in this work, using eigenvalues and eigenvectors of unitary cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allowing pst occurring in its unitary cayley graph. moreover, we have some developments when $r$ is a product of local rings.

The aim of the present paper is to provide a unitary exposition on possible representations for positively homogeneous functions in the dual space. Starting from the classical Minkowski–Hörmander duality result of convex analysis, we develop a scheme in order to associate to a positively homogeneous function families of sublinear functionals representing it. Such dual description allows us to c...

‎We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups‎. ‎We prove Arthur's conjecture‎, ‎the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group‎, ‎for quasi-split special unitary groups and their inner forms‎. ‎Furthermore‎, ‎we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms‎. ‎This w...

After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a...

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit highly reducible unitary positive energy representations in a Fock space. The multiplicity of their irreducible componen...

Quantum chaos in hermitian systems concerns the sensitivity of long-time dynamical evolution to initial conditions. The skin effect discovered recently non-hermitian reveals spatial boundary condition even deeply bulk. In this letter, we show that these two seemingly different phenomena can be unified through space-time duality. intuition is duality maps unitary dynamics non-unitary and exchang...