An interesting problem

Posted on 10/05/2012 by Souradeep

I was given the following interesting problem, of which a non-computation proof was desired, and I got some kind of proof that does not involve direct computation, but isn’t the proof the book (Arthur Engel’s Problem Solving Strategies) recommends, but anyway (It will be fun to generalise too) here is my (perhaps shortenable further) problem+proof:

Given the set of n elements S = {1,2,3,…,n}, where 1≤k≤n, construct all possible subsets of S having k elements. Now, by the Well Ordering Principle, each of these sets has a minimum value element. Prove that the average of all such (n,k) minimum value elements is (n+1)/(k+1).

This problem does at first glance seem to be challenging, but we can attack indirectly searching for patterns. What I did was to not bother about the arithmetic mean as such in the beginning, but about the sum, that is, the arithmetic mean multiplied by the number of elements.

This paragraph is an addendum after I noticed my proof doesn’t quite work in the visual sense for n=1 or n=2, with respective values of k, and also for the general case k=1, so to kind of preserve the post (which anyway shall see better methods that are more complete and encompassing) I shall merely comment that the proofs for those are straightforward: for n=1, there is only one subset {1}, and its AM is obviously 1. We can plug that into the given relation and indeed see that it holds true. For n=2, the subsets are {1} and {2} for k=1, and {1,2} for k=2; the respective values are 3/2 and 1 respectively, and they too fit the given relation for n>2. And finally for general n and k=1, the sets are {i}, i ranging from 1 to n, and so the AM is (n+1)/2, which as we shall see also holds. Anyway now to give the general proof as I did it for n,k>2, k≤n:

Let f(n,k) be the required AM of minima of k-cardinality subsets of {1,2,3,…,n}. Let g(n,k) denote the sum of the minima, so that

$g(n,k)= \begin{pmatrix} n \\ k \end{pmatrix}f(n,k)$

We shall examine this g(n,k) now. What I did was investigate what happens when n is replaced by n+1 (or should we say the set S is increased by an element n+1). Clearly the new sum g(n+1,k) will contain all possible minimum-elements of the k-subsets from the n elements 1,2,3,….,n, in addition to all k-subsets that contain n+1 as an element. The former sum is g(n,k), and the latter is g(n,k-1) because we can see easily that n+1 will never be the minimum element and so taking all k-subsets of {1,2,….,n+1} containing n+1 results in the same thing as taking all (k-1)-subsets of {1,2,…..,n}. So then we have the identity

$g(n+1,k)=g(n,k)+g(n,k-1)$ ………………………………..(1)

Having done that we seek other relations. It occured to me that the element 1 might have a key role to play, as in, it will appear as a minimum element whenever something else isn’t. So I decided to partition the different cases of k-sets summing up to g(n,k) into those that contain 1 and those that do not. Let us examine the k-sets where 1 is fixed. Since if 1 is a fixed element, the remaining k-1 elements will be selected out of n-1 elements, and the contribution to the sum of minimums from these sets will be the number of sets multiplied by 1, and so the sum of minimum elements of the k-subsets not containing 1 is

$g(n,k)- \begin{pmatrix} n-1 \\ k-1 \end{pmatrix}$ .

This sum is the set of all minimum elements of sets like {2,3,5,7,…}, {5,6,9….}, etc. each of k elements. Now we employ a simple trick that frankly I was happy to discover. I am not sure how to put it into words, but see this: the minimum element of {2,3,4} is 2, and that of {1,2,3} is 1. So why not be able to see that all k-subsets that don’t contain 1 are bijectable with all k-subsets containing 1 of the parent set {1,2,3,….,n-1}? It is like shifting all the elements backward by 1. This backward shift results in our above computed partial sum of non-1 k-subsets of {1,2,3….,n} to be equal to the partial sum of 1-included k-subsets of {1,2,3,….,n-1}, increased by the number of such sets. So we finally get

$g(n,k)- \begin{pmatrix} n-1 \\ k-1 \end{pmatrix}=g(n-1,k)+ \begin{pmatrix} n-1 \\ k \end{pmatrix}$

which reduces to, on applying derived relation (1) and a standard combinations’ identity,

$g(n-1,k-1)= \begin{pmatrix} n \\ k \end{pmatrix}$

So we get,

$g(n,k)= \begin{pmatrix} n+1 \\ k+1 \end{pmatrix}$

The required AM answer is hence

$f(n,k)= \frac{ \begin{pmatrix} n+1 \\ k+1 \end{pmatrix}}{ \begin{pmatrix} n \\ k \end{pmatrix}}= \frac{n+1}{k+1}$

I note that it will be similar to find the value of an AM if the minimum is replaced by maximum.

Principles of evolutionary biology

Posted on 07/04/2012 by Souradeep

[I want to elucidate in this post how evolutionary biology works. You may find this blog to be a more expressed version of the biology part under Abstract Thoughts of Connectivity. But I will be introducing some new stuff as well, hopefully.]

Many people in the world question evolutionary biology because they say they don’t see it in action, and that it postulates ‘too slow’ processes that have been going on for hundreds of millions of years. Well they are wrong, for theoretical evolutionary biology agrees to almost all experimental findings of the past, and more are being proven day by day. And, it is about the best explanation we can have, the others being somewhere down the line erroneous. And, people might question about things not answered by evolutionary biology, as being a reason for their skepticism; to them I’d say, evoluionary biology is a science and like all sciences the main intention in formulating it is to give answers by building on new theories with old, and anway, we take it as an axiom that nothing unscientific can ever happen (as if it has) and that theories are always malleable. Further, the lifespan of a living being is of no consequence to the various organisational processes that keep on happening in a vast universe. I shall now begin.

