, however, that we are assuming thousands of marbles, transitions, to get it! Fig. 2 illustrates the direction in which all physical processes tend to run: from more order to more disorder.[1] By definition, in physics the equilibrium (or ‘equalized’ distribution) state is the one with the greatest disorder.

An interesting analog to the above is based on polymers, using Monte Carlo simulations in what is called ‘the two space algorithm’.[2] In one such computation it is found that a characteristic scale size for the polymer (given by a defined ‘radius of gyration’) fluctuates about some final equilibrium value. Such simulations would seem to have direct bearing on work[3] that provides a model for pre-biotic evolution. In particular, that in given environments a single macromolecule can emerge to dominate, with all others fluctuations only[4].

In part (1) we examined magnetic permeability and the degeneracy function. Accessing such quantities enables us to depict specific types of bifurcations.   The term appears abstruse but simply means a particular dynamical problem solution splitting into two parts. An actual example from fluid dynamics is the famous ‘pitchfork bifurcation’, which has nothing to do with demons! It arises by considering the complex interactions of a controlled water channel whose flow continuously recycles. At some point, beyond a critical value of the Reynolds number (R_c), the single flow relinquishes its symmetry and two stable flows result. These are shown in the figure below:

Fig. 3   Pitchfork bifurcation.

The critical value (where the vertical dotted line intersects the abscissa) turns out to be 40.5. As before, with the Ising model, we see that hidden complexities manifest in a kind of order or self-organization for each of the bifurcation paths. Indeed, each one of these paths can be thought of as mirror images of the same Markov process, tending to some new ‘equilibrium’ displaced from the original one.

An interesting but slightly more complex example from plasma physics is the ‘two stream instability’. In this case we have the plasma dispersion function F(w) which leads to two bifurcation ‘paths’, including a split symmetrical one and a symmetrical one similar to that shown below- but in a different direction relative to coordinate axes.

Fig.  4.   Profile for two-stream instability. 



In this case we have the plasma dispersion function F(w) which leads to two bifurcation ‘paths’, including a split symmetrical one and a symmetrical one similar to that shown- but in a different direction relative to coordinate axes. 

In finding conditions under which it operates, one considers treating a dispersion relation for plasma waves such that, viz.:

 F(w) =   (m_e/m_i)/ (  w / w _e)² + 1/ [(w / w _e )  ² - (k V_o/ w _e)²  ]

(N.B.   A dispersion relation implies that a relationship exists between the plasma frequency w and the wave number k.)

Where (m_e/m_i) denotes the electron to ion mass ratio, and  for further analysis (i.e. to find applicable roots) we can  define the variables x, y as follows:

x = w / w _e

or the ratio of the plasma frequency to electron plasma frequency.

Meanwhile:

y = k V_o/ w _e

or the ratio of the product of the wave number k by the electron thermal velocity (V_o) to the electron plasma frequency.   Plotting the graph on the axes yields a bifurcated graph with 4 roots (Fig. 4). It will always feature a local minimum F_m such that: 0   <   F_m    <  x=y.

When: F(x_m, y) = F_m  < 1 there will be four real roots.   (Fig. 5)

When F_m  > 1 there will be two real and two complex roots with suitable approximations for the latter , e.g. 

k² V_o ²    £  w _e ² then the 2 complex roots are found to be:

w / w _e  =   - ½ + i((Ö3/2),   and     - ½ -  i((Ö3/2) 

These will give the limits for the instability for when F_m >1


Problems:

1) For the plasma bifurcation, when: F(x_m, y) = F_m  >  1 there will be two real and two complex roots. Find the two real roots. 

2) Explain how the plasma bifurcation above occurs mathematically.

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[1] This, of course, is from a thermodynamic viewpoint. In order to elicit thermodynamics from Markov processes, one must somehow assure that the probabilities in question approach that for the Boltzmann distribution, viz. P = exp(-E/kT)/Z, where Z is the partition function.

[2] See, e.g. Yaneer Bar-Yam, 1997: Dynamics of Complex Systems, Addison-Wesley, p. 499.

[3] See, e.g. Eigen, M. and Schuster, P.:1979, The Hypercycle, Springer-Verlag.

[4] The magnitude of a fluctuation can be obtained from probability parameters, in this case multiplying the rms deviation [N]^1/2/2) by 1/N, or in the case of the polymers, [R_G]^1/2 /2) by 1/[R_G]^1/2.

