"God is a mathematician."

The more advanced the topics I deal with in physics, the more stark I observe the divergence from philosophy and mathematics to be. While one seems to drill right down to the bedrock of all things existential, the other assumes disturbingly abstract overtones, often requiring multiple interpretations to seem to possess any semblance of meaningfulness.

This is where the strength of the mind is tested: an ability to make sense of fundamental concepts in various contexts and to recall all of them at will so that complex associations don't remain complex but instead break down under the gaze of the mind's eye to numerous simple associations.

While computation theory would have us hold that a reasonable strength of any computing mechanism could be measured as the number of calculations it can perform per second, when it comes to high-energy physics, the strength lies with the quickness with which new associations are established where old ones existed. In other words, where unlearning is just as important as learning, we require adaptation and readjustment more than faster calculation.

In fact, the mathematics is such: at the fringe, unstable, flitting between virtuality and a reality that may or may not be this one.

One could contend that the definition of mathematics in its simplest form - number theory, fundamental theories of algebra, etc. - is antithetic to the kind of universe we seem to be unraveling. If we considered the example of physics, and the divergence of philosophy from theoretical physics, then my argument is unfortunately true.

However, at the same time, it seems to be outside the reach of human intelligence to conceive a new mathematical system that becomes simpler as we move closer to the truth and is ridiculously more complex as one strays from it toward simpler logic - not to mention outside the reach of reasoning! How would we then educate our children?

However, it is still unfortunate that only "greater" minds can comprehend the nature of the truth - what it comprises, what it necessitates, what it subsumes.

With this in mind: we also face the risk of submitting to broader and broader terms of explanation to make it simpler and simpler; we throw away important aspects of the nature of reality from our textbooks because people may not understand it, or may be disturbed by such clarity, and somehow result in the search seeming less relevant to daily life. Such an outcome we must keep from being precipitated by any activity in the name of and for the sake of science.

On Monday, I attended a short lecture by the eminent Indian particle physicist Dr. G. Rajasekaran, or Rajaji as he is referred to by his colleagues, on the Standard Model of high-energy physics and its future in the context of the CERN announcement on July 4, 2012. While his talk itself straightened a few important creases in my superficial understanding of the subject, two of its sections continues to nag at me.

The first was his attitude toward string theory, which was laudatory to say the least and stifling to say the most. When asked by a colleague of his from the Institute of Mathematical Science about constraints placed on string theory by theoretical physics, Rajaji dismissed it as a political "move" to discredit something as exotic as the mathematical framework that string theory introduced.

After a few short, stunted sniggers rippled through the audience, there was silence as everyone realised Rajaji was serious in his allegation: he had dismissed the question as some political comment! Upon some prodding by the questioner, Rajaji proceeded to answer in deliberately uncertain terms about the reasons for the supertheory's existence and its hypotheses.

Now, I must mention that earlier in his lecture, he had mentioned that researchers, especially of high-energy/particle physics, tended to dismiss new findings just as quickly as they were ready to defend their own propositions because the subject they worked with was such: a faceless foe, constantly shifting form, one moment yielding to one whim, one serendipity, and the next moment, to the other (ref: Kuhn's thesis). And here he was, living his words!

The second section was his conviction that the future of all kinds of physics lay in the hands of accelerator physics. That experimental proof was the sole arbiter for all things physical he summarised within a memorable statement:

God is a mathematician, but even he/she/it will wait for experimental proof before being right.

This observation arose when Rajaji decided to speculate aloud on the future of experimental particle physics, specially considering an observable proof of the existence of string theory.

He finished ruing that accelerator physics was an oft ignored subject in many research centres and universities; now that we had sufficiently explored the limits and capabilities of SM-physics, the physics to follow (SUSY, GUT, string theory, etc.) necessitated collision-energies of the order of 10¹⁹ GeV (the "upgraded" run of the LHC in early to July 2012 delivered a collision energy of 8,000 GeV).

These are energies well outside the ambit of current human capability. It may well be admitted at this point that an ultimate explanation of the universe and all it contains is not going to be simple, and definitely not elegant. Every step of the way, we seem to encounter two kinds of problems: one cardinal (particle-kinds and their properties) and metaphysical (why three families of particles and not two or four?).