The beginnings: It is indeed almost impossible to conceive of a mathematical model for life, because of the phenomenal number of particles involved. But to a good approximation one can build probabilistic models that under simplified conditions gives one results consistent within allowable limits, that is, be able to say how it began without saying exactly what happened. The early earth was full of different compounds of nitrogen and oxygen and sulphur, some of which dissolved in the early seas. By gradual chemical reactions the molecules began to get more and more complex. The laws of chemical kinetics apply, of course, and so such processes like the formation of complex molecules were probabilistic, but showing general trends in concentration wrt time if a large enough sample of reactants was taken. It so happened that some of the molecules began to acquire the ability to chemically catalyse ‘copies’ of itself or of other molecules from the other moleules around. Rather like for example how a zeolite catalyses in its pores. Now, what happens in such a system of kinematic processes is over time the concentration of molecules that are stable or self-generate to a good approximation, if not perfectly, will tend to be greater than of those molecules which are unstable or fail to catalyse copies of themselves. So, fast forward this rather slow process for millions of years, and one will end up in seas teeming with increasingly complex molecules and also groups of molecules, more complex than the ones that started with, that within statistical aberrations, are able to maintain their populations. The groups of molecules are now complex enough to allow us to name their selforganisational behaviour as life. Over time the ‘living beings’ grow more and more complex, adapting to their surroundings, till a stage comes when wordpress blogs like this are created. More detail on this in the biology part of Abstract Thoughts of Connectivity.

Analysis of the principles of evolutionary biology:

The first thing to be clear about is what ‘adaptation to surroundings’ means. I find the best way to do this is through a complicated example, whose template appears in most such similar detective-questions. For example, lets take up a thought I had yesterday evening when observing a pack of dogs as I was eating noodles at a shop. I noticed how some of the dogs were slinking away from humans but in general, when no attention was paid to them, were coming up and standing or sitting near humans in hopes of having a bit or two of food being thrown to them. That happened because they had learnt to associate humans with food, and is more or less a mark of moderate intelligence on part of the animal. However, that’s not my point, my actual thought was on the fear of snakes or such other similar creatures that most mammals have hardwired into their genes, something that is pretty instinctive. Two days ago I was reading on how lemurs in Madagascar still exhibit alarm responses to raptor activity near them, though raptors have been extinct on that island for about five hundred years now. That’s because before humans exterminated them, eagles were an apex predator in the region, and over time the lemurs had evolved to ‘fear’ these raptors who might at any moment snatch juveniles away from the canopies. But the genetic imprint of this fear means that it has become coded in the animal’s algorithmic instructions. This kind of reaction is not a product of observation and inference. So, why the lemurs evolved such a reaction is kind of well understood now, at least in terms of assuming the main aim is adaptation, but the main, more astute question is, how did the lemurs get such a trait? Did suddenly one of them observe an eagle snatch its kin away and develop such a fear that by some magic of epigenetics its genes were altered (epigenetics does work, and is fairly well studied, but not at this scale, and in this manner) so that its descendants also were born with an eaglephobia and hence got the chance over time to make their genes get spread in the group of lemurs because such a gene clearly stabilises the species population by minimising a hazard? No, that seems rather farfetched, as if such things happened in daily life there is a strong chance many kids would be born with a phobia not to put their fingers into power sockets or something like that. (I shall return later to a more rigorous dissection of the why in terms of mathematics, but let me finish the how first) We need an explanation that connects to our molecular theory of probabilistic processes as I detailed in the paragraph above. And we are now armed with the fact that genes are unchangable at a moment’s notice, or only slightly modulated over generations (switched off or switched on in the manner of epigenetics), and so too at an experimental level, from famous rat-tail and dog-tail severing experiments that failed to produce a tailless dog over several generations. We try instead this alternative hypothesis: what if a mutation happened down a generation that induced eaglephobia in the resultant lemur offspring? Such a lemur would be, by virtue of its fear of eagles, avoid canopies, and survive, and etc etc over time these gene would be common in the lemur community? This seems to be a more logical hypothesis, particularly as it involves no ‘fishy magic’ of genes suddenly changing. We can still have some ‘fishy magic’ here too: someone might argue “Hey, nature knows somehow that eagles are detrimental to lemurs, so something somehow happened so that this fear comes along” which again smacks of intelligent design hypotheses, and clearly can’t be true. Our human emotions of “what’s right” are not the explanations, but are themselves the product of algorithms driven deep into us that make the task of social communication (for neurotypical humans) easier. They follow the chemical-mathematical laws as much as the system of lemurs and snakes that we are trying to investigate does. We then have no other alternative but to say this: a mutation indeed was the result, but it was purely accidental and had nothing to do with the snake’s presesence initially, but once it came in it helped rather than harmed. A fair question is why only this mutation happened that survives to this day. Why not something like eaglephilia? The answer is that many different mutations happen in many different organisms of many different kinds, but only a small fraction of them turn out to have any significant survival value or devalue on their own! And, it so happens that such mutations (as I elucidated in my twice-hyperlinked post) can carry over as vestigeal remnants when they are not needed, yet are not harmful either, and might adapt entirely to any other purpose. As for example halteres in crane flies or pelvic remnants in pythons and whales. So, we can conclude what most probably happened to early lemurs: a ‘bug in the brain’ lets call it, a form of phobia, developed at some point in mammalian history (specifically in early lemur history), plausibly when dinosaurs roamed the planet, to stay away from gigantic arthropods and reptiles, and that has carried onto us even today, as a vestigeal remnant to the extent that most of us humans find a tiny spider more ‘alien’ than a big male lion. A third way, is through the development of a ‘fear processing centre’ that records things that have harmed but not fatally so, and exists in only more developed animals and inhibits the act in the future (this is what happens to kids who put their fingers into power sockets). But such a way always works in combination with hormonal and genetic pathways of fear processing. There are often discrepancies in the human brain’s ‘reptile’ and ‘modern’ areas, and that lead to things like addiction and god and murder.

