Friday, 28 December 2012

Game Theory- The Rock-Paper-Scissors Algorithm

From the roll of a die and the shift of a pawn to the spin of a cue and the flip of a card, its principles come into action through the players, their strategies and their payoffs. A premise that entails the methodology, skill and technique behind even the most primitive forms of human leisure and makes something serious out of the business of games. One where equilibria are sought, risk and choice are evaluated and conflicting interests are observed in a manner of critique; a platform where competition and completeness are deduced and location, segregation and randomisation are considered. A formulation of not only the players but the games themselves in a means that quantifies the tricks and flicks to winning and the shortcomings to losing. Game theory is to games as calculus is to change, for games offer a wealth of profundity regarding the mechanics of human skill and strategy as well as the strategic basis to decision making; extending well into frontiers such as economics and evolutionary biology where the flux of mutation and competition play themselves out like a well choreographed game. Consider the prisoner's dilemma, where two suspects are apprehended on the suspicion of thievery, if a bargain is offered to each suspect with the promise of mitigated jail time in return for cooperation the end result will be unfold with defection by both parties due to the notion of strict dominance despite the seemingly logical outcome of cooperation by both parties. Furthermore, the classic battle of the sexes takes form in game theory whereby a man and woman desire boxing and ballet respectively however they simultaneously desire to be together. In order to go about such a game, one necessitates the utility of the Nash equilibrium in its mixed strategy form; an equilibrium in which there exists a set of strategies, one for the man and one for the woman such that no player has an incentive or advantage to change his or her strategy given what the other players are doing. Analogous to a law that no one would break despite the absence of law enforcement or capital punishment. A more familiar circumstance would be that of rock-paper-scissors, whereby Nash equilibrium may be reached via mixed strategy given the zero sum nature of the game. An overall nexus of balance and conformity driven towards the satisfaction of both parties and the achievement of a realistic and level headed outcome, comparable to an algorithm that plays the game over and over until scissors and paper fit hand in hand.

Thursday, 27 December 2012

Chaos- The Butterfly Effect

A butterfly flutters its wings in Guatemala and triggers a tornado in Guantanamo. While such stands  as nothing short of a miracle, a perfect yet powerful expression of roll, pitch and yaw by a humble arachnid; it remains the metaphorical ideal for an entire dynamics bent on the eccentricities and capriciousness of nature. The bold and presumptuous notion that randomness is merely a mirage, that order and disorder are intimately linked and that small changes result in vast differences. The notion of chaos usually conjures the image of randomness and entropy whereas it is a manifestation of unpredictability and fluctuation played by the laws of nature. Everything from the emergent ethology of flocks of birds and the formation of the weather to the evolution of stock markets and economic crashes unveils a non-linear, irregular and unpredictable platform. One where the deterministic view collapses and makes room vacant for a chaotic framework of dynamically changing systems provoked by minute and seemingly trivial modifications. A double pendulum will assume a sporadic and capricious motion when released due to minuscule differences in the initial configuration, just as the weather varies due to fluctuations in atmospheric conditions such as convection as quantified by the Lorenz attractor. Chaos may be both observed and envisaged as a facet of nature; for mountains are not cones neither are clouds perfect spheres nor are trees flawless cylinders. It is the non-Euclidean geometry of the physical world that is addressed by the chaotic nature of fractals, essentially the byproducts of dynamically changing systems that convey a feedback loop of infinite regression, self-similarity and locally and globally oriented irregularity. Even simple contraptions such as a Poincare's three body astronomical system indicate that small errors in calculation result in chaotic subsequent outcomes, way out of proportion of the initial mathematical estimates and premises. Others such as the Belousov non-equilibrium thermodynamic reactions probe the chaos of chemical oscillation from a clear state to coloured to clear again; indicating a chemical basis of morphogenesis centred on the reaction-diffusion paradigm expounded by Turing. Chaos is kaleidoscopic; every small fluctuation, modification and variation constitute to the outcome and output. And however unpredictable, dubious and erratic that outcome may be, it is alway reducible to simplicity and lucidity regardless of whether its a Mandelbrot set or a Romanesco broccoli!

Sunday, 23 December 2012

Calculus- The Integral Differentiation

Its incalculable complexity reduces arithmetic to child's play while its unsurpassed applicability belittles algebra with the rest of its regularity. A mathematical scheme bent on the diversity of change, the correlation of the input with its output and the continuity of functions. Since its inception via Newton and Leibniz, it has emerged as a union between the celestial and the terrestrial; a language of fluxions and fluents and the manifestation of the laws of nature. Calculus is a dimorphic synthesis of integration and differentiation linked via its fundamental theorem. The former of the two, seeks the area between the curve of a function f(x) while the latter pursues the way a function behaves as its input differs such as the of instantaneous rate of change. While the regularity, uniformity and discreteness of institutions such as algebra remain most apparent, calculus intervenes through the annihilation of such regularity and uniformity,assuming the curliness of slopes and curves. Calculus remains the gateway to realising the spread of vector and tensor fields permeating and changing through space and time, the medium towards the fathoming of continuums beyond that of the discrete. For instance, in realising limits for functions f(x), we may realise their behavior near certain values and extrapolate them for understanding the behavior of particle frequencies near magnetic fields, dummy variables and their jump discontinuity or even the interference pattern of photons via double slits. The elliptical orbit of the celestial bodies, the logarithmic spiral of a chambered nautilus shell and Zeno's paradox; whereby an infinite number of halves may be crossed in a finite amount of time, all share this common denomination of change and fluctuation as quantified by calculus. Trigonometric functions  included, as their differentiation reveals the rate of change of such variables relative to one another.  But the applicability of calculus exceeds that of the two dimensional, path integrals give rise to higher dimensions of application such as integration above curves while volume integrals consider the everyday reality of three dimensional spatial freedom. It is a ever encompassing framework, played by certain simple rules yet fulfilled by a nature's laws and governing dynamics; a formulation of maxima and minima, of input and output and of limit and continuity.