Thanks To Grade Inflation University 'Cum Laude' Honors Are Now Meaningless

In May, 2013, I cited a study by Stuart Rojstaczer and Christopher Healy, published in the prestigious Teachers College Record , revealing that about three-fourths of all grades awarded at university level are “A”s or “B”s.   As I observed at the time, this makes those As and Bs next to useless precisely become of the very commonality. An 'A' used to stand for academic excellence, but it can't if so many are getting them! It also renders the achievements of truly exceptional students ho-hum. How in the world can they truly stand out if middling or loser students get the same grades they do? It's preposterous!

I went on to point out that if one went back to the mid 1960s such a college level grade anomaly would not be found, unless it was at some "Flunky U" or maybe a junior college. What did one find, say at Loyola University- New Orleans, or the University of South Florida?  Well, the proportion of As were fairly stable at about 10 percent, with Bs at 20 percent, and 'gentleman's Cs' right at around 40 percent where they ought to be.  Hence, in a General Physics class of 400 students - say at USF-  one expected to see maybe 40 As, 80 Bs, 160 Cs, 80 Ds and 40 Fs.

In other words, grade frequency conformed to the standard Gaussian distribution or normal curve. Similarly, at the other end of the curve Ds would make up 20 percent and Fs 10 percent. Even in the typical astronomy class, say astrophysics - the grade distribution tended to follow the same pattern. So for maybe 10 students in AST 443, 1 would receive an A,  two got Bs, four received Cs,  two received Ds and there was one F.

But what do we find today? Barely 5 percent Ds, if that many,  and zero percent Fs! Essentially, college teachers today - tenured profs as well as adjuncts- have given away the "grade store" and sold out . This has occurred either from capitulating to threats of bad teacher evaluations, or kowtowing to pesky ass parents who insist Junior and Missy not be docked points on a critical test.

So, given this shabby situation it's no surprise one reads (e.g. 'You Graduated Cum Laude? So Did Everyone Else',  WSJ, p. A2, July 3):

"Honors designations have become close to the norm at many top schools, according to a Wall Street Journal review of the criteria for earning honors and the percentage of the senior class that got the designation in the top 50 of the WSJ/Times Higher Education Ranking.

The share increased to 44 % from 32%  in the past decade at USC, which requires a GPA of at least 3.4 for the lowest honor- cum laude- and to 44 % from 39 % at Lehigh where students need at least a 3.4."

Adding:

"At Wellesley College, 41 percent of this year's graduating class completed their degrees with Latin honors which means  GPA of at least 3.6"

Meanwhile, at Middlebury College  the proportion of students graduating with Latin honors was "north of  50 percent this spring" - which is ridiculous.

We are also informed "most elite schools cap the share of graduating class that can receive academic honors"  but "the caps vary widely".   For example, from 25 % at Columbia, to 60 % at Harvard. Again, this is preposterous, but at least Harvard re-calibrated from its "91 percent in 2001, highlighted in a Boston Globe article about generous honors policies".

And my essential point, reinforced in spades (ibid.):

"Academic researchers say the uptick is a sign of grade inflation, not smarter students."

Correct, because you simply cannot get that many smarter students! In many ways this farcical phenomenon resembles the claimed "Flynn effect" where most Americans are allegedly getting smarter and smarter each year as gauged by standard IQ tests. The phenomenon of incrementally increasing IQ first circulated in 1984, following a study in that year by James R. Flynn, purporting to show that citizens in advanced nations like the U.S. have experienced massive IQ gains over  time.

Thus Americans – for example – have gained 3 IQ points per decade from the early 1900s to today, as reflected in both the Stanford –Binet and Wechsler Intelligence scales. By another test’s standards (the Raven’s Progressive Matrices) – for which scores go back to people born in 1872- the gains disclosed amount to 5 IQ points per decade. So a guy with the same genetic background who was born in 1910 and had an IQ of 100 then, would attain an IQ of 160 by 1970 thanks to the Flynn effect, or equal to the (accepted) IQ of Einstein. Of course, the Flynn effect is supposed to apply to a statistical ensemble, not individuals - but I use the example of an individual measure over decades to make the point the claim is nonsense.