Other ways selforganisation occurs is for example through sexual selection. It started with the appearance of sexual dimorphism, and a two-way race begins: the males (in almost all cases) over time acquire traits the females over time acquire the algorithms to perceive as indicator of good genetic fitness. Somewhere down the line something called a sexual handicap principle also develops. I mention this in passing, for I’m pretty tired by now, it being 1:20 am here, but I save some energy for the final paragraph, which will be the most significant I guess.

What it all means, mathematically:

A system of living creatures changes due to perturbations that occur in the external nonbiotic system, or in itself, that affect it, like weather or geological changes or solar activity. The changes don’t occur due to some external agent controlling from outside, but naturally as an outcome of events that don’t get perturbed by the changes being gradually accumulated over time. The mutations can happen in about any manner imaginable, but only the ones that don’t severely impale the organism’s chances of thriving get represented in future populations. There is no scope of any ’personified’ explanation here, for its pure combinatorics and statistics! If such mutations don’t happen in a species, or don’t get a chance to happen, it goes extinct. It is not clear as of now, because of the human-intelligence-limitations-induced difficulties of interconnecting physics successfully with all of natural science, why something that is favoured (produces in us the emotional reaction of ‘good’ or ‘favoured’) is favoured. The answer probably has to do with entropy in thermodynamics, which ultimately governs the directions of chemical reactions, which again governs the concentrations, etc of the reactants and products as as dynamic function of each other should perturbations occur. Life is not exactly a reversible chemical reaction, it is more like two or more nonreversible reactions that have opposing effects happening nonsimultaneously. We might even conjecture that it is meaningless in the scientific sense to ask why evolution occurs: it is surely the postulates and laws of the physical world that result in what we see as evolution, and that all other explanations are baked by creative social minds? An important thing to note that a thinking mind, while governed by the same algorithmic limitations as a genecode or hormonal drive, has the capacity to extend somewhat, even out of its limitations. It is obvious that such ‘noninstinctive centres’ once developed would almost certainly never be phased out, for they would any way help in advanced ‘fluid’ adjustment to situations. However, with it comes potential peril: an ‘intelligence’ developed exclusively for one purpose might fail miserably at another, which is why while lay humans can interact with each other so well, they fail by comparison at being scientific and comprehending science, which results in superstitions as we see today. Perhaps natural selection will phase that out too one day.

Gaussian integers from axiomatic-build up from integers

Posted on 06/04/2012 by Souradeep

[This post aims to introduce rigorously the gaussian integers from the integers without any direct implications to complex numbers. Pure mathematical fun, basically. I follow Tao, btw, in my method. Intuition is of course applied to make the laws result in a world that 'ought to' exist.]

We assume the natural numbers and from them the integers are developed axiomatically. We also assume the properties of addition (+) and multiplication (• or simply just the two numbers written side by side, by the property of commutativeness and associativeness) are defined on the integers, as we intuitively know them to be. We shall also closely mimic a vector space here (which it indeed is).

Define a gaussian integer to be of the form (a,b), where a and b are integers. We associate two operations associated with the gaussian integers (which I shall show in typical Tao fashion to be equivalent to two integer-defined functions later on). The functions are say ⨁ and ⨂. We then define the gaussian integers to follow the following rules:

For any two gaussian integers (a,b) and (c,d),
(a,b) ⨁ (c,d) ≔ (a+c,b+d) ………………………………….(1)
For any two gaussian integers (a,b) and (c,d),
(a,b) ⨂ (c,d) ≔ (ac-bd,ad+bc) ………………………………….(2)
For any integer k, we define k•(a,b) to be (ka,kb), where the • represents ordinary scalar multiplication when applied as a binary operator to pairs of integers. …..(3)

These rules (1), (2), (3) can be worked with straightaway to obtain some interesting properties.

We can easily use (1) and (2) to show that the elements (a,b) are commutative, associative, closed and distributive under ⨁ and ⨂.

Also putting a = k, b = 0 in (2) yields (k,0) ⨂ (c,d) = (kc,kd), which we can interpret from (3) as being that the set of gaussian integers (k,0) is equivalent to the set of integers k (note that we still can’t remove the ⨂, for that would be premature).

We see from (2), putting b = d = 1, a = c = 0, that (0,1) ⨂ (0,1) = (-1,0) = -1. We can now venture to drop the ⨂, and replace it by •, and introduce an object i, so that (0,1) ≔ i, and i•i = -1. Note that we can now write (0,k) as k(0,1) = ki

We can similarly replace the ⨁ in (1) with +. Hence ai+bi = (a+b)i, etc. We define the negation of a gaussian (a,b) as -(a,b) ≔ (-a,-b). We also define (a,b)-(c,d) ≔ (a-c,b-d). We plug those into (1) to be able to replace (a,b)+[-(c,d)] with (a,b)-(c,d).

Finally, (3) allows to write (a,b) = (a,0)+(0,b) = a+bi, (a,-b) = (a,0)+(0,-b) = (a,0)-(0,b) = a-ib. The system of gaussian integers is now ready for use if any, or exists at an abstract mathematical standpoint.

I think we can from here advance in real and complex analysis side by side, with reals of course having some additional properties like trichotomy, ordering, etc.

Calculus ‘potentials’ in extremisation

Posted on 01/04/2012 by Souradeep

[A technique in variational calculus that I self rediscovered.]