While the mathematics is "reconfigured" to include such new findings, the philosophy acquires a rupture, a break in derivability, and implications become apparent ex post facto.

The fallacy of attaining infinity

I made a lot of trips to The Hindu office on Mount Road this past week, and I made all of them using the suburban railways. When standing still inside a train that's moving at around 80 km/hr, I'm also moving at the same speed in the same direction. When walking ahead inside the train, I'm moving faster than the train inside the train. When walking toward the back of the train, I'm moving slower than the train is. All this is boring relative-motion stuff. How about when I'm moving sideways inside the train?

When I'm moving sideways at a speed of, say, 2 m/min, the train will have moved forward a distance of 22.22 m in that same minute. If there were an imaginary path that I was inscribing on the ground, then my sideways-one won't be perpendicular to the train's path: it will be adjacent and separated by an angle (like in the diagram shown below).

In the diagram, b is 22.22 m long, a is 2 m long, C is 90°, and A is tan(a/b) = 0.0015°. Now, the time taken by the train to traverse 22.22 m is 1 s. Let's keep that fixed; instead, in that same second, let's move faster and faster from point A to point B (i.e., my sideways motion). If I move 3 m instead of 2, the angle A becomes 0.0022°. If I move 5 m, the angle's value climbs to 0.0039°. At some point, where the train's speed is too high, the value of A has to move toward around 0°, and c, toward b. In other words, if I move really fast along the breadth of the train and the train has also sped up to a great velocity, I can get from one side of the train to the other as if I simply vanished at this point and materialized at that.

For that to happen, let's make some hypothetical modifications to the train: let the breadth be 2 km instead of a few metres, and let it be accelerating toward around 2,000 km/hr. Assuming that at some point of time the train has stopped accelerating and attained a constant velocity of 2,000 km/hr, if I move 2 m sideways in 1 s, the value of A stands at 0.0000628° and c at 555.5536 m. To make c smaller, let's say the train has sped up to 2,100 km/hr and I move at 1 m/s. This makes A 0.0000298° and c, 583.3309 m. If I move so much as 0.1 m, A becomes 0.00000298° and c, 583.3300002 m. At this stage, A is as good as 0° and c almost equal to b.

For someone watching me move from inside the train, I will have moved sideways at 0.1 m/s. However, for someone looking at the imaginary path (i.e., from a relativistic reference frame), it will be non-existent! This is because I will have moved from A to B in a train so fast that my path will become a, as if there were two parallel lines (one beginning at A and the other at B) and I moved from one to the other along a path that is parallel to both lines. This situation is a mathematical improbability, and thus must correspond to an improbable assumption in the real-world. What is it?

The simplest wrong assumptions are always associated with almosts and nearlies. Saying 583.33 m is almost equal to be 583.3300002 m is different from saying 583.33 m is equal to 583.3300002 m. In the real world, for as long as we don't hit relativistic velocities (i.e., those close to that of light), there will always be these extremely small but furiously persistent inconsistencies - they might seem valid for mathematical and practical approximations but they will always translate into very real differences.

The cause-effect paradigm

Some people find differential calculus very easy. Others find vector algebra very easy. However, given that our education system is firmly unidirectional for many justifiable reasons, the calculus-folk would have had to suffer vectors before they came across what they liked. This happens to most students. Unfortunately, the process is so rigorous that such students may be driven to lose focus or interest in the subject as a whole. There could be no other way to do it, but that doesn't mean there's no better way to teach such subjects inside classrooms.

From time to time, students and teachers alike need to be reminded that each topic in a subject is weak by itself, and only with the assistance of other topics is anything achieved. Instead of going from specifics to the larger picture, why not come from the larger picture to the specifics? After all, and this is just an (convenient) example, mathematics is a powerful but singular set of tools used to solve problems in the real world: every problem is application driven, including in string theory and loop quantum gravity, where, without the verification of their hypotheses by experiments, each remains just a strongly-defended opinion.

[caption id="" align="aligncenter" width="510" caption="The tools of multilateral thinking can be used within classrooms as well to improve efficiency and productivity."][/caption]

I must concede that some problems are better solved using some tools than others, but keeping in mind why the problem is being solved like that is important. Even if calculus provides a circuitous route to a solution, what's wrong with its being adopted by the calculus-lovers to get there? When they get there, the relationship between the problem and the solution becomes clearer: there is a better cause-effect relationship established than when a student struggles through vectors and is exhausted by the end, reluctant to take it up again.

