Anomaly of discrete symmetries can be defined as the Jacobian of the path-integral measure. We assume that an anomalous discrete symmetry at low energy is remnant of an anomaly free discrete symmetry, and that its anomaly is cancelled by the Green-Schwarz (GS) mechanism at a more fundamental scale. If the Kac-Moody levels ki assume nontrivial values, the GS cancellation conditions of anomaly mo...

Compactified string theories give rise to discrete symmetries which are essential if they are to provide a realistic low energy theory. We find that in a class of four dimensional string theories these symmetries are constrained by similar conditions to those discrete anomaly cancellation conditions found in the case the discrete symmetry is a residue of a spontaneously broken gauge symmetry. S...

The development of image processing technology is well developed today, such as object recognition in an images. Algorithm to obtain accurate results in object recognition continues to be developed. One example is the handwriting recognition application. This application is usually used to archive records or documents from physical form such as a notebook or a letter in the form of files digita...

We show that more than two generations of quarks and leptons are required to have an anomaly free discrete R symmetry larger than R parity, provided that the supersymmetric standard model can be minimally embedded into a grand unified theory. This connects an explanation for the number of generations with seemingly unrelated problems such as supersymmetry breaking, proton decay, the μ problem, ...