We study the effect of lifting the degeneracy of vortex modes with a parity time (PT) symmetric defect, using discrete vortices in a circular array of nonlinear waveguides as an example. When the defect is introduced, the degenerate linear vortex modes spontaneously break PT symmetry and acquire complex eigenvalues, but nonlinear propagating modes with real propagation constants can still exist...

The forward and backward scattering off linear systems with discrete rotational symmetries R(z)(2π/n) with n ≥ 3 are shown to be restricted by symmetry reasons. Along the symmetry axis, forward scattering can only be helicity preserving and backward scattering can only be helicity flipping. These restrictions do not exist for n < 3. If, in addition to the n ≥ 3 discrete rotational symmetry, the...

We present two important developments for understanding nk Boolean landscapes: First, a discrete analogue to the continuous phase portrait for analyzing Boolean landscapes and, second, an evolutionary alternative to mutation and recombination. Applications include evolution in biological systems, social networks, electrical flow on power grids, business organizations. The Curie Principle states...

We present "twin Higgs models," simple realizations of the Higgs boson as a pseudo Goldstone boson that protect the weak scale from radiative corrections up to scales of order 5-10 TeV. In the ultraviolet these theories have a discrete symmetry which interchanges each standard model particle with a corresponding particle which transforms under a twin or a mirror standard model gauge group. In a...

A systematic procedure is developed for constructing fermion systems in discrete space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we derive constraints for the number of particles. In the physically interesting case of many particles and even more spacetime points, this result shows that the permutation symmetry of discre...

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate an...

An effective theory is proposed, combining the standard gauge group SU(3)C ⊗ SU(2)L ⊗ U(1)Y with a horizontal discrete symmetry. By assigning appropriate charges under this discrete symmetry to the various fermion fields and to (at least) two Higgs doublets, the broad spread of the fermion mass and mixing angle spectrum can be explained as a result of suppressed, non-renormalisable terms. A par...

We find that massless Higgs doublets at the GUT scale can be the natural result of a discrete symmetry. Such a mechanism does not require elaborate fine tuning or complicated particle content. The same discrete symmetry will also protect against proton decay and flavor changing neutral currents. However, this mechanism always predicts non-minimal standard models. An explicit example of how this...

If gaugino condensations occur in the early universe, domain walls are produced as a result of the spontaneous breaking of a discrete R symmetry. Those domain walls eventually annihilate with one another, producing the gravitational waves. We show that the gravitational waves can be a probe for measuring the gravitino mass, if the constant term in the superpotential is the relevant source of th...

We go beyond the Standard Model guided by presymmetry, the discrete electroweak quark-lepton symmetry hidden by topological effects which explain quark fractional charges as in condense matter physics. Partners of the particles of the Standard Model and the discrete symmetry associated with this partnership appear as manifestations of a residual presymmetry and its extension from matter to forc...