As far as laying the groundwork is concerned, teaching students everything is the way to go: at some point later, then, they will be better equipped to make a choice - between what they think they ought to stick with and what they think they can afford to avoid. However, in this order of things, the problems solved using tool-set A and tool-set B, even if in different terms, could be the same, or related in some way so that even what seems difficult could be better understood in terms of what seems easy.

These are only musings concerned with the different ways through which students can convert information into knowledge. The point is: as long as we're here to solve problems, let's have fun doing it.

Visualization calibration

What would a musical vector look like? Vectors have magnitude and direction; music possesses an amplitude (volume) and a frequency (pitch). If the directive parameter of the vector is substituted with the frequency of some noise and the magnitude of the vector substituted with the amplitude, and if the origin of the vector is held fixed, then it would move around that pivot, pointing in a certain direction for a given frequency and stretching in that direction according to the amplitude.

The next step is to model the direction according to the frequency: given that the noise playing could be at any frequency between 20 Hz and 20,000 Hz, it would be quite a mundane exercise to manually calibrate a scale and have the vector point at the appropriate positions. Instead, it would be more interesting to ditch the cylindrical coordinates normally taught in classrooms at the middle school level and take up the circular coordinate system. Here, instead of the X and Y axes, there's the radial vector and the angular position: if I stand at a particular point, instead of being so much to the left and so much toward the front, I will be some distance from an origin and inclined at some angle against a baseline.

[caption id="attachment_20680" align="aligncenter" width="300" caption="A circular, or polar, coordinate system"][/caption]

Now, let's fix the frequency conversion first. 20 Hz to 20,000 Hz is a range of 19,980 Hz. Dividing that value by 360 degrees, we get 55.5 Hz per degree: this means that starting at 0 degrees, each subsequent degree represents an increment of 55.5 Hz, as in 0 Hz, 55.5 Hz, 111 Hz, 166.5 Hz, and so on. Therefore, as the noise plays out, the vector will point in the corresponding direction. In order to make it more visually captivating, the timestep can be incremented to 0.5 seconds. In other words, the vector will correspond to the frequency only once every second instead of corresponding continuously. With suitable fade-in and fade-out effects, a smooth flashing motion can be visualized.

Before fixing the amplitude conversion, let's look at the following wave representation of some noise.



Demarcating it into three sections,



If the red line was to be held as the baseline, then the net displacement from it of each point of the green curve (with a timestep of 0.5 seconds) can be computed and a standard deviation (SD) arrived at. Now, the value of the SD is going to be different for different sections, the reasons behind which are evident. Now, instead of computing the deviations separately, section after section, it can be done continuously. Since the value of the SD is equal to the average value of all measured deviations in that section, the section under consideration can be moved with a timestep of 0.5 seconds and a range of 5 seconds.

For example, let's assume that the range of A is 5 seconds. This is the original section. Now, as the noise begins to play, we wait for the first 5 seconds to transpire. At 5.5 seconds, we move the head of the section we're considering to coincide with the the position at which the noise is playing - like a slider along a rail - while we bring up the rear, constantly ensuring that the range remains at 5 seconds. In this moving range, we continuously compute the SD and use this changing value as the radius of the circle we're using to visualize the vector in.

If the noise playing is a continuous and uniformly pitched beep, the vector is going to point in one direction all the time and the radius of the circle is going to be constant throughout. If a sine wave is playing out, then the radius of the circle will rise and fall according to the frequency of the wave and the vector will oscillate between two points on the perimeter of the circle. Here again, a latency can be effected by introducing a lag component to the vector's movement, ensuring that it moves, say, 0.25 seconds later than right then. The final step to calibrating a visualizer is the graphic effects: since we've assumed a circular coordinate system, the equation for the Archimedean spiral can be employed to assign each point, or pixel, within the circular a particulate color.

r = a + b . θ

'a' is the gradient of the coloring; 'b', the number of pixels on the radius of the circle; and 'r', the coloring function that has been employed. The total number of pixels in the circle will be πr² which will also then be the number of colors to be assigned overall. Using a loop counter to increment the hex colors (and assigning them to the value of 'r'), the moving vector can be colorized depending on where it points to and to what distance within the circle (while θ is increased from 0-360 degrees). Since the radius of the circle, 'b', is going to keep changing, it would be better to colorize the entire canvas, superimpose the image of the circle on it, mask the colors, and then use the vector to unmask the colors on its "skin".