Even if we just stick to ensembles, it is foolish. Using the Raven’s and scored against today’s norms,  our ancestors in 1910 would have an average IQ of 70, or about moron level. By comparison, our mean IQ today – that is, disclosed within the ‘hump’ portion of a Gaussian distribution – would range from 130 to 150 depending on the test. For reference, 130 basically gets you into Mensa (accepting the top 2% of IQs) and 150 marks you as a “genius”. Are we all geniuses on our way to becoming Über-Geniuses? I don’t think so!  Neither are all the cum laude honors recipients at Harvard really cum laude quality.  They achieved that honor via a skewed grading system.

Most importantly, the ranks of neither Mensa or Intertel have increased markedly - nor have the potential members who would be accepted.  (Taking into account all those who've no interest in joining either one). Select a random sample of the populace, say 10,000 or 50,000 – and dispatch them to sit the Mensa and Intertel IQ tests. Those making the cut will still be only 2% and 1%, respectively, as has been the case for decades.  You will not find 50 percent getting through, or even 20 percent.

And you will certainly not find the mammoth proportions now receiving cum laude honors! An anomaly that suggests that either the courses taken are way too easy, or the professors - TAs are marking assignments, exams too easy.

Make no mistake, because of grade inflation, students use 'Rate My Professor' to  avoid professors who believe the grade of “C” is the average grade and who set up standards that require students to do more than show up, read a couple of hundred pages, and answer a few questions. This then translates into fewer students in the more rigorous courses that also feature more no nonsense profs.

Now, it's true some departments traditionally grade tougher than others, say like science and engineering departments. In my experience and at least through 1985, all tended to have lower overall grade averages than those in social sciences and humanities. The reason is that they adhered to rigorous Gaussian curve models for each test and homework grades.. If then 50 people took a class, say in complex algebra, and the grades ranged from a maximum of 80-85%  - which 5 got, and a minimum of 40 -45% which another 5 received, then those limits defined the extent of As and Fs, respectively.  The large central tendency bunch (say 28 in number)  that scored between 50% and 70% would ALL receive Cs,  no question. All the rest would get Bs and Ds, depending on whether they fell between 70 and 79 or below 50.

By contrast to the above, we've learned that typical Education, Early Education programs tend to have the highest course grade averages. It’s not unusual for the average grade in elementary education courses to be an A-minus, and in secondary education to be a B-plus. That means either our future teachers are brighter than a supernova—or that their profs don’t know there are more than just two letters in the alphabet. More likely, it means very sub-average or average students are taking relatively easy courses. The proof of this? Looking at years of GRE  test scores taken by Ed. majors vs. those of science majors. In year after year, Ed.  major grads' GRE averages seldom crack the 950 total for both math and verbal, while science grads routinely crack 1200-1250. Case closed!

Lastly, one has to factor in the role of parental expectations and pressure into the ongoing inflation of grades. Because every parent believes his kid will either be another Fortune 500 CEO, or a Billionaire hedge fund owner, he mandates to every university staff member that Jr. is expected to get As. Not to perform to the highest standards, but ...get those As. And woe betide the sorry butt of any prof who doesn't cooperate. He will be beseiged by angry emails or phone calls.

Administrators play into this absurd game, because they are the ones who've made the sine qua non index for their universities attaining a critical mass of students who consistently garner high enough grades to remain - to keep the money pumping in from Mommy, Daddy or more often now, the private lender. Better this critical inflated grade mass of lazy ass students than that they actual develop critical thinking skills, and oh yes...absorb some knowledge.

Sadly, there's no sign the trend is going to reverse any time soon, but there are solutions, none of which entitled students and their helicopter parents are likely to accept. The most plausible change would be to the grading scale. Janice herself has often stated how "stupid" the American grade scale is, as it is "so squeezed at the top".   I.e. "passing" only begins at a 70 percent mark with all 4 passing grades compressed into 30 points, (70- 100).

As she puts it: "Why not have at least 50 marks allocated to passing, so you fail only if you get below 50?"  At Harrison College where she attended, for example, anything below 50 % was an F, no arguments. Grades from 50-60 rated a D, from 61-70 a C, and From 71-80 a B. Marks higher than 80 earned an A but virtually no student attained a 93 % the usual starting point for an A in the U.S. The reason is that exams at HC were genuinely difficult - so much so that even the 'cream of the crop' seldom  scored even 90%.