A common form of extremisation problems in euclidean geometry is of the following form:

Given k points $X_i$ in euclidean space, and a set of k functions $f_{i} :{ \mathbb{R}}^{n} \to \mathbb{R}$ , each of which depends on the corresponding $X_i$ , to find the extremum value if feasible under the circumstances of $\sum_{i=1}^{k} {f_{i}( \textbf{PX} _{ \textbf{i}})}$ , where P is a point in the space and the $\textbf{PX} _{ \textbf{i}}$ are of course directed vectors. For ease of notation, we imply P is associated with all such vectors and replace the $\textbf{PX} _{ \textbf{i}}$ by $\textbf{X} _{ \textbf{i}}$ .

The general method to do this is to imagine giving each of the vectors $\textbf{X} _{ \textbf{i}}$ a displacement $\textbf{ds}$ . Also let $x_{j}$ denote the n coordinates. Then each $f_{i}$ changes as

$\delta f_{i}= \sum_{j=1}^{n} \frac{ \partial f_i}{\partial x_{j}} \delta x_{j} =( \nabla f_i) \cdot \textbf{ds}$

We require that

$\sum_{i=1}^{k} \delta f_{i}=0$

These two relations, together with the isotropy of $\textbf{ds}$ , mean that for all possible cases, we must have

$\sum_{i=1}^{k} \nabla f_{i} = \textbf{0}$ …………………………(1)

or alternately,

$\nabla ( \sum_{i=1}^{k} {f_{i}}) = \textbf{0}$ …………………………(2)

We can now either work with (1) and find an equation relating the vector functions $\nabla f_{i}$ , which in some symmetrical cases yields immediate results, or we could work with (2) and get n differential equations to solve. It is interesting that we can view such problems in a physical way, when $\nabla f_{i} =k \textbf{X} _{ \textbf{i}}$ (conservative fields), as problems regarding the extremisation of electrostatic or gravitational potential when the mechanical systems set up are constrained in some way. Of course, saddle points also might come up.

Such problems in general are not very trivial to solve, and general cases are by and large beyond my skills at the moment, but a few symmetric problems are treated related to triangles, as examples:

Given a triangle ABC, find P such inside it that PA + PB + PC is mimimal

This is trivial, here the scalar functions related to PA, PB and PC are proportional to each of PA,PB,PC in magnitude, and so their gradients are ‘forces’ (physics analogy working well) of the same constant magnitude in equillibrium at P. Clearly this happens (Lami’s theorem if you are a formula enthusiast) when each of APB, BPC, CPA is 120º.

Given a triangle ABC, find P inside it such that PA² + PB² + PC² is minimal

This is also trivial, for here we deal with ‘potentials’ that are analogous to three springs as it were, connected to P from the vertices. The three ‘forces’ are in the ratio PA:PB:PC, and they add to produce a null vector at P. In mechanics this happens if P were to be the barycentre of the system ABC with three equal masses at A,B,C, rhe result being their respective moment vectors are proportional to and direcred along PA,PB,PC; so the answer to our problem is the centroid. [Note: if we apply the moment of inertia parallel axis theorem, with three masses at A,B,C, and the centroid at G, we get a proof of this, and also of this result: for every point M, MA² + MB² + MC² = GA² + GB² + GC² + 3MG², something that takes good visualisation to prove in plane geometry using only 'geometry' axioms: see Challenge and Thrill of Pre-College Mathematics.]

This problem is a little more astute, for it necessiates actual determination of the angles by algebra if one wants to find the solution easily:

Given a point P inside a triangle ABC, find the minimum of aPA + bPB+cPC, where a,b,c are positive reals.

We again employ the potential method and arrive at this following scene: three forces of the ratio a:b:c in equillibrium at P. If x, y, z are the angles BPC, CPA, APB respectively, we get

$sin \, x : sin \, y : sin \, z=a:b:c$

subject to the constraints (from the fact that P lies inside the triangle)

$x+y+z=2 \pi ; \, \, x,y,z< \pi ; \, \, x>A, \, y>B, \, z>C$

and so we can write the equations in the form

$sin \, x =ka, \, \, sin \, y=kb, \, \, sin \, z=kc$

where

$k>0; \, \, ka,kb,kc<1$

There is the following identity satisfied by angles that add up to a complete rotation, in this case x,y,z, with the summation and product implied over all cyclic terms:

$\sum sin^{4} \, x-2 \sum sin^2 \, x \, sin^2 \, y+4 \prod sin^{2} \, x =0$

and using that we get

$4k^{2}a^{2}b^{2}c^{2}=2(b^{2}c^{2}+c^{2}a^{2}+a^{2}b^{2})-(a^{4}+b^{4}+c^{4})=(a+b+c)(a+b-c)(b+c-a)(c+a-b)$

We get the value of k by solving the quadratic equation, neglecting the negative solution, and then we have to examine whether the conditions outlined restricting the angles are satisfied. It turns out that a,b,c have to be sides of a triangle, for the expression (a+b+c)(a+b-c)(b+c-a)(c+a-b) must have a positive value. If k turns out to result in the values of ak,bk or ck to be greater than 1, or the angles not to fall in the restricting limits, then the problem has no local minimum solution, (in the language of differential geometry there exists no 2D ball surrounding P where the scalar function is greater than that at P) and a solution for the particular triangle will exist on the boundary, that is, on the sides of the triangle. This special case is also interesting in its own right, for it can be solved geometrically.

Schwarz triangle:

Given a triangle ABC, find the inscribed triangle that has maximum perimeter.