[caption id="attachment_20677" align="aligncenter" width="474" caption="A still of a visualization on Windows Media Player, achieved by using more subtle gradients, fading effects, and multiple layers of images."][/caption]

Visualization calibration

What would a musical vector look like? Vectors have magnitude and direction; music possesses an amplitude (volume) and a frequency (pitch). If the directive parameter of the vector is substituted with the frequency of some noise and the magnitude of the vector substituted with the amplitude, and if the origin of the vector is held fixed, then it would move around that pivot, pointing in a certain direction for a given frequency and stretching in that direction according to the amplitude.

The next step is to model the direction according to the frequency: given that the noise playing could be at any frequency between 20 Hz and 20,000 Hz, it would be quite a mundane exercise to manually calibrate a scale and have the vector point at the appropriate positions. Instead, it would be more interesting to ditch the cylindrical coordinates normally taught in classrooms at the middle school level and take up the circular coordinate system. Here, instead of the X and Y axes, there's the radial vector and the angular position: if I stand at a particular point, instead of being so much to the left and so much toward the front, I will be some distance from an origin and inclined at some angle against a baseline.

[caption id="attachment_20680" align="aligncenter" width="300" caption="A circular, or polar, coordinate system"][/caption]

Now, let's fix the frequency conversion first. 20 Hz to 20,000 Hz is a range of 19,980 Hz. Dividing that value by 360 degrees, we get 55.5 Hz per degree: this means that starting at 0 degrees, each subsequent degree represents an increment of 55.5 Hz, as in 0 Hz, 55.5 Hz, 111 Hz, 166.5 Hz, and so on. Therefore, as the noise plays out, the vector will point in the corresponding direction. In order to make it more visually captivating, the timestep can be incremented to 0.5 seconds. In other words, the vector will correspond to the frequency only once every second instead of corresponding continuously. With suitable fade-in and fade-out effects, a smooth flashing motion can be visualized.

Before fixing the amplitude conversion, let's look at the following wave representation of some noise.



Demarcating it into three sections,



If the red line was to be held as the baseline, then the net displacement from it of each point of the green curve (with a timestep of 0.5 seconds) can be computed and a standard deviation (SD) arrived at. Now, the value of the SD is going to be different for different sections, the reasons behind which are evident. Now, instead of computing the deviations separately, section after section, it can be done continuously. Since the value of the SD is equal to the average value of all measured deviations in that section, the section under consideration can be moved with a timestep of 0.5 seconds and a range of 5 seconds.

For example, let's assume that the range of A is 5 seconds. This is the original section. Now, as the noise begins to play, we wait for the first 5 seconds to transpire. At 5.5 seconds, we move the head of the section we're considering to coincide with the the position at which the noise is playing - like a slider along a rail - while we bring up the rear, constantly ensuring that the range remains at 5 seconds. In this moving range, we continuously compute the SD and use this changing value as the radius of the circle we're using to visualize the vector in.

If the noise playing is a continuous and uniformly pitched beep, the vector is going to point in one direction all the time and the radius of the circle is going to be constant throughout. If a sine wave is playing out, then the radius of the circle will rise and fall according to the frequency of the wave and the vector will oscillate between two points on the perimeter of the circle. Here again, a latency can be effected by introducing a lag component to the vector's movement, ensuring that it moves, say, 0.25 seconds later than right then. The final step to calibrating a visualizer is the graphic effects: since we've assumed a circular coordinate system, the equation for the Archimedean spiral can be employed to assign each point, or pixel, within the circular a particulate color.

r = a + b . θ

'a' is the gradient of the coloring; 'b', the number of pixels on the radius of the circle; and 'r', the coloring function that has been employed. The total number of pixels in the circle will be πr² which will also then be the number of colors to be assigned overall. Using a loop counter to increment the hex colors (and assigning them to the value of 'r'), the moving vector can be colorized depending on where it points to and to what distance within the circle (while θ is increased from 0-360 degrees). Since the radius of the circle, 'b', is going to keep changing, it would be better to colorize the entire canvas, superimpose the image of the circle on it, mask the colors, and then use the vector to unmask the colors on its "skin".

