A model of decay of student living standards

Abstract: We propose a new theory for the decay of student living standards and compare it with previous efforts in the field (namely, the theory of Linear Extinction). Our theoretical model, Serial Subtraction (SS), takes a new approach on the issue of standards’ degradation and appears successful in predicting most of the observed phenomena even in the vicinity of the human-to-animal crossover. A complete analysis of the principles of the theory is given for many components of student living, including clothes and dish washing, food preparation and flat maintenance.

I. Self-disorganised Criticality
Relaxation dynamics in student lifestyles has been a long standing problem in the field of self-disorganised criticality [1-3]. The main question that has concerned scientists for ages is the following. How can students regress to subhuman living standards yet avoid a catastrophic collapse by equilibrating around the minimum effort human-animal transition? That is to say, how does the system disorganise itself in such a manner as to sustain a subhuman existence up to the degree ceremony (and in many cases [4] even beyond that) without the living standards descending all the way to the pigsty singularity?

The main quantity of interest is the evolution of the degradation index, K, defined for any everyday process (such as cooking, washing clothes etc.) as the sum of the effort put in each individual component of the process:

K= (Σ wi2)1/2 (1)

where the sum runs over all n components and wi is the effort of the ith component, usually, a ridiculously complicated function of caloric consumption subject. For example, for a typical 4-dimensional activity such as the dish-washing process, the components would be soaking, scrubbing, drying and replacing-on-shelf, each component carrying an associated effort. It is often useful thinking of K as the magnitude of a vector K in n-dimensional effort-space (notice that higher values of K represent lower degradation degrees, a slightly misleading convention adopted by the pioneers of self-disorganised criticality and one which remains the norm to this day [5]).

So far, little understanding has been gained of the evolution of K throughout the relaxation process. The only theoretical attempt at tackling this puzzling issue, the Linear Extinction theory (LE) of Vindero et al [6], has been only partially successful, reproducing most of the decay patterns observed yet failing to account for a low-effort minimum at which the system appears to equilibrate. LE theory predicts that the efforts, and consequently K, go asymptotically to 0 (the pigsty singularity), in clear contradiction with empirical evidence which support a long-time freezing of K at a finite value Kc around the human-animal transition.

The structure of this paper has as follows. In the next chapter we introduce the general principles of both the LE and SS theories together with short discussion on their implications. In chapter III, we examine the predictions of SS in more detail and then conclude the paper with a summary of our main findings.

II. Serial Subtraction vs. Linear Extinction

Let us now examine the principles of each of the two theories starting with LE using the paradigm of clean clothes’ procurement. In the whole discussion we will assume a male Drama student living alone with no loss of generality and 5-dimensional washing comprising of: (i) white/coloured clothes segregation (ii) washing (iii) hanging (iv) ironing (v) folding.

As mentioned briefly in the Introduction, LE assumes that the decay of the degradation index (again, this means a relaxation through successively more decadent states) proceeds through the linear decrease of the efforts [6],[7]. So, if a student initially spends 1 hour, say, on the ironing stage, he will begin to spend less and less time until w4 goes to zero in a finite time (and correspondingly for the rest of the n-1 components). Unfortunately, whilst this way of thinking does follow the decay closely for large K, for small K the predictions of the theory are much worse and, most importantly, the eventual fate of K is the collapse to the K=0, pigsty state. This means, that LE suggests that the student eventually stops washing clothes altogether.

SS, on the other hand, assumes that the change in the various efforts is negligent during the decay and that the latter is effectively achieved through the incremental subtraction of particular components. It is the basic premise of SS theory that a test student instead of spending less time in each stage of the cycle, he or she just drop certain stages completely. Although the off-critical (high K) results of SS are not as good as those by LE, SS successfully predicts both the behaviour of the degradation index in the critical regime and the existence of a human-animal transition at finite K in the long-time limit. The exact value of Kc is then determined by the effort in the few or only component left with noteworthy accuracy.

So, the main difference between the two theories is one of incremental against linear decay. It appears that the former captures better the intricacies of student dynamics in the interesting critical limit though, understandably, an interpolation between the two may be required at higher values of K.

III. Incremental decay in a range of household activities

In this section we illuminate the workings of SS by touching upon incremental disintegration of student living standards in a variety of situations. Our schematic listing of the decay sequence can and has been reproduced by numerous experiments [8].