Two sample problems from one of my own Calculus Physics final exams are given below;

1)Find the root mean square velocity of a molecule of hydrogen at a temperature of -20 C on Mars’ equator if the atmospheric pressure is 0.0056 bar. (Earth’s is 1.0 bar). Take the molecular weight of hydrogen as 2.016 g/mol.

b) A group of 4 astronauts lands on Mars with solar radiation collection material of total area 2000 m2. If the efficiency of the material is 30%, and the ambient night time temperature on Mars (for their base location at Isidis Planitia) is -40 C (10C day time), will they have adequate collecting material if the solar constant on Mars is 620 W/ m2  ? (Assume insulating material with a thermal conductivity of 0.08 W/mC, and a need to keep the inside area of their domecile at least at 10 C, requiring solar radiant energy collected of at least 1,200 W per minute for an area of 10 m x 10 m.)

c) Estimate the thickness of insulating material they're likely to need in order to make it work. Comment on whether this expedition is even feasible given the limits of their materials, and that no more than 100 m ³  of insulating material can be taken.

2) Examine the pendulum system shown in the diagram below.

Here, h2 = 1.7 m and h1 = 1.5 m. Given a pendulum length, L, write out the Lagrangian for the system, i.e. difference between its kinetic and potential energy, using: g, L, h2, h1 and the deflection angle Θ.

b) Hence or otherwise compare the velocities of the pendulum bob of mass m (= 0.1 kg) for the same values of h2, h1 if an experiment evaluating energy change is conducted on both Mars and Earth at the same time, for the same deflection angle Θ and length, L = 1.0 m. (Take the acceleration of gravity on Earth as 9.8 N/kg,and on Mars as 3.7 N/kg) Compare also the potential energy in each case. Why or why not would these be different? (Take the deflection angle to be 15 degrees in each experiment.)

Five other problems of similar difficulty were included with the student having to choose 4 of the total and complete the exam in 90 minutes.    The highest score obtained for this exam was 71%, for a class of exceptionally bright students. 

Note that the level of difficulty of the physics exams are pretty well the same as for Chemistry, Biology, English Lit, History, Math etc.  For example, two problems (of 7) for an HC Math final where 5 in all had to be answered in 90 mins.:

1)(a) The coordinates of the points L and N are (5, 6) and (8, -2), respectively.
(i)State the coordinates of the midpoint M of the line, LN.

(ii) Calculate the gradient of the line LN and propose, with justification, the equation for a parabola to which the line LN would be tangent at the point (8, -2)

(iii) Determine the equation of the straight line which is perpendicular to LN and which passes through point M.  Propose, with justification, the equation of the circle that is tangent to this line.

(b) An aircraft leaves Jamaica at 13:55 hrs. and travels to Barbados via Antigua. The average speed of the aircraft is 420 km/hr. It arrives in Antigua at 16:45 hrs. local time. Given Antigua is ONE hour AHEAD of Jamaica, compute the distance between Jamaica and Antigua.

2) (a) Using Fig. 1 and the information therein, calculate (giving reasons)


i)  The Angle MSQ

ii) The Angle RSP

iii)  The Angle SPN

b)   The matrix R =

(cos(Θ)......-sin(Θ))
(sin (Θ)......cos(Θ))

i) Determine the coordinates of the image (1, 2) under the transformation R when Θ = 90 degrees.

ii) If the point (p, 3) is on the line (L) given by: x + 2y = 5, calculate the value of p.

iii) Given the point (1,2)is on L, determine the image of L (L') of the line L under the transformation R.

iv) Write the matrix equation to represent the pair of simultaneous equations given by L and L'.

No surprise that highest honors at Harrison College were awarded for course grade averages of 70 percent or higher. (No GPA point grades were used.)  In no way and at no time did the percentage of highest honors ever exceed 1 percent in a given year. Further at no time did the equivalent of U.S. 'cum laude' honors ever exceed 5 percent.

At HC honors implied a genuine distinction - not being one of a large, similar achieving pack that benefited from grade inflation!

"Super High" IQ Scores - Why They Are Impractical & Meaningless

Parade  magazine contributor Marilyn vos Savant was once claimed by The Guinness Book of Records to have the highest IQ ever recorded. By one measure- a test first taken by her in 1956 - it was claimed to be 228 or a mental age of 22 years and 10 months at the age of 10.  A recent article ('The 21 Smartest People In The World')  at salon.com also claimed to have a list of the "world's highest IQs" with a number even higher than vos Savant's and many with IQs higher that Albert Einstein's (160). But is this really possible? Do such high IQ scores even have any credible meaning?

