Identification Compactifies

Identification Compresses

Identifying two ends of a line forms a loop. When starting point and ending point (including \(-\infty\) and \(\infty+\)) of a homogenous line is identified, it becomes a circle (loop). We must identify Aryabhata’s discovery of \(0\) as a symbol of symmetry (or equilibrium) to be the Fundamental Law of all Languages. When all structures (Languages) are reduced to composition of simple groups, they can further be seen as a composition of \(C_2\), the Cyclic group of order two which is represented by a homogenous straight line with two dual points- one starting point and the other ending point (in CS, this is Binary Language). Identifying these two end points gives the trivial group- having only one element which is the identity. This is represented by \(0\) as the Fundamental Identity.

• \(0\) is the fundamental Origin of all structures (Languages).

Identification of asymmetry represents both sides in equilibrium with each other at their meeting point. This is why Identification Compactifies. The Real Number line is not compact but the Circle which identifies ends of Real Line is compact. Homeomorphisms (flows preserving boundaries of closed structures) preserve compactness. The charts (homeomorphisms) used to represent Manifolds as composition of Euclidean Spaces form partitions of the Manifold. That is, the Manifold acts as the Identity (symmetry) which is expressed in asymmetric forms through the charts (homeomorphic flows projecting the manifold to Euclidean Spaces). This involves going from a symmetry (the manifold) which is compact to a composition of relatively less compact forms. This is similar to how in Group theory, Identity is unique and needs only one symbol to express itself whereas all other elements need its inverse symbol to compose together and form the identity. Circle (compact) cannot be represented as the Real Line (non-compact) using a single chart. Similarly, the partitions of unity involve expressing \(1\) (the identity) into partitions.

• Identify the \(\partial\) symbol as corresponding to English symbol (,) which means “only one side (,) there’s more”.

• Identifying multiple flows (trajectories) arising in a vector field can be expressed as a Differential equation which is compact form of the various Integral flows.

• Identifying various Lie Algebras with their corresponding Identities (origins) composed in a particular way gives a compact Lie Group.

• Equations only involving Laplacian (composition of divergence and gradient (both are duals to each other) capture Spherical and Elliptic behaviour - which is a projection of sphere along an equilibrium cone. The behaviour can be captured in a compact form by Spherical Differential Equations. Spheres correspond to completely uniform metric, that is the metric is \((1,1,1,\ldots)\). For example the Euclidean Metric which is the completely symmetric metric giving Sphere as the Identity.

  • With the metric \((1,1)\), The Measure in the plane becomes \(1\,\,x \,\, 1\,\,x \,\,+\,\,1\,\,y\,\,1\,\,y\). When this is equated with the Identity, it gives \(1\,\,x \,\, 1\,\,x \,\,+\,\,1\,\,y\,\,1\,\,y =1\) which is a circle.

  • Allowing “Asymmetries to flow” with time eventually gives the Euclidean Metric as all the asymmetries cancel out each other to leave only the ultimate symmetry which is the Sphere (symmetry= same metric= sphere) in the corresponding space.

• Expressions which involve composition of both \(\nabla \,\, \cdot\) and \(\nabla \,\,\times\) to form the scalar triple product (the determinant) give the entire measure (of the space) which is called “Determinant”. This composes all symmetries together and hence is the “top form” which measures to the Identity (total measure of the space).

• Reducing “space” or asymmetries or increasing symmetric are equivalent. Hence Identification leads to compactification.

• In Calculus Language, Identification (Integration) is unifying elements from lower degree of freedom to a boundary of measure 0 in higher dimension. Line is identified to a circle, which is its compactification.

• Identifying Operator (continuous) with its Spectrum (discrete) gives an Eigenvalue Equation. The solutions are Eigenfunctions which are compactified Identities to represent the remaining functions. This is explored in Harmonic Analysis and more specifically, in Fourier Analysis.

  • Electron orbits are discrete (Bohr’s Model) which gives elliptic levels of energy as the Spectrum. These elliptic orbits are also seen as discrete orbits of Planets and in Number theory (elliptic curves). The Electron Wavefunction densities are continuous. The composition of spectra with the Operator gives Schrodinger Equation as the Eigenvalue equation. The solutions are Eigenfunctions which are the Orbitals.

  • Theory of Light is incomplete without recognizing the Colours (Frequencies) as the spectrum of the waveform.

  • Sound waves and the fundamental frequencies of vibrations are to be identified.