(a) Clothes’ washing

Dimensionality in ideal form: 5
Stages: (i) white/coloured clothes segregation (ii) washing (iii) hanging (iv) ironing (v) folding
Weakest link: Ironing
Decay sequence: Almost invariably, ironing is the first stage to be dropped from the process with folding following closely behind. The asymptotics of the degradation index is predominantly dictated by the washing process (zeroth mode) with hanging regulating the first higher-order correction, although this stage can also be dropped in favour of the much more effortless in-machine drying.

(b) Dishes’ washing

Dimensionality in ideal form: 4
Stages: (i) soaking (ii) scrubbing (iii) drying (iv) on-shelf replacement
Weakest link: On-shelf replacement
Decay sequence: A counter-intuitive outcome of studying this process is the vulnerability of the soaking stage. That is, although soaking involves a high gain-to-effort ratio it is usually set aside because of considerations of student memory spans and also the intricate discipline and synchronisation which is required in providing a proper soaking bath after late-night student meals. Again, the critical index Kc can be often calculated by including only the scrubbing stage with drying being sometimes significant but evidently not to the hungry student. Restoring dishes to the appropriate storing unit is also bypassed especially given the small number of cutlery and crockery a student would on average posses.

(c) Meal preparation

Dimensionality in ideal form: 4
Stages: (i) ingredient procurement (ii) pan and pot procurement (iii) cooking (iv) serving
Weakest link: Both procurement phases are equally susceptible
Decay sequence: Because of the position of the weak links in the process relative to the preparation algorithm and the abnormally low gain-to-effort ratios involved at all stages, traditional meal preparation is reserved for special events only and is swiftly scrapped in favour of frozen chicken burgers and Pot Noodles.

IV. Conclusion

We have been able to show that the relaxation of student standards when left to live on their own is characterised by the incremental removal of stages in all household tasks instead of the proposed linear degradation mechanism where all stages are affected simultaneously. This is particularly true of first year students, especially Mathematics and Drama undergraduates.

It was also demonstrated that the ground state reached through the relaxation mechanism (commonly known as human-to-animal transition) is not, as previously assumed, the K=0 pigsty singularity but rather one whose K value is dictated by the least-effort stage of the respective composite process. Although we found some exceptions to this rule, namely dish washing, the general principle of SS seemed to apply well for the majority of household operations studied.

A comprehensive approach to the 2+2 problem

Abstract: The problem 2+2=p where p is some non-trivial integer has long been the Holy Grail of integer sum mathematics. Well, we report the Grail seized. Working on a number of new analytical, experimental, numerical and quantum mechanical techniques we fix the value of p to 4 with higher accuracy than ever before. In the process, we set earlier conjectures that p=24,886 to rest.

I. Introduction
Integer number sums have been with us for a long time now. Two, three and even higher term sums have been observed and studied in the context of a plethora of every day phenomena with a great deal of approximate and exact results having been published [1]. Still, despite ongoing efforts from Mathematicians and Theoretical Physicists the vital question remains: What is the result of the 2+2 problem?

Many techniques have surfaced in the past for tackling this highly non-trivial sum, going back to arguments published on prehistoric caves [2] on what the actual outcome of adding a 2 to an existing 2 may be. In the 16th Century, a number of theories were put forward, but none was successful at providing a conclusive solution to the problem, followed by further approximate and hand-waving (as well as hand-counting) arguments in the 17th and 18th Centuries. Early in the 19th Century, modest progress was achieved by showing that the result of the sum must itself be an integer [3], but this is pretty much how things stand to this very day.

In this paper, we attempt to settle the question once and for all with a strategic attack on all aspects of the problem. We shall be presenting theoretical arguments, numerical simulations and experimental results by the end of which it shall become apparent that 2+2 equals 4 as previously suspected with a high degree of confidence.

II. The Hoicin transformation

It has been noted by many researchers on the field, that the theoretical methods employed in the very similar 1+1 sum can not be applied to the 2+2 sum [4] hence it must be through the development of new tools that a successful theory of the 2+2 sum may emerge. One such tool is the Hoicin transformation [5], a little known technique which, although limited in its power, can at least constraint the number of possible answers.