 In the case of vos Savant, according to Wikipedia:

" Alan S. Kaufman ,a psychology professor and author of IQ tests, writes in IQ Testing 101 that "Miss Savant was given an old version of the Stanford-Binet (Terman & Merrill 1937), which did, indeed, use the antiquated formula of MA/CA × 100. But in the test manual's norms, the Binet does not permit IQs to rise above 170 at any age, child or adult. And the authors of the old Binet stated: 'Beyond fifteen the mental ages are entirely artificial and are to be thought of as simply numerical scores.' (Terman & Merrill 1937). ...the psychologist who came up with an IQ of 228 committed an extrapolation of a misconception, thereby violating almost every rule imaginable concerning the meaning of IQs."

 The list of the "world's smartest people" featuring over a dozen with higher IQs than 160 (for Einstein) is also daft and a classic example of what author Charles Seife would call "proofiness", discussed in his excellent book, : 'Proofiness: How You're Being Fooled By the Numbers'.  In page after page Seife decries the use of numbers not merely to lie but to baffle with bullshit. No better example of such proofy twaddle can be seen than in the website below:

http://members.shaw.ca/warmbeach/INDEX3.htm

But alas, super high IQ scores are also rife with proofiness. Further understanding of what I am about can be grasped by reference to the standard (normal) Gaussian distribution or "Bell curve"  on which the distribution of IQ scores is based, e.g.



Note the percentages of the population decreasing at both the high and the low ends, corresponding to the number of standard deviations  (stds) from the zero point or mean.

We see, for example, that by a 2.2 std dispersion we are effectively at a population fraction of 2 percent - which is the cutoff threshold for qualifying in Mensa. Generally, this is taken as an IQ of 133-35. (Depending on the test used). Note also, how the corresponding population applicable is decreasing as the number of stds gets larger from the central point. At about 2.6 std we are near the threshold (or just slightly beyond it) for the top one percent or Intertel members. At 3 std (about 145 IQ) one qualifies for the Poetic Genius Society.

Using a base of 300 million as the U.S. population, the Bell curve proportions would translate into 6 million who'd qualify for Mensa entrance, and 3 million who'd qualify for Intertel.  By about 3.3 stds we reach the 0.1 percent cutoff and the threshold for the Triple Nine Society. The population qualifying is one-tenth that for Intertel or 3 million/ 10 = 300,000. Divide 300,000 by 300,000,000 and you get 1,000 - hence the Triple Nines are also known as the "One in one thousand society".  

Note how already we are near the effective upper limit of the Bell curve in terms of being able to discriminate between the IQ groups. At near 4.5 stds, however, there is the Mega Society which will have a threshold at the 0.0001 % level. Or doing the same math as I showed in the previous paragraph, only 1 in a million qualifying. This would imply 300 in the U.S. population of 300 million - and with an IQ of 171. But what does that even mean, and is it useful?

As the relevant article in Wikipedia points out:

"No professionally designed and validated IQ test claims to distinguish test-takers at a one-in-a-million level of rarity of score. The standard score range of the Stanford-Binet IQ test is 40 to 160.The standard scores on most other currently normed IQ tests fall in the same range. A score of 160 corresponds to a rarity of about 1 person in 30,000 (leaving aside the issue of error of measurement common to all IQ tests), which falls short of the Mega Society's 1 in a million requirement."

In other words, there aren't even any IQ tests that currently exist which achieve the level of discrimination to identify a would-be Mega Society member! The highest score achievable (and hence measurable) within the constellation of intelligence allows no higher than 160 - which was Einstein's IQ.

Consider the mean or 0 mark which sets the "average IQ" at 100. This means half score above this level and half below. At about 4.5 std, the threshold for the Mega Society we are already in uncharted territory. This is why it is likely impossible to design a test adequate to ferret out those 300 potential members in the U.S. The Mega Society insists it has "unsupervised IQ tests that the test author claims have been normalized using standard statistical methods" but this is very doubtful.  Most psychometricians concur that by 3 std (145) to 4 std (160 IQ) one is already at the limit of the measurable validity for testing IQ in any useful way. Beyond 4 std (again, Einstein's level) the difference between such scores blurs.

Again, in considering such stratospheric IQs one ought to look not only at the std dispersion from the mean but the area under the curve. At 4.7 std the area is essentially nil. What about claims of a 300 IQ? This is even more preposterous. We are now talking about 13.33 std from the mean. As one contributor put it on an IQ- statistics site:

"An IQ of 300 has no useful meaning because there aren't enough people in the world (or intelligent beings in the history of the cosmos) to make it useful".