We suppose the triangle whose perimeter is to be minimised is DEF, which is inscribed in the given triangle ABC. We draw AD, BE, and CF, (note I put them as concurrent, but they might not be, and I shall prove that they are, so I put the point G as well.) and as construction draw two perpendiculars FM and EN upon BC. Let X be where EF intersects AD. The physical analogy is to imagine three special stretched springs along DE, EF and FD, whose force on the points D,E, and F (at which one might put frictionless beads free to slide on the frame of the triangle) is constant in magnitude. Then the minimisation of potential occurs when the system is unmoving: the two forces acting on each bead must have equal and opposite components along the direction of constrained motion of that bead (along the side). Let us take the point D as an example: we get that

$cos \, \angle FDM =cos \, \angle EDN$

This means that

$\frac{FD}{DE}= \frac{MD}{DN}$

Elementary geometry tells us that

$\frac{FX}{XE} = \frac{MD}{DN} = \frac{FD}{DE}$

and hence DX bisects the angle EDF of the triangle DEF. Hence by symmetry the lines AD, BE and CF are the angular bisectors of the triangle DEF, and hence must be concurrent at a point (mentioned in brackets as G xD) which we shall call G. It turns out due to properties of angles that G is the incentre of the triangle DEF which hence must be the pedal triangle, and hence G is the orthocentre of ABC, meaning that D,E,F must be the feet of the perpendiculars so that DE+EF+FD is to me minimised.

Some Properties of Cycloids

Posted on 20/02/2012 by Souradeep

[Some properties of cycloids I found fun to play with. Apologies for the paucity of diagrams for I have no requisite geometry software.]

The Tautochrone

An inverted cycloid is the locus of the path on which which a ball constrained to oscillate under constant gravity will execute SHM about the middle inflection point, when the gravitational force is constant. The simple approximation all people encounter to SHM is the simple pendulum, which however is accurate enough for human purposes for only small angles.

Let us suppose we have a cycloid generated by the circle $x^2+(y-r)^2=r^2$ rolling upside-down in the positive x direction on the line $y=2r$ . The resultant inverted cycloidal hump has its minima at $(0,0)$ and end-points at $( \pm \pi r,r)$ . Let us take, in the first quadrant, a point $(x,y)$ on the cycloid, at which the angle the generating circle has moved forward by is θ. We immediately get the parametric equations:

$x=r \, sin \, \theta +r \theta \\ y=r-r \, cos \, \theta$

From these we get

$\frac{dy}{dx} = \frac{sin \, \theta}{1+cos \, \theta}= \frac{ \frac{y}{r}}{ \sqrt{1-( \frac{y}{r} -1)^{2}}}$

If $s$ is the distance covered at time t, (the starting point is irrelevant as of now) then

$\frac{dy}{ds} = \frac{ \frac{dy}{dx}}{ \sqrt{1+( \frac{dy}{dx})^{2}}} = \sqrt{ \frac{y}{2r}}$

We can integrate that, assuming as initial conditions $s=0,y=0$ , to give

$s= \sqrt{8yr}$

Let us assume that a particle is shot out from the centre of the cycloid at some initial speed (whose value will come in the parameter). Friction is neglected. We see the acceleration as being directed towards the origin, along the cycloid, so we can write, employing our derived relations along the way,

$\frac{d^{2}s}{dt^2} =-g \frac{dy}{ds} =-g \sqrt{ \frac{y}{2r}}=- \frac{g}{4r} s$

This equation represents SHM, and its general solution is, representing the directed distance $s$ travelled across the cycloid by the particle

$s= a \, sin( \sqrt{ \frac{g}{4r}} \, t+ \phi )$

where the parameters a and Φ account for all possible initial configurations and amplitudes of motion. This gives the time period of the system as

$T=4 \pi \sqrt{ \frac{r}{g}}$

We hence get this property of the tautochrone: If we constructed a cycloidal ramp using only half of a cycloidal hump, and placed it so that the original generating line were perpendicular to the gravitational force, from wherever on the cycloid we let a particle go down to the bottommost point, it would take the same time $\pi \sqrt{ \frac{r}{g}}$ . A wikipedia applet here.

The Brachistochrone

The brachistochrone is defined as the curve from one point to another following which if a particle moves under the action of gravity in a conservative field it will do so in the shortest time. To find it may use either the Euler-Lagrange equation of stationary action, or plain simple geometry, the fermat principle of least time and knowledge of conservative fields. Both lead to the same conclusions, though here the second method will be followed. For earth’s constant gravity scenario the curve is a cycloid, as can be proved:

Let us suppose the particle has to move from $(x_{1},y_{1})$ to $(x_{2},y_{2})$ , where $x_{2}>x_{1}, \, y_{1}>y_{2}$ . We pick up the story when the particle is at $(x,y)$ . Since the field is a conservative one, the velocity v is related, at coordinates $(x, y)$ , to the change in height as $v= \sqrt{2g(y_1-y)}$ . Also, we know from the Fermat principle of least time, that if the velocities of a body in two media separated by a planar boundary be $v_1$ and $v_2$ , we have $\frac{v_1}{v_2}= \frac{ \sin{ \theta_1}}{ \sin{ \theta_2}}$ , where $\theta_1$ and $\theta_2$ are the angles of refraction and refraction respectively, for minimum time. We assume, in the typical way of calculus, that as the ball moves down its path the velocity changes in increments, in both direction and magnitude, over infinitesmial changes in ordinate. In this way the angles of refraction and incidence also change, at first increasing, then if they go past a right angle, decrease back. For the particular path the ball traverses each such infinitesmial change, or segment of path, must be in accordance with the principle of least time, for then we could modify each segment path to make the total path even shorter. This gives us, from the mathematical relations between the angle and the velocity, that $\frac{v}{cos \, \alpha}=k$ , where k is constant throughout the experiment but depends on the final endpoint and the value of g, and $tan \, \alpha =- \frac{dy}{dx}$ . We get

