Minimal Representations of a Face of a Convex Polyhedron and Some Applications

In this paper, we propose a method for determining all minimal representations of a face of a polyhedron defined by a system of linear inequalities. Main difficulties for determining prime and minimal representations of a face are that the deletion of one redundant constraint can change the redundancy of other constraints and the number of descriptor index pairs for the face can be huge. To reduce computational efforts in finding all minimal representations of a face, we prove and use properties that deleting strongly redundant constraints does not change the redundancy of other constraints and all minimal representations of a face can be found only in the set of all prime representations of the face corresponding to the maximal descriptor index set for it. The proposed method is based on a top-down search strategy, is easy to implement, and has many computational advantages. Based on minimal representations of a face, a reduction of degeneracy degrees of the face and ideas to improve some known methods for finding all maximal efficient faces in multiple objective linear programming are presented. Numerical examples are given to illustrate the method.

The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

Neuronal Development of Hearing and Language: Cochlear Implants and Critical Periods

The modern cochlear implant (CI) is the most successful neural prosthesis developed to date. CIs provide hearing to the profoundly hearing impaired and allow the acquisition of spoken language in children born deaf. Results from studies enabled by the CI have provided new insights into ( a) minimal representations at the periphery for speech reception, ( b) brain mechanisms for decoding speech presented in quiet and in acoustically adverse conditions, ( c) the developmental neuroscience of language and hearing, and ( d) the mechanisms and time courses of intramodal and cross-modal plasticity. Additionally, the results have underscored the interconnectedness of brain functions and the importance of top-down processes in perception and learning. The findings are described in this review with emphasis on the developing brain and the acquisition of hearing and spoken language.