Or, again, no test that could possibly discriminate that single potential member from 6 billion people. (Other estimates put the proportion in even more rarefied terms, i.e. 1 in 50 billion - or nearly 1 out of every other person who's ever walked the Earth).  In effect, it means that any article bragging on one or more people with supposed IQ scores exceeding Einstein's is preposterous. Even the Mega Society entry threshold is likely a Macguffin, given that in reality there is no living human with an IQ that high that can be satisfactorily verified at the necessary level of statistical confidence.

Even if such a "Mega" person really existed (irrespective of how many claim to be Mega Society members) would one really see a qualitative difference, say from an "Einstein-level" IQ person? I doubt it. I can't see a Mega Society member being able to do any more than Einstein did, including developing the tensor calculus to use in his General Theory of Relativity.

Thus as one of the web psychometricians put it, even a 150 IQ person (near the top of most IQ test standards) would not see a significant difference between himself and a 160 or 170 IQ person. Where the real differences would be seen is where the areas registered for standard deviations are the most.  For example, between a normal IQ person and a Mensa (upper 2 percent) IQ person. Their interests, and even vocabulary would likely be so different as to approach the analogy of an alien trying to communicate with a typical earthling.

But this is precisely why Mensa as an organization was created in the first place, to establish a communal civic space where gifted individuals could meet and converse without being put down by pejoratives (Dweebs, or "geeks") from "normals".  Intertel was launched for similar reasons, though it is interesting to note that many Intertel members also belong to Mensa - simply because it is vastly larger and there are more opportunities for social exchanges.

As an interesting side note, at Monsignor Edward Pace High in N. Miami one of the first  segregations transpired after we all took an IQ test (Stanford -Binet) in 10th grade. In the immediate aftermath I noticed a loss in the diversity of the student body with many of those perceived "slower" (but often more interesting, joking and friendly) leaving Pace for good. I was also encouraged to leave, but for a different reason:  to attend Nova High where I could advance at my own pace and not be held back by a less challenging curriculum. I decided not to take the advice because it would mean 100 extra miles a day total transporting by my dad.

The takeaway from this blog post? Don't trust any claims of people with IQs over 160!

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Sample questions from the "Sigma Test" similar to the one to qualify for the Mega Society

1) Several faucets were used to fill up six tanks. For one hour, all the faucets discharged water in a reservoir, which distributed it between four of these tanks: A, B, C and D. After that, for one hour, the faucets discharged water in a double funnel which directed half of the water to tanks E and F and the other half to the reservoir which, in turn, continued to distribute its water between tanks A, B, C and D. With this, tanks A, B, C and D were full. To fill tanks E and F up, it was necessary to use one faucet, which, for two hours, distributed its water between tanks E and F. After this all the six tanks were full. What was the number of faucets initially used? (Note: all the faucets had the same water flow rate and all the tanks had the same volume).
           

2)  Several rectangles are drawn on a plane surface in such a way that their intersecting lines form 18,769 areas not further subdivided. What is the minimum number of rectangles that must be drawn to form the described pattern?

           

3)  Several straight line segments are drawn on a plane surface in such a way that their intersecting lines form 1,597 areas that are not further subdivided. What is the minimum number of line segments that must be drawn to form the described pattern?

           

4)  1 + 10^1,234,567,890 triangles are drawn on a plane surface. What is the maximum number of areas, not further subdivided, that can be formed as these triangles intersect each other? (Contributed by Rodrigo de Almeida Rodrigues)

           

5)  According to Fermat’s Last Theorem, a^n + b^n = c^n has no solutions for n > 2 (a, b, c and n must be positive integers). In 1992, I proved this in a simple, yet incorrect manner. This was my reasoning: Fermat’s Theorem is a generalization of Pythagoras’ Theorem, which asserts that the sum of areas of the squares drawn on the legs (short sides) of a right triangle equals the area of a square drawn on the hypotenuse of the same right triangle (a^2 + b^2 = c^2). If we try to generalize that theorem, going from 2 to 3 dimensions (a^3 + b^3 = c^3), we have a triangular prism formed by displacement of a right triangle along an axis perpendicular to its face, as illustrated by the figure below. 