[caption id="attachment_20677" align="aligncenter" width="474" caption="A still of a visualization on Windows Media Player, achieved by using more subtle gradients, fading effects, and multiple layers of images."][/caption]

Exploring a clustering technique using mathematical group theory

Step I: Identification of ICM

[caption id="attachment_20428" align="aligncenter" width="596" caption="A scatter plot"][/caption]

The x-axis, identified by the base marker, represents numerical values from 1 to 100 (integers). The y-axis, identified by the decimal marker, represents numerical values from 0 to 1 (decimal, 2 significant digits).

From all the data, two points are selected at random, marked in the image by a black rectangle around them. They are labelled as the initial cluster markers (ICM).

Step II: Geometric tagging

The Euclidean distance between each point in the scatter plot and the ICM is calculated. In this case, there are 48 points (after excluding the ICM): x₁, x_2, . . ., x₄₈. Therefore, the distances between x₁ and ICM₁ and ICM₂ are calculated; between x₂ and ICM₁ and ICM₂, and so on until x₄₈ and ICM₁ and ICM₂.

If the distance between some x_i and ICM_a is lower than x_i and ICM_b, then x_i is grouped with ICM_a and tagged as x_i^a. This gives rise to a clear demarcation; the data set is now split amongst x_i^a and x_j^b. Each section is identified as S{x_i,j^a,b}.

Step III: Averaging

Before any average is computed, it must be determined as to which value is to be considered. Three cases are presented below.

1. x-value: if the x-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line x_i = µ_i


2. y-value: if the y-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line y_i = µ_i


3. Alien value: consider the following table.

Base marker (1-50)	Decimal marker (0-1)	Anonymous marker (1-100)
4	0.71	20
13	0.61	38
14	0.98	7
18	0.03	4
11	0.68	55
37	0.26	67
30	0.04	46
21	0.22	57
37	0.48	90
47	0.94	43

The x- and y-axes represent the values in the first two columns while the third column shows a set of values not considered in the construction of the scatter plot. Since each row in this table is denoted as a point in the plot, each such point is also associated with a certain “anonymous value” as shown in the table.

These can be considered in the averaging process, whereby all the “anonymous” values of all the points in each cluster are averaged separately:

S(x_i^a): A(x_i^a)

S(x_j^b): A(x_j^b)

These new points, A_a and A_b, are plotted in their respective clusters.

Step IV: Convergence

A_a and A_b become the new ICM, and steps I to III are repeated.

The iterations stop when the new A_a and A_b converge with the A_a and A_b from the previous iteration.

Result

After the values have converged, the two S{x_i,j^a,b} now presented by the machine are two distinct groups of data, the emergence of which also signals that the machine has “learnt”. If a very large database is supplied as an input to the machine, there is an advantage as well as a disadvantage.

• Advantage: the machine learns better


• Disadvantage: data may not converge at all due to improper choice of clusters

In order to prevent the second possibility, a suitable number of clusters has to be determined for n, the number of data-points. As a rule of thumb,

k = (n / 2)^1/2

Here, ‘k’ is the number of clusters.

If a dataset of 50 points is input, then k = 5 clusters are suggested. Once the iterations have been performed and convergence has been attained in five separate clusters, the centroids of each cluster (or, the average value of the data-points in each cluster) can now be used to arrive at 2 super-clusters.

Exploring a clustering technique using mathematical group theory

Step I: Identification of ICM

[caption id="attachment_20428" align="aligncenter" width="596" caption="A scatter plot"][/caption]

The x-axis, identified by the base marker, represents numerical values from 1 to 100 (integers). The y-axis, identified by the decimal marker, represents numerical values from 0 to 1 (decimal, 2 significant digits).

From all the data, two points are selected at random, marked in the image by a black rectangle around them. They are labelled as the initial cluster markers (ICM).

Step II: Geometric tagging

The Euclidean distance between each point in the scatter plot and the ICM is calculated. In this case, there are 48 points (after excluding the ICM): x₁, x_2, . . ., x₄₈. Therefore, the distances between x₁ and ICM₁ and ICM₂ are calculated; between x₂ and ICM₁ and ICM₂, and so on until x₄₈ and ICM₁ and ICM₂.