The clever trick is to appreciate the symmetry of the sum. One can convince oneself that both terms in the sum – namely the left 2 and the right 2 – are the same number. This is extremely important for the Hoicin transformation to work since, after some tedious algebraic manipulations and topological arguments that are beyond the scope of this paper, one can show that the sum can be transformed to a product thus:

2+2 = 2*2 (1)

Now, it is widely acknowledged that products are, in general, much more complicated than sums but in this particular case the reformulation of the problem as a product is highly advantageous for the following reason. Since the product is one of 2 with another integer (and, in fact, this is a doubly robust statement due to the presence of not one but two 2s), it naturally follows that the answer must not only be an integer but an even integer. Consequently, an answer such as 2+2=5 can be discarded as ludicrous, although another like 2+2=24,886 is, in principle, entirely consistent with our conclusions thus far. After the examination of further evidence in the chapters to follow, it shall also be shown that 24,886 is, however appealing at this stage, entirely ludicrous as well.

III. Quantum mechanical analysis

We also attempted the construction of the wavefunction of the result in two ways. Initially, we operated with the summation operator on two distinct 2s and, then, operated twice on a third 2 with the ladder operator.

In this latter instance, we got a result 4 when we transformed back to ordinary space but we are not entirely convinced that our operation (essentially, 2+1+1) is formally equivalent in quantum arithmetic to the original problem. This result, although tempting, will require more work which we have already undertaken.

In the original summation operation, we got the result 6+f(2)*i where f(2) is a hideously complicated function of 2 and i is as usual (-1)1/2. Although there is no reason to discard this result, we believe that the outcome of the summation must be a pure, real number. Since the calculation, numerous colleagues have pointed out to me in private communications that the discrepancy may arise for one of two reasons: either the identical nature of the two 2s leads to some yet undiscovered quantum mechanical effect or one or both of the 2s were not prepared in the state 2 at all.

IV. Numerical simulations

We have performed extensive simulations of the 2+2 sum in a variety of Operation Systems and with a number of programming languages, since, by its very nature, this problem probes the very fundamental computational operations of any given programming environment. Our simulations consisted of 105 samples where for each sample the 2+2 sum was calculated over 1017 realisations and are, to the best of our knowledge, the most thorough simulations of the sum in the scientific literature.

In the first case, we used a C++ code running on Linux to get the result of the sum to 3.998+/-0.005, which for all practical purposes must be taken as a very accurate measurement. The most unreliable results came from running the same algorithm on Windows XP were a range of results was obtained. 71% of the simulations produced a value very similar to the one quoted above, 16% of the samples observed came up with a different numeric result � namely 2+2=779 � whilst the rest 13% of the data seem to suggest that 2+2=Crash. Unfortunately, the two latter answers can not be trusted since, by the Hoicin argument of the previous chapter, 779 is not an even number and Crash is inconsistent with the Van Graat [6] principle which states that the sum of two integers can never be a letter, phrase, small vegetable or miniature fire engine.

V. Experimental results

Experimentation on sums being as complicated as it usually is, we chose to set up several distinct experiments by which to test the 2+2 sum.

Firstly, 19 postgraduate students from our group were instructed to raise two fingers from the five in their right hand. Then, a further two fingers were raised from the left hand (although the same hand can be used without compromising the outcome of the experiment) and the total number of raised fingers was recorded. The procedure was repeated again and again over a 72 hour period with very interesting results.

In the first couple of hours of readings, the overwhelming majority of data seemed to converge to 2+2=4. However, as the experiment went on some discrepancies were recorded. Periodic fluctuations were observed by all experimenters at or around pub closing times and some alphanumeric results (most, of a rather explicit nature) were also recorded as physical exhaustion begun to set in. When the experimenters were approached towards the end of the 72 hour period, most experiments had collapsed with the postgraduates raising only one of the two sets of fingers and waving them menacingly against members of staff whilst repeating a two-word mantra.

In subsequent spectroscopic, chromatographic and telefluromatic experiments, a value of 4 was reached in good agreement with numerical results.

VI. Conclusion

We have used several alternative approaches to resolve once and for all a fundamental mathematical problem. Our data seem to suggest that 2+2=4 and it would be tempting to assume at this stage that this may indeed be the case.