$tan \, \alpha =- \frac{ \sqrt{k^{2}-v^{2}}}{v}$

And consequently,

$\frac{dy}{dx}=- \frac{ \sqrt{ k^2-2g(y_1-y)}}{ \sqrt{2g(y_1-y)}}$

$=- \frac{ \sqrt{ \frac{k^2}{2g}(y_1-y)-(y_1-y)^2}}{(y_1-y)}$

It is here that we have to make some intuitive and clever substitutions, in terms of parameters. We know we have to show it to be the cycloid, and some of the terms in the above equation look analogous to the derivations for the tautochrone. We first set $\frac{k^2}{4g}=r.$ . We see that

$0 \leq (y_{1}-y) \leq 2r$

because

$\frac{v^2}{k^2}= \frac{2g(y-y_{1})}{k^{2}} = cos^{2} \, \alpha \leq 1$

and that enables us to write that

$\frac{y_{1}-y}{r} \equiv 1-cos \, \theta$

The symmetry of trigonometry means we need not bother about where θ is constrained to lie, or whether we need to cosntrain at all. We can intuitively make our own assumptions where needed, later on, when we shall see that such taking of cos θ actually has some physical meaning.

We can then perform simplifications to obtain

$\frac{dy}{dx}=- \frac{ \sqrt{1-(1- \frac{y_{1}-y}{r})^{2}}}{ \frac{y_{1}-y}{r}}$

which, again employing an evasive symmetry argument in the numerator, becomes, by the substitution of the parameterised expression,

$\frac{dy}{dx}=- \frac{sin \, \theta}{1-cos \, \theta}$

Also, from the same parameterised expression, we get

$\frac{dy}{d \theta}=-r \, sin \, \theta$

And hence we can get, from the chain rule of derivatives,

$x=r( \theta -sin \, \theta)+c$

The constant c can be found out from the initial value of x to be $x_{1}$ . Hence, we have the two parameterised equations:

$y_{1}-y=r(1-cos \, \theta) \\ x-x_{1}=r( \theta -sin \, \theta)$

We can hence from these equations see that the curve is a cycloid produced by a circle of radius $r$ rolling upside-down from left to right, along the line $y=y_{1}$ , with θ being the angle of rotation wrt the vertical, in the anticlockwise direction. We hence see that θ indeed has a physical significance, as does the other parameter.

We can solve for $r$ and θ by plugging in the final point in the set of equations:

$y_{1}-y_{2}=r(1-cos \, \theta) \\ x_{2}-x_{1}=r( \theta -sin \, \theta)$

We can set

$m \equiv \frac{y_{1}-y_{2}}{x_{2}-x_{1}}= \frac{1-cos \, \theta}{ \theta -sin \, \theta} \geq 0$

the last inequality being because the gradient is always negative for the path, except if the initial and final points are at the same vertical level.

My skills at numerical analysis of transcendentals are not good, and I tried my hand at getting some limiting values or similar analytic arguments without absolute success, so I shall resort to some computer help and assumption here. A graph of the equation $z= \frac{1-cos \, \theta}{ \theta -sin \, \theta}$ , from Wolfram Alpha, is given below, with z as ordinate and θ as abscissa:

Its obvious that small values of m, representible by a line $z=m$ give multiple solutions of θ. However, it is equally obivous that, for all m there is only one solution of θ in $[0,2 \pi ]$ (the ‘single piece cycloids’), since the first minimum is at $\theta = 2 \pi$ , and it seems almost certain (though I can’t prove so) that the single-piece cycloid through two points is the one that is the path of minimum time, the others being local minima. The uniqueness of a cycloidal curve through the two points that gives the minimum time is hence established.

Area under a hump, and arc length of a hump

Again consider a cycloid, this time formed by a circle of radius r rotating on the x-axis, with θ being the angle rotated from the vertical, in the anticlockwise direction. The following equations are obvious:

$x=r \theta - r \, sin \, \theta \\ dx=r(1-cos \, \theta) \, d \theta \\ y=r(1-cos \, \theta)$

The area under a hump is then

$A= \int_{0}^{2 \pi r} {y \, dx} = r^2 \, \int_{0}^{2 \pi} {(1-cos \, \theta)^{2} \, d \theta}$

which is trivial to compute, giving

$A=3 \pi r^2$

The arc length can then be computed using the relation derived for the tautochrone, $s= \sqrt{8yr}$ , setting y=2r, and considering each half twice, to give

$L=8r$

What is life

Posted on 05/02/2012 by Souradeep

[Controversy.]

Life is physically a very complicated system of molecules interacting with one another. Distinct associations are most often seen among these interactions….such associations of molecules are regarded as individual living beings. I’m poor at sensational paragraphic oratory so I shall directly launch into the topic: what is the goal of life and why do we live anyway?

It may seem that the universe is pointless, and that it is us humans who attempt to apply emotions and suggest that we are something special and that things are made for us, somehow. I hold that as true, because I rationally find it so, and continue on to rationally justify so.

What prompted the current accepted views on evolutionary biology (I imply accepted by scientific people of course, not unscientific fundamentalists) are experiments done by Urey and others, showing that quite bulky amino acids are formed if a simple mixture of gases is exposed to an electric discharge and left to stand for many days (to simulate early earth conditions when nitrogen and hydrogen were in plenty and lightning was probably more than it is now). Paleontological and geological findings also show intriguing patterns that initiate theories in a theorist as to why the organism had such a body structure, what it looked like, what made it evolve to look like it did, where it lived, how it lived, why it lived the way it did, and what made it die out. Many a time erroneous conclusions maybe arrived at, which are however (unless the concluder is a nutcase) modified or changed on the arrival of fresh information. It is like an infinitely-detailed crime investigation. For example: one can on seeing the hipbone structure of say a dinosaur say whether it was a sauropod or ornithopod, from the teeth and fossilised dung/stomach contents, herbivore or carnivore, from estimated muscular distribution and energy usage and circulatory system (for these things are rarely preserved) estimated lifespan and pattern of life, and so on. Sometimes it gets pretty complex, as in cases where finding the connection requires a lot of analysis of different data and some deep thinking to fit in a theory that accounts for everything within random statistical population variations.