If the distance between some x_i and ICM_a is lower than x_i and ICM_b, then x_i is grouped with ICM_a and tagged as x_i^a. This gives rise to a clear demarcation; the data set is now split amongst x_i^a and x_j^b. Each section is identified as S{x_i,j^a,b}.

Step III: Averaging

Before any average is computed, it must be determined as to which value is to be considered. Three cases are presented below.

1. x-value: if the x-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line x_i = µ_i


2. y-value: if the y-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line y_i = µ_i


3. Alien value: consider the following table.

Base marker (1-50)	Decimal marker (0-1)	Anonymous marker (1-100)
4	0.71	20
13	0.61	38
14	0.98	7
18	0.03	4
11	0.68	55
37	0.26	67
30	0.04	46
21	0.22	57
37	0.48	90
47	0.94	43

The x- and y-axes represent the values in the first two columns while the third column shows a set of values not considered in the construction of the scatter plot. Since each row in this table is denoted as a point in the plot, each such point is also associated with a certain “anonymous value” as shown in the table.

These can be considered in the averaging process, whereby all the “anonymous” values of all the points in each cluster are averaged separately:

S(x_i^a): A(x_i^a)

S(x_j^b): A(x_j^b)

These new points, A_a and A_b, are plotted in their respective clusters.

Step IV: Convergence

A_a and A_b become the new ICM, and steps I to III are repeated.

The iterations stop when the new A_a and A_b converge with the A_a and A_b from the previous iteration.

Result

After the values have converged, the two S{x_i,j^a,b} now presented by the machine are two distinct groups of data, the emergence of which also signals that the machine has “learnt”. If a very large database is supplied as an input to the machine, there is an advantage as well as a disadvantage.

• Advantage: the machine learns better


• Disadvantage: data may not converge at all due to improper choice of clusters

In order to prevent the second possibility, a suitable number of clusters has to be determined for n, the number of data-points. As a rule of thumb,

k = (n / 2)^1/2

Here, ‘k’ is the number of clusters.

If a dataset of 50 points is input, then k = 5 clusters are suggested. Once the iterations have been performed and convergence has been attained in five separate clusters, the centroids of each cluster (or, the average value of the data-points in each cluster) can now be used to arrive at 2 super-clusters.

Plays of the day

The "Club 27" apocrypha

Do rockstars who die at the age of 27 change the way we look at rock n' roll?

*

Necker cubing

Some time ago, during a certain event, it so befell that I had to get up on stage and speak over a mic; I say that because what I said isn't important. As I started to speak, I became aware of two voices: my voice pre-amplification (pre-A) and my voice post-amplification (post-A). I had to be aware of the pre-A so I wouldn't raise my voice unnecessarily, and I had to be aware of the post-A so I could listen to what I was saying.

Over the course of the next few minutes, I could often be caught trying to listen to my pre-A and check for the loudness of my voice using the post-A, which didn't work at all, leading to a constantly varying amplitude of the output - more often than not at increasing volumes. Then again, I let my audience laugh at me: I've found that distracts people enough to let me carry on with my work. Anyway, the experience was like trying to drive a motorcycle precisely over the center of the road at all times.

Consider the following schema.

Here, S stands for the source, A for the amplifier, V for the volume (or quantity) and I for information (or quality).

Are there any hormonal systems or neural networks that function on this principle? Because this reminds me of the McGurk effect in interdisciplinary cognition.

*

Communist journalism

Does constantly asking "How is the common man being wronged?" foster a Communist proclivity?

*

The calculus affair

My textbook of differential equations and their applications was finally delivered by Flipkart (and then to me by D.). When I was looking for the book first, I chanced upon a textbook of algebraic topology, which would've been perfect for the Conway's game of life problems I've been looking at. While browsing through the first few pages on Amazon's preview, I had a shock when I realized I'd lost with my calculus. Of course I bought the book on differentiation immediately!

I was never so good at solving problems I was asked to solve inside the classrooms, but when it came to differential calculus, I could solve the toughest problem in a jiffy. What a dejection it must have been when I took more than 10 minutes to figure out the differential of e^x was e^x. When in Dubai doing engineering, I wanted to study journalism so much. Now, at ACJ, my fingers itch everyday for a challenge in calculus.