We can construct a cube on one of the three quadrangular faces of that prism. Two of those faces correspond to the legs of the right triangle (ADFB, BFEC) while the larger face corresponds to the hypotenuse (ADEC). It is possible to construct a cube on one of the faces, implying that the 4 sides of that face have the same length. This affects the whole prism, causing the cube constructed on the other face to have the same size than that constructed on the first, for if AB=BF and BF=BC, then AB=BC. In that way, no cube can be constructed on the third face, for if AC represents the hypotenuse, then AC cannot be equal to to AB. Therefore, a^n + b^n = c^n has no solution for n=3. Following the same line of reasoning, we can show that it has no solution for any number of dimensions larger than 2. What is the error in this proof?

6) A certain gear system consists of 5 concentric, superposed discs: A, B, C, D and E, which are mounted on a solid platform, taken as a stationary reference. The discs have different sizes and spin at different speeds. All the discs spin at constant rates, some clockwise, some anticlockwise. Each disc has a red dot on its surface, and initially all these red dots are not lined up. At a given moment, all the discs start to spin simultaneously, each at its own speed, without any contact between them. It takes 7 minutes for disc A, 13 minutes for disc B, 17 minutes for disc C, 19 minutes for disc D and 23 minutes for disc E to complete a full 360-degree spin. After a certain time, all the red dots were aligned, disc A being in the same position that it was 2 minutes after the discs started to spin, disc B being in the same position that it was 3 minutes after the discs started to spin, disc C being in the same position that it was 4 minutes after the discs started to spin, disc D being in the same position that it was 7 minutes after the discs started to spin, and disc E being the same position that it was 9 minutes after the discs started to spin. How much time elapsed from the moment the discs started to spin until the discs reached that configuration for the first time?

7) In 1993, in an essay about Science and Religion, I described a project regarding the possibility to build an “invisibility machine”. On describing the details, I realized that some problems were insolvable, not only because of technological limitations but also for physical reasons imposing theoretical and possibly insurmountable limits. The project starts from the central idea that in order to make an object invisible, it is necessary for an external observer looking in its direction to visually stop noticing its presence. This can be done in the following way: A sphere is constructed, and its whole external surface is covered with minute, high-resolution TV cameras and monitors. Millions or even billions of cameras and monitors are to cover the whole sphere in such a way that each monitor transmits the image captured by a camera located in the point diametrically opposite to that monitor. The result will be as shown in the figure below. 

The image of the object (blue square) is captured by a camera located in point A, which transmits the image to a monitor in point M. As a result, an observer in point O will see the blue square as if there were nothing in front of him. In that way, everything inside the sphere will be invisible to the external observer. But this scheme presents two problems. One of them can be solved in theory while the other one is insoluble. Indicate those two problems and explain why one of them can be solved but the other one cannot..

8) The porous and gray “lead” inside a pencil consists of a mixture of graphite and clay. The ratio of graphite to clay is not known. On writing on a sheet of paper, a fine layer of “lead” remains on the surface of the sheet. Describe a method for calculating the mass of “lead” in the dot of the letter “i”. You may use only US$10 to buy the material needed for the experiment..

9)  We have a cylinder with a radius of 50 cm and a tape measure 0.01 cm thick. The height of the cylinder equals the width of the tape measure. The thickness of the tape measure is invariable and one of its wider sides is inextensible. What is the minimum length of tape necessary to wind it around the cylinder 9 times, all rounds overlapping, as in a roll of scotch tape. The top and base of the cylinder may not be covered with tape. The solution must be given with 14 significant digits and it is not allowed to cut the tape or cut or deform de cylinder.

10)  Describe a practical and fast method that can be used with good precision to determine the number of words in a person’s vocabulary..     

Grade Inflation Continues to Render Most College Scholastic Honors, Achievement Meaningless

The depressing recent study by Stuart Rojstaczer and Christopher Healy, and published in the prestigious Teachers College Record is enough to make any educator shake his or her head in despondent resignation. Their finding: About three-fourths of all grades awarded at university level are “A”s or “B”s.

Of course, this makes those As and Bs next to useless precisely become of the very commonality. An 'A' used to stand for academic excellence, but it can't if so many are getting them! It also renders the achievements of truly exceptional students ho-hum. How in the world can they truly stand out if middling or loser students get the same grades they do? It's preposterous!

There is NO way in a real universe, there can be such a preponderance of high grades! Go back now to the 1960s, before the emergence of the surreptitious blackmail device known as "teacher evaluations". What did one find, say at Loyola University, or the University of South Florida?  Well, the As were at about 10 percent, with Bs at 20 percent, and 'gentleman's Cs' right at around 40 percent where they ought to be - if conforming to the standard Gaussian distribution or normal curve. Similarly, at the other end of the curve Ds would make up 20 percent and Fs 10 percent. But what do we find today? Barely 5 percent Ds and Fs and Cs marginally higher because college kids consider those failing grades!  This is nuts!