We see from forensic evidence that our data shows a gradual simplification of the ecosystem the further back we go in time. And indeed the earliest lifeforms were merely protein molecules capable of acting as catalysts. We here shall forget we know everything about life, and in an attempt at rigorous analysis theorise a bit (good chances of the theory being off-track by a large degree, but I merely want to present an example), and finally get to a conclusion as to what life is, for it gets pretty involved with information theory and cybernetics, and physics.

At the beginning, after the young earth had cooled down enough, the early seas had formed, due to precipitation, or as some reports suggest, bombardment by water-bearing asteroids and comets. And in those seas a number of complex molecules were forming from the ammonia, nitrogen and hydrogen already present. It so happened that some of these molecules had the property of being able to catalyse the formation of copies, or analogies, of itself. Some of these copies were unstable and short-lived, and a tiny few were actually better at catalysing than the parent molecule. The mathematics here gets pretty complex and chaotic, but one can intuitively say that over time (and long amounts of it) the number of such self-replicating molecules will have grown, both in number and diversity. Pretty soon molecules began to join together in ever more complex shapes and formations, all of which could self-replicate by simple or complex ways, resulting in the first viruses and, eventually, cells. It is important to realise that evolutionary changes like this were not happening because the molecules somehow had a ‘mind’ and could sense environmental catastrophes. They were random mutations and glitches in the copying process that happened to be of the right sort to increase the chances that the assortment of molecules could catalyse. The ecosystem is a dynamic one, with solar energy incident on the earth being radiated back into space by the earth, and under such conditions we can’t say whether the formation of such molecules was a high-entropy or low-entropy process. Intuition suggests that such complex molecules were more ‘disordered’ than the rocks and liquid and gas they came from, and hence entropy could be said to have increased with time, as is consistent with the second law of thermodynamics. However another view is that the earth has cooled down and stabilised over time from its high-temperature origins; the frequency of asteroid impacts is much less here. That seems to imply that entropy decrease actually favours life. The most probable answer to this paradox perhaps is that the second law of thermodynamics only says about entropy decrease with time being a thing of the entire system as a conglomerate. There can exist local regions where entropy decreases with time. Perhaps the birth of a star and the formation of its solar system was such an event where entropy initially increased too much during the formation of the star and now is decreasing back to normal with the stabilisation of its solar system, like a fluctuation. Such oscillations are found everywhere in the universe. Fast forward two billion years. We now are so advanced with our interacting systems of molecules that we can refer to them as organisms, and the immensely complex chemical reactions they take part in either directly or indirectly as life. (Should you like me to elaborate a bit on evolutionary biology, I recommend you to read the biology section of Abstract thoughts of connectivity.) Life consequently can be thought of as a roughly oscillatory sequence of processes, much like starbirth from planetary nebula, which itself comes from a dying star. An organism similarly dies when at a stage in time it becomes unable to perpuate the processes in it, due to increasing inability to keep reactions going, either as a result of insufficient raw materials or as a result of irreparable machinery errors.

What then is the goal of life? That now seems similar to ask the goal of a burning matchstick, or a lightning bolt. Human anthropocentric flawed logic invents reasons like ‘life exists so that all animals live and do their work and die in peace’ or ‘life is god’s will’ or ‘life’s goal is to be happy and content and help others.’ We tend to misuse the algorithms that have been hardwired into us to propagate our species, when we attempt to apply these algorithms in situations they are not built for. It is rather like attempting to use a word processor to open a .mp3 file. The output you’d get would be meaningless to large degree, and illogical. The biosphere is then just like any other interacting system, with the exception of being sufficiently complicated to allow complex pathways of responses to stimuli to develop. No external thinking agent urged it to become like it is (I contradict you if you are a theist, but I am not, and I have logical reasons for notbeing so), and neither does it have a purpose of goal defined for it. It is we humans (who purport to be the most intelligent creatures, and are so by a considerable margin) who apply our (very often flawed) definitions of good, bad or goal, or worthiness. The physical processes just go on irrespective of whatever interpretations they live in a higher-order thinking mind. Everything reduces to physics in the end, for chemistry and biology are essentially applied physics to certain special conglomerates of matter, that were once considered separate disciplines because people didn’t know enough to understand the interconnections. That’s probably the most amazing thing about the universe, that it exists and evolves according to certain mathematical laws, and that (probably from the Godel incompleteness theorem) no intelligent organism can know everything about it. Consciousness is a very sophisticated process, however, and I would like to agree to Penrose’s view that it involves quantum-scale phenomena. (Though I understand and know very little of both neurobiology and quantum mechanics at present xD). Life is emotionless, goalless, and perpetuating.

Mathematics : The Abstract vs the Physical

Posted on 30/01/2012 by Souradeep

[On whether abstract mathematical defintions and uses of number systems, or other systems, are needed to use numbers to portray the real world. And of course I know little and hence it might be shallow.]