*

Symmetrical itching

I was sitting with a couple of friends outside on the lawn when one of them, A, itched two sides of her face at the same time. It was curious because she said it happened often, i.e., symmetrical itching, sometimes on her shoulders, sometimes on her hands. I quickly made a note of it, much to the amusement of my friends who thought I was being curious about nothing.

Now, I find that there's nothing concrete to explain symmetrical itching (even though the itch itself as the cause of concern has been widely debated) and most answers on the Web are centered around the "wiring" of the CNS and a possible eczema infliction. This shalt be pursued.

Plays of the day

The "Club 27" apocrypha

Do rockstars who die at the age of 27 change the way we look at rock n' roll?

*

Necker cubing

Some time ago, during a certain event, it so befell that I had to get up on stage and speak over a mic; I say that because what I said isn't important. As I started to speak, I became aware of two voices: my voice pre-amplification (pre-A) and my voice post-amplification (post-A). I had to be aware of the pre-A so I wouldn't raise my voice unnecessarily, and I had to be aware of the post-A so I could listen to what I was saying.

Over the course of the next few minutes, I could often be caught trying to listen to my pre-A and check for the loudness of my voice using the post-A, which didn't work at all, leading to a constantly varying amplitude of the output - more often than not at increasing volumes. Then again, I let my audience laugh at me: I've found that distracts people enough to let me carry on with my work. Anyway, the experience was like trying to drive a motorcycle precisely over the center of the road at all times.

Consider the following schema.

Here, S stands for the source, A for the amplifier, V for the volume (or quantity) and I for information (or quality).

Are there any hormonal systems or neural networks that function on this principle? Because this reminds me of the McGurk effect in interdisciplinary cognition.

*

Communist journalism

Does constantly asking "How is the common man being wronged?" foster a Communist proclivity?

*

The calculus affair

My textbook of differential equations and their applications was finally delivered by Flipkart (and then to me by D.). When I was looking for the book first, I chanced upon a textbook of algebraic topology, which would've been perfect for the Conway's game of life problems I've been looking at. While browsing through the first few pages on Amazon's preview, I had a shock when I realized I'd lost with my calculus. Of course I bought the book on differentiation immediately!

I was never so good at solving problems I was asked to solve inside the classrooms, but when it came to differential calculus, I could solve the toughest problem in a jiffy. What a dejection it must have been when I took more than 10 minutes to figure out the differential of e^x was e^x. When in Dubai doing engineering, I wanted to study journalism so much. Now, at ACJ, my fingers itch everyday for a challenge in calculus.

*

Symmetrical itching

I was sitting with a couple of friends outside on the lawn when one of them, A, itched two sides of her face at the same time. It was curious because she said it happened often, i.e., symmetrical itching, sometimes on her shoulders, sometimes on her hands. I quickly made a note of it, much to the amusement of my friends who thought I was being curious about nothing.

Now, I find that there's nothing concrete to explain symmetrical itching (even though the itch itself as the cause of concern has been widely debated) and most answers on the Web are centered around the "wiring" of the CNS and a possible eczema infliction. This shalt be pursued.

Mathematizing The Lord Of The Rings

Fundamental entities

Set of parameters: P(θ) = {θ₁, θ₂, ..., θ_n} = P^θ

Set of constants: C(φ) = {φ₁, φ₂, ..., φ_n} = C^φ

Constitutional entities

Problem space: I_P(x, y, z, t) = I_{P(x, y, z, t)}(P_i^θ, C_k^φ)

Solution space: I_S(x, y, z, t) = I_{S(x, y, z, t)}(P_j^θ, C_l^φ)

Active entities

F(θ, φ): I_P(x, y, z, t) --> I_S(x, y, z, t)

E(x, y, z, t) = Ē

Game definition

Players

P^F: Frodo Baggins / P^S: Samwise Gamgee / P^P: Peregrin Took / P^M: Meriadoc Brandybuck / P^G: Gandalf / P^A: Aragorn / P^L: Legolas / P^I: Gimli / P^B: Boromir / P^R: Faramir / P^D: Denethor / P^S: Saruman / P^Z: Sauron / P^H: Gollum

Fixtures

C^S: Shire / C^B: Bree / C^W: Weathertop / C^R: Rivendell / C^C: Caras Galadhon (Lothlorien) / C^M: Moria / C^O: Rohan / C^H: Helm’s Deep / C^G: Gondor / C^L: Morgul Vale / C^Z: Mordor / C^U: Orodruin / C^I: Isengard

Problem function

I_P, _Ē = I_P(x, y, z, t, u)[({P^F, P^S}, {P^R, P^H}), ({P^P, P^M, P^G, P^A, P^L, P^I, P^B, P^D}, {P^S, P^Z})]

Solution function

I_{S, Ē} = I_P(x, y, z, t, u)(P^F, C^U)

As simple as that.