It would be interesting to see how the techniques we developed for the 2+2 case can be used in tackling higher order sums. Currently, we are already working on the 3+3 sum as well as asymmetric sums, for which the Hoicin principle does not apply, but it is difficult to see how more complex sums could be calculated. Some topological techniques are being developed by various research groups, however it is the authors view that an exact solution to higher-than-4 sums is still beyond the grasp of current mathematical understanding. It could be the case that the new supercomputer under construction in Cambridge may just about have enough power to also brake into three-term sums of the lowest order.

Fascinating tetrahedron

What is tetrahedron?

I am tetrahedron. I have many facets. Four to be exact…
So you’re a four sided shape?
There are four sides…internally. And another four externally, of course. To achieve that I have six edges and four vertices, which makes me a three-dimensional object with a vertex configuration {3, 3, 3} so I have all the symmetry elements required to be a member of the tetrahedral symmetry group.
Are you real?
Tetrahedron exists in a realm of geometric space not normally perceived by humans. For obvious reasons communication is by means other than social.
Are there others like you?
Tetrahedron is a Fellow of the Society of Four-sided Objects, so there must be.
Are you the only pyramid?
No. But there are a few members who are indeed not pyramidal. Several open-ended objects, for instance.
Do you have a front and a back? Or top and bottom?
Obviously, being a regular three-dimensional object, it doesn’t matter which way up I am.
So, how do you see, for instance?
I can see from any of my four faces. This of course, gives me a unique perspective on things.
Do all your faces look the same? Are they the same colour?
My faces phase, as it were, into different colours, depending on my position relative to any electromagnetic radiation that should happen to be present.
You must be very beautiful?
Ah yes, if only you could see me…
How big are you, exactly?
Exactly, I’m not sure. It’s difficult to compare yourself to anything when your existence is somewhat tenuous. But I would say that I am not overly large. Quite small in fact.
If you’re that tenuous, does gravity have any effect on you?
I don’t have any need to be at rest on any surface, if that’s what you mean.
So, does that mean you can fly?
Perhaps fly is the wrong word. I tend to float about a lot, that’s for sure.
Where do you live?
I have no specific place of rest, though I tend to frequent certain areas within space-time which are familiar to me.
How do you type?
I roll around the keyboard. This is where being pointy has its advantages.
So you must be quite small then?
Yes, if by small you mean small enough to roll around on a keyboard.

So, what can you do with a tetrahedron, you may ask?

You can elongate tetrahedron, stretching me (a bit of a strain though).
You can stand me on edge or balance me on a vertex (or point, as prefer to call it).
You can join two tetrahedra together to form a tetrahemihexahedron (mmm must try that!).Tetrahedron can join with another this way too, forming a stella octangula, really a stellation of an octahedron (in fact, the only stellation). The tetrahedron is the only simple polyhedron with no polyhedron diagonals and it cannot be stellated, which is a bit annoying as I’ve heard it’s good exercise.

You can stack a few of my colleagues to form a tetrix.
You can slice tetrahedron so I’m truncated (though this would be rather cruel).
Open me out. Look I’m sunbathing!
You can squeeze tetrahedron for a hyperbolic version of my usual Euclidean self. I’d get rather puffed out though.
A number of compound tetrahedrons can be constructed by rotating tetrahedron about the centre of each face. Feel a bit dizzy.
You can stick an even number of tetrahedrons together to form a ring. The centre of the ring wiggles in or out showing different sides (sounds like a party!).
Inflated tetrahedron to form curved edges (I’d be inflate-a-hedron or tetraflate).
You can divide me into two trapezoid thingies (ouch!).
The tetrahedral way
A polyhedron is a three-dimensional figure bound by polygons. A polygon is a two-dimensional shape bound by straight lines. It is said to be regular if the edges are of equal length and meet at equal angles.
The simplest regular polygon is the triangle, therefore the simplest possible regular polyhedron is – you’ve guessed it – me!
You might consider the tetrahedron a humble polyhedron. Only four points in space mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron. It’s the simplest of all solids. If each face is an equilateral triangle, the result is a regular tetrahedron.
This is a good diagram of a tetrahedron. You can see my four faces, four vertices and six edges that puts me in a special group called Td. I’m the prototype of the tetrahedral group, of course.
There are only five regular polyhedrons known collectively as the Platonic Solids. Of course, some of my closest friends are in this little select group. Let me see, there’s Cube or Hexahedron as they sometimes prefer. Then there’s Octahedron, Dodecahedron & Icosahedron who always hang around together. I’m a loner myself…

The tetrahedrite series
Much to my delight, I have discovered that there is a group of minerals that have a similar profile to my own; geometrically speaking of course. The general formula for this group is A12X4B13 and substitution is widespread, forming an extensive solid solution series. The ‘A’ can be silver, copper, zinc, mercury or iron, the ‘X’ can be arsenic, tellurium or antimony and the ‘B’ can be sulphur, selenium or tellurium. The most common of these is tetrahedrite – copper antimony sulphide or Cu12Sb4S13.