Usually it is seen that certain concepts of mathematics are used quite intuitively in describing real-life phenomena. Saying that there are n objects, or that the wavelength of a particular photon (or, in terms of waves, a beam of monochromatic light) is $\lambda$ , doesn’t in general require us to know what exactly n or $\lambda$ might mean. We concern ourselves, in the natural sciences, with the magnitude of the quantity, and the dimensions of the quantity, and what the quantity represents. We have no need to ask what the quantity means. In abstract mathematics, however, numbers are built from scratch, starting from the natural numbers and their defined properties (the Peano Axioms). And then on progress is made by introducing certain (well-chosen to comply for the major part with reality) axioms, so that we can, in turn, account for subtraction, division, exponentation, tetration, continuity and other such properties of the number systems, and we progress from $\mathbb{N}$ , through $\mathbb{Z}$ , $\mathbb{Q}$ and $\mathbb{R}$ till we come to $\mathbb{C}$ . (There exist systems like quarternions and octonions, but for the major part they aren’t used very frequently, and are chiefly of a theoretical interest.) We also use more abstract concepts like vectors, operators, matrices, groups, etc, in the sciences, and frequently so. While it is true that mathematical rigour is not very essential in physics, to apply a certain system in a physical theory to account for observations and to predict more observations does need prior generalisation and examination of the number system thus needed, so that the physical system becomes as workable and accurate as it can.

The discovery that mathematics can be treated rigorously and can be built piece-by-piece by abstract axioms is historically rather recent. The origins of numbers were nothing significantly abstract. The concept of numbers probably arose out of need in prehistoric populations, to keep tabs on material possessions. At the beginning the notation for numbers was rather simplistic, like 5 might consist of four parallel lines cut by a fifth (or say the Roman Numerals which are still in use as novelty items today). People soon discovered that such notations were cumbersome to carry out calculations in, and the concept of a base developed. The decimal system prevalent today probably has its roots in humans having (normally) five digits on each appendage, making a total of ten on the two hands. Whether or not 10 is a useful base has little to no correlation with its unanimous usage as a base. Another system extremely used in computing and electronics is the binary system, which is so because electronic circuits tend to work efficiently and suitably only if information is discretised, and can only detect the presence (1) or absence (0) of a current. Similar needs necessiate the use of the hexadecimal, octal and other systems. Algebra historically has, to generalise, denoted numbers by unknowns, of the form of other symbols, like a,b,c,x,y,z……in order to express generalisations and equations in a compact and suitable notation. With the gradual introduction of abstractness we have gradually expanded our mathematical terminology to include many other abstract devices, with the result that ‘being a number’ doesn’t count as much. In essence, numbers and operators have been in use for long before they were formally defined. I want to address, in the next two paragraphs, the question “Is it necessary to formally define numbers before using them in physics?”.

Intuitively the need of numbers is best explained by geometry: We need to compare, measure and perform operations on things around us in reality, or an analogous space created in our minds and existing in blueprint only. The natural numbers have been in use since time immemorial. They can be easily used to compare integral multiples of things. for example, if Stick A is considered to be 1 unit long, then two sticks laid end-to-end AA would be 2 units long. And so on. Fractions came about to express parts rather than whole. One part out of three parts of a pie could not be expressed as 1/3. Negative numbers, similarly came about because of people’s overwhelming need of symmetry and ‘viewing things from the other end.’ Something like: “If Jack gave John 5 apples, lets say that John gave Jack -5 apples.” Real numbers finally came when people realised that merely ‘dividing units by one another’ was insufficient to give measures to certain quantities, like the ratio of circumference to diameter, or the measure of hypotenuse of an isosceles triangle of unit sides. It was the concept of the real numbers that enabled humans to finally justify that their observable space (real or imaginary) was limitless in detail, that one could go smaller and smaller, or larger and larger, and still never run out of numbers to describe magnitude. It is true that actual spacetime is not euclidean, and not even fully continuous, but those perturbations are barely registerable to the human senses, and definitely were not discovered when such number explorations were going on. Had, for example, observable geometry been 4D, or discretised, our ‘numbers’ might have been significantly different compared to what we have today. And so, perhaps, would have been our entire mathematics, if ‘continuity were not allowed’. I have little idea about whether an intelligent being could conceive of continuity, or ‘as small as it can go’ in such a world, but since we can to a good degree use higher dimensions in our mathematical and scientific pursuits despite being unable to visualise them effectively, I think that the chances are high such an intelligent being would. Similarly, if our space was curved in the manner of elliptical geometry, all geodesics would warp back to where they started, and our concepts of infinity would be in peril. We humans being moderately intelligent beings (and the people who are at the forefront of research and original thinking rather much more so) have been able to create algebra spaces where commutativity need not exist, or which is discretised, or which extends to 11 dimensions…the list is pretty large. But are our creations merely attributable to there already existing a template to build from in nature? Would it have been possible to arrive at the same concepts of numbers by abstract axiomatic approaches alone, without any reality to guide one? I don’t think so, or at least I think the chances of that happening would have been slim. However, it is almost certainly possible for a sufficiently lifeform to generalise to other concepts provided the initial template has been given by nature.

When we axiomatically construct numbers, we, as we did before, start with the natural numbers. It is true that one can progress in whatever direction and methodology one wants to….there are no barriers to create new systems in which there are things to operate on, and operations are defined. However, one has to take care that one’s axioms do not create logical contradictions. As for example, were the axiom of universal specification included in a theory of sets, (roughly speaking it means that one can have a set of all possible elements, including the set itself) the famous Russell’s paradox would emerge. And, one generally strives to keep as close to usuability in explaining the phenomena of nature as possible, for the greater majority of the people who are not pure mathematicians would have little interest in something that doesn’t work. However, in the end it results that, even if there are an abstract build-up and a physical-intuitive build-up of the same system, one need not feel forced to classify them in terms of superiority. For although axioms are rigorous, they are still chosen, and are perfectly malleable to suit our needs, and are human creations anyway.

aspieigen

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