[caption id="" align="alignleft" width="240" caption="Making sense of Middle Earth"][/caption]

I Just Really Suck At Calculus!

Skin head, brain dead
Everybody gone mad
Situation, bad division
Everybody calculation
In the root, on the pews
Everything is wrong, dude
Damn damn, turned to lead
Everybody's gone mad

All I wanna say is that
I just really suck at calculus
All I wanna say is that
I just really suck at calculus

Beat me, scold me
You can never teach me
Will me, kill me
You’ll never instil in me
Fool me, sue me
Everybody screw me
Kick me, bite me
Don't you integrate me

All I wanna say is that
I just really suck at calculus
All I wanna say is that
I just really suck at calculus
Tell me what has become of my life
I have a textbook with seventy-five pages
I am the victim of Leibnitz’s genius, now
I'm tired of having to calculate
You're rapin' my equations wide
Oh, for God's sake
I look to tuition to fulfill its prophecy...
Set me free

Skin head, brain dead
Everybody gone mad
Integration, evaluation
Everybody calculation
In the root, on the pews
Everything is wrong, dude!
Stupid man, stupid male
Your theories are all stale

All I wanna say is that
I just really suck at calculus
All I wanna say is that
I just really suck at calculus

Tell me what has become of my rights
Am I all wrong just because I can’t see
The blackboard from the back bench, now
I'm tired of bein' the victim of shame
They're throwing me in a class with a bad name
I can't believe this is the book from which I frame
You know I do really hate to say it
The professor don't wanna see
But if Einstein was livin'
He wouldn't let this be, no, no

Skin head, brain dead
Everybody gone mad
Situation, aggravation
Everybody calculation
In the root, on the pews
Everything is wrong, dude
Damn damn, turned to lead
Everybody's gone mad

All I wanna say is that
I just really suck at calculus
All I wanna say is that
I just really suck at calculus

Some signs in life they just don't wanna see
But if Ramanujam was livin'
He wouldn't let this be

Beat me, scold me
You can never teach me
Will me, kill me
You’ll never instil in me
Fool me, sue me
Everybody screw me
Kick me, bite me
Don't you integrate me

All I wanna say is that
I just really suck at calculus
All I wanna say is that
I just really suck at calculus

All I wanna say is that
I just really suck at calculus
All I wanna say is that
I just really suck at calculus

Entropy And Determinism

 

[caption id="" align="aligncenter" width="180" caption="Wittgenstein's Poker: The Story of a Ten-Minute Argument Between Two Great Philosophers"][/caption]

 

Philosophy has often been characterized as the recourse for thinkers, even as consideration for the value it seeks to add is discarded in favor of a more methodical approach toward understanding each of its myriad contributions. While there persists an affectation ofexistential subjectivism, one of objectivism would perhaps beseem the content as well as aid in better contextualizing the lessons of the work at hand in our lives. Before I close this digression, I would like to add that the role philosophy plays, whilst observing from an ontological vantage point, is much greater and demands a compulsion, if not an interest, to discover and establish annealing affinities between any number of entities that we do contemplate to be in the possession of logical similarities.

Whilst a philosophical expediency is predominantly empirical, the opportunity of science reaffirms that character and proceeds to favor establishmentanarianism as an assimilation of conclusions drawn from empirical observations; that our pronouncements on the macrocosm we occupy would be so primordially limited in scope and faith is abundant proof toward the corroboration of determinism. If not for entropy, the deterministic quality of our surroundings would be fait accompli simply because, in our attempts toward the evaluation of this universe, the premise and the calibration have both been quantitatively estimable; there is absolutely no sustainable argument that our conclusions can not be so.