This paper mainly studies the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice. By utilizing the tensor product of matrices and the diagonalization of block circulant matrices, we derive closed-form formulas expressing the Laplacian-energy-like invariants of these lattices. In addition, we obtain explicit asymptotic values of...

in this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. some relations among these invariants will be exhibited. special attention will be paid to monotonicity of them.

Extending work of Hochman, we study the almost-Borel structure, i.e., the nonatomic invariant probability measures, of symbolic systems and surface diffeomorphisms. We first classify Markov shifts and characterize them as strictly universal with respect to a natural family of classes of Borel systems. We then study their continuous factors showing that a low entropy part is almost-Borel isomorp...

We give an invariant classi cation of orbits of the fundamental representations of exceptional groups E7(7) and E6(6) which classify BPS states in string and M theories toroidally compacti ed to d = 4 and 5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classi cation. The cubic and quartic invariants...

A bstract In order to study the pseudo entropy of timelike subregions holographically, previous smooth space-like extremal surface was recently generalized mix and time-like segments area becomes complex value. This paper finds that, if one tries use such kind piecewise surfaces compute entanglement is not unique in general. We then generalize original holographic proposal spacelike pick up a f...