This paper describes some recent developments which have made it possible to effectively classify homogeneous systems having infinitely many generalized symmetries, both in the commutative and the noncommutative case. It discusses the program that has to be carried out next to come to an automatic classification mechanism.

Chemical hypergraphs and their associated normalized Laplace operators are generalized studied in the case where each vertex--hyperedge incidence has a real coefficient. We systematically study effect of symmetries hypergraph on spectrum Laplacian.

We review classical properties of harmonic-oscillator coherent states. Then we discuss which of these classical properties are preserved under the group-theoretic generalization of coherent states. We prove that the generalized coherent states of quantum systems with Lie-group symmetries are the unique Bell states, i.e., the pure quantum states preserving the fundamental classical property of s...

In this paper the basic results involved in the application of NOETHER'S theorem relating symmetry groups and conservation laws to the variational problems of homogeneous elastostatics are outlined. General methods and conditions for the existence of variational and generalized symmetries are presented. Applications will be considered in subsequent papers in this series.

We consider the relation of symmetries and subspace controllability for spin1 2 networks with XXZ couplings, subject to perturbation of a single node by a local potential (Z-control). The Hamiltonians for such networks decompose into excitation subspaces. Focusing on the single excitation subspace, it is shown for single-node Z-controls that external symmetries are characterized by eigenstates ...

We describe the construction of CMC surfaces with symmetries in $\mathbb S^3$ and R^3$ using a quadrilateral fundamental tetrahedron tessellation space. The piece is constructed by generalized Weierstrass representation geometric flow on space potentials.

We apply the Painlevé test for integrability of partial differential equations to a system of two coupled Burgers-type equations found by Foursov, which was recently shown by Sergyeyev to possess infinitely many commuting local generalized symmetries without any recursion operator. The Painlevé analysis easily detects that this is a typical C-integrable system in the Calogero sense and rediscov...

Recently we have proposed a new method for establishing correspondences and computing canonical descriptions [6], based on the idea of describing objects by their generalized symmetries, as defined by the object’s vibration or deformation modes. In this paper we extend this method to the problems of object recognition and categorization. Examples of recognizing hand tools and airplanes are shown.

We propose construction of a unique and definite metric (η +), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, CPT invariant and PT(CPT)-norm is indefinite (definite). Here, P and C denote the generalized symmetries : parity and charge-conjugation respectively. The limitations of the other current approaches have been brought out.