Tetrahedrite forms a solid solution series with the rather rare mineral tennantite – copper arsenic sulphide. The two share the same crystal structure but they differ in the percentage of arsenic and antimony.

In case you hadn’t guessed, tetrahedrite is named for its common crystal form, the tetrahedron. Tetrahedrite forms interesting (rather like me) mineral specimens. The crystals are opaque and can be black through steel grey or silver with a metallic lustre. They have an average specific gravity for metallic minerals. Crystal habits of course include the tetrahedron but sometimes modified by the dodecahedron and tristetrahedron, giving the crystals multiple facets while still retaining the basic tetrahedral shape.

Twinning is occasionally seen, as are massive and granular habits. The crystals don’t cleave but they fracture conchoidally, that is leaving a curved dent, like glass. Unfortunately they are not very hard and they tarnish to a greenish shade. I’m glad that doesn’t happen to me. The multi-faceted tetrahedral crystals as well as the flat faced simple tetrahedral crystals can be very striking, naturally, just like me.

It is an ore of copper and a minor ore of silver as it can contain a small percentage of silver. Thankfully I’m not a minor ore for anything.
They can be found in Peru, Australia and Mexico and are associated with other minerals such as quartz, pyrite, galena, chalcopyrite and other sulphides. I must make a point of visiting these places…

A tetrahedral life
Recently, I found myself inside a cylindrical chamber. It was metallic and had many perforations. It had a glass door, which was promptly shut after some fabric had been stuffed inside. I decided it was an excellent opportunity to see what precisely occurs inside this machine; I had seen the white cuboid shake about violently many times before, but its true purpose was not clear to me.

Water began to rush in, and the cylinder proceeded to spin from side to side, tossing the fabric pieces about in the water. It was then that my vision was affected by a mass of foam quickly filling the cylinder. The foam was fragrant and I concluded that it was soap powder that was responsible. After much swishing and spinning in this foam the cylinder finally emptied its last load of water, rinsing away the remnants of foam.

For a while, the cylinder lay dormant, occasionally doing a slow turn. I realized that this was the end of the process, where the fabric is gently flipped in what is termed the ‘anticrease’. I wondered if this was some subatomic particle I hadn’t heard of since there is also a ‘crease’. They seem to annihilate each other inside this machine…

On another close encounter with water, tetrahedron has observed the human ritual of bathing and I have tried it for myself. I assumed that this was also for the purposes of washing. Of course, tetrahedron doesn’t require any washing but it appeared as though it might be a more pleasant experience than the white cuboid so I quickly dived in before the bather. There was some foam here but not nearly as much, though I did notice that my presence caused a strange rippling effect. I’m not sure if this is normal or if it was just me…

In the interests of science I threw myself into a glass full of this yellow substance and saw the rising bubbles forming a frothy off-white head which rapidly diminished. I closely examined this foam.

Unlike other foam, which seemed to me to be many semi-coalesced bubbles of different sizes, the champagne foam was mainly uniformly sized spherical bubbles of less than 0.1 millimetres in diameter, suspended in the fluid itself.

By viewing objects that were behind these, it was clear that they were acting as tiny lenses. Their spherical shape presumably diverge light because the gas they contained has a lower refractive index than the surrounding fluid. As a result, the light entering the surface of the foam scattered in different directions by multiple encounters with the bubbles. Reflections from the bubbles’ surfaces also contribute to this scattering, some of the light finding its way back to the surface. This must be why the foam looks off-white even though the fluid is yellow.

Underneath this disappearing layer I found isolated bubbles, still rising. Curiously they appeared to rise in an ordered fashion in a column from the bottom of the glass. I quickly left as the glass was about to be utilized and I was starting to feel a little light headed…