In mathematics, the Klein bottle ([kla?? n]) is a non-orientable surface, informally, a surface (a two-dimensional manifold) with no identifiable “inner” and “outer” sides. Other related non-orientable objects include the Mobius strip and the real projective plane. Whereas a Mobius strip is a two-dimensional surface with boundary, a Klein bottle has no boundary. (For comparison, a sphere is an orientable surface with no boundary. ) The Klein bottle was first described in 1882 by the German mathematician Felix Klein. In physics/electro-technology: as compact resonator with the resonance frequency which is half that of identically constructed linear coils * as inductionless resistance. * as superconductors with high transition temperature In chemistry/nano-technology: * as molecular knots with special characteristics (Knotane) * as molecular engines * as graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism. *

In a special type of aromaticity: Mobius aromaticity * Charged particles, which were caught in the magnetic field of the earth, can move on a Mobius band. The cyclotide (cyclic protein), active substance of the plant Oldenlandia affinis, contains Mobius topology for the peptide backbone. In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent (i. e. homeomorphic) to a Mobius strip. The term ‘cross-cap’, however, often implies that the surface has been deformed so that its boundary is an ordinary circle. A cross-cap that has been closed up by gluing a disc to its boundary is an immersion of the real projective plane.

Two cross-caps glued together at their boundaries form a Klein bottle. In chemistry, a molecular knot, or knotane, is a mechanically-interlocked molecular architecture. Examples of naturally formed knotanes are DNA and certain proteins. Lactoferrin has an unusual biochemical reactivity compared to its linear analogue. Other synthetic molecular knots have a distinct globular shape and nanometer sized dimensions that make them potential building blocks in nanotechnology.

APPLICATION OF BOOLEAN ALGEBRA IN TRAFFIC SIGNALS There are six lights to operate. The Red, Amber, and Green lights in the North-South direction will be designated as R1, A1, G1. Similarly, the lights in the East-West direction will be called R2, A2, and G2. When the digital signals are in the Logic-1 state they turn their respective lights on, otherwise the lights are off. A digital clock signal will be supplied and at each clock pulse the lights should.

The design of the circuit that produces the clock pulses at appropriate times will not be considered here. There are two types of road crossing: quiet crossings that use a simple sequence, and busy crossings require a longer (delayed green) sequence. One digital input signal called J (for junction type) will indicate whether the road crossing is considered quiet. J=0 denotes a busy junction and J=1 a quiet one. Thus, we have a one-input, six-output synchronous system to design.