Essentially, college teachers today - tenured profs as well as adjuncts- have given away the grade store and sold out.  And it's irrespective of whether we're talking about State U. or Harvard. Intimidated by little wet behind -the -ears punks delivering solemn, negative judgments via teacher evaluations, they've decided timidity and wussing out are the better parts of valor.  But in the process they've essentially wrecked all academic credentials, judicious comparisons and objective assessement.

Oh, make no mistake, it was done with the best of intentions. College administrations sought a cheap, expeditious way to evaluate their staffs, and 'Voila!' Some genius thought of a teacher's evaluation. Just hand out a little one page eval form to little Missy or Sonny and let them have at it. What they obviously didn't consider is that neither Missy or Sonny had the maturity to do a proper, objective assessment. No, by the time they received the forms they were already grating at low marks they'd received during the year and now, and NOW....this was the time for payback! And payback is always a bitch!

So, for the few curmudgeons who continued to demand standards as opposed to giving out A and B freebies, it was game over. For those profs who insisted that their students EARN their As instead of expecting them for just showing up, well, it was 'hasta la vista'. The college administration had to inform the uncooperative fool that this was the end of the line.

Meanwhile, for those timid souls that capitulated, the majority, the sky was the limit - not only were they popular, oh so popular, especially for the easiest, 'crib' courses like media, or deconstruction of films by Steven Spielberg. They were the ones that most often notched promotions and even got tenure. But at what cost? Well, at turning our university grading system into a global laughing stock. Because no where else in the world do kids get rewarded merely for turning in labs or papers half done.

Make no mistake, because of grade inflation, students avoid professors who believe the grade of “C” is the average grade and who set up standards that require students to do more than show up, read a couple of hundred pages, and answer a few questions. This then translates into fewer students in physics classes, say, and usually results in questions from administrators who may claim they believe in academic rigor and integrity, but whose slavish devotion to teacher evaluations refutes it.

Now, it's true some departments traditionally grade tougher than others, say like science and engineering departments. In my experience and at least through 1985, all tended to have lower overall grade averages than those in social sciences and humanities. The reason is that they adhered to rigorous Gaussian curve models for each test and homework grades.. If then 50 people took a class, say in complex algebra, and the grades ranged from a maximum of 80-85%  - which 5 got, and a minimum of 40 -45% which another 5 received, then those limits defined the extent of As and Fs, respectively.  The large central tendency bunch (say 28 in number)  that scored between 50% and 70% would ALL receive Cs,  no question. All the rest would get Bs and Ds, depending on whether they fell between 70 and 79 or below 50.

This sort of distribution was consistently applied.

By contrast to the above, we've learned that Education programs tend to have the highest grade averages. It’s not unusual for the average grade in elementary education courses to be an A-minus, and in secondary education to be a B-plus. That means either our future teachers are brighter than a supernova—or that their profs don’t know there are more than just two letters in the alphabet. More likely, it means very sub-average or average students are taking relatively easy courses. The proof of this? Looking at years of GRE  test scores taken by Ed majors vs. those of science majors. In year after year, Ed.  major grads' GRE averages seldom crack the 950 total for both math and verbal, while science grads routinely crack 1200-1250. Case closed!

Lastly, one has to factor in the role of parental expectations and pressure into the ongoing inflation of grades. Because every parent believes his kid will either be another Fortune 500 CEO, or a Billionaire hedge fund owner, he mandates to every university staff member that Jr. is expected to get As. Not to perform to the highest standards, but ...get those As. And woe betide the sorry butt of any prof who doesn't cooperate. He will be beseiged by angry emails or phone calls.

Administrators play into this absurd game, because they are the ones who've made the sine qua non index for their universities attaining a critical mass of students who consistently garner high enough grades to remain - to keep the money pumping in from Mommy, Daddy or more often now, the private lender. Better this critical inflated grade mass of lazy ass students than that they actual develop critical thinking skills, and oh yes...knowledge.

Sadly, there's no sign the trend is going to reverse any time soon. Too many, including colleges, students, profs and the parents, are addicted to the phony exaggerated grade system- just like coke.

So, as we approach the period of standard commencements across the nation....let's give three cheers for more bogus Summa Cum Laude's and oh.......Magnas too!