Thursday, 13 December 2012

String Theory- A Primer

A grand contender for a final theory of everything. One where the macrocosm and the microcosm unite in synchrony, humming to the tune of vibrating yet infinitesimal filaments of energy. Since the ancient Pythagoreans and their love affair with the octaves, thirds and fifths of the lyre; to the contemporary developments of modern physics, string theory has emerged as an controversial contender for the unification of the forces of nature. It is a presumptuous idea, that all phenomena, matter and energy are bound by the dynamic fluxes of vibrating strings in hyperspace.Such strings behave much like the strings of a cello or violin, except producing particles as opposed to musical notes; therefore quarks can be envisaged as strings vibrating in a single pattern such as an A-flat or even neutrinos as filaments oscillating in a C-sharp. Now down to the details. In 1968, Veneziano 'accidentally' discovered that the beta function of Euler mirrored some aspects of the strong force that acted upon hadrons. This lead to its physical interpretation involving vibrating one dimensional strings that modelled point-like particles. Unlike quantum field theory which experiences infinities as a result of viewing particles as dimensionless points and by consequence bringing those points close together to create infinitely strong forces, strings can avoid this anomaly. A string my be open with its ends fluttering or closed into a loop; originally, open strings were found to masquerade as a spin-1 bosons. But a string is not an object in space, it is literally a piece of space and unlike a violin, whose oscillation is deadened as it transfers it energy to the outer environment, a piece of string is isolated with its energy contained, thus its oscillation isn't deadened. And since there is an infinite degree of harmonics or resonant modes, such strings can imitate an infinite number of types of particles, endowing each with specific energy.

Moreover, because patterns of string vibration arise in pairs, varying by around half a unit of spin, it creates an internal supersymmetry (SUSY). SUSY allows one to merge bosons and fermions given that they return to their original states after 360 and 720 degree rotations respectively, if one can define such transformations as manifestations of a greater multidimensional geometry, each of the known particles can be related by heavier superpartners. The additional dimensions are for mathematical consistency and Kaluza was the first to realise that adding a fifth dimension to spacetime mirrored Maxwell's equations, the extra dimensions curled up like a cartesian plane passing via each point in space. But among the hallmarks of string theory included the ability to mimic spin-2 gravitons with closed strings, allowing the possilibity of describing gravity quantum mechanically but without invoking fields directly. The only free parameter in question is the tension of the string, which can be calibrated more precisely using the graviton to meet the strength of the strong force. As a result, the minimum energy of a string, the planck energy, can be multiplied many times over by the amount of wavelengths in each mode or oscillation to give a large resultant, such may be cancelled out by quantum fluctuations leading to a massless string. As for other particle properties, spin simply becomes the aspect of how a string vibrates, as these vibrations can extend over curled up dimensions it becomes relatively simply to imagine a fermion moving in a gyrating fashion in another dimension, thus gaining a 720 degree rotation in the 3D space to resume its original position. So if mass and spin can emerge from a multi-dimensional geometry, what about the other forces?

 By the 80's, the 5 string theories which all paired their bosons and fermions differently were united by Witten into M-theory into a context of an 11 dimensional Calabi-Yau space-time (10+1). It incorporates one dimensional strings in conjuction with branes (membranes), allowing gravity and the non-gravitational forces to unite at a single energy. This wasn't previously possible, given that the gravitational coupling constant refused to match the quantum forces except until an energy density that exceeded the unification energy; in M-theory it matched neatly. But we are talking ridulous scales on the order of 10^-35m, around 10^-20 the diameter of a proton; at such tiny scales gravity competes with the other forces so we need to unite quantum field theory with general relativity. Such a stark incompatility between the two arises because of the Uncertainty principle; in general relativity, perfectly flat space arises in the absence of a significant mass (such as the quantum vacuum) and thus the value of the gravitational field should be exactly zero. However, the Uncertainty principle needs only a mean value of zero and so the value of the gravitational field can fluctuate in a random 'foam'; therefore we get infinities (or ultraviolet divergences). These infinites come about because in quantum field theory, you deal with points that act as sources of fields, but with one dimensional strings infinities never occur because the energy of a vibrating string is dependant on its frequency as well as the amplitude at which it oscillates (closed strings and loops rely on circumference) and more frenzied modes of vibration produce more energy than placid ones. Much like a Feynman diagram, which capture the history of a particle, a 'worldsheet' capture snippets of a bunch of strings and recording their history as a static network of connecting tubes in spacetime; when the loops interact, two loops may merge and split into a different figuration. Such allows gauge bosons to be expressed as aspects of vibrating strings rather than the conventional view of seperate fields for each charge and particles as point-like bundles of energy osillating in those fields.