A family of non-Hermitian but ${\cal PT}-$symmetric $2J$ by toy-model tridiagonal-matrix Hamiltonians $H^{(2J)}=H^{(2J)}(t)$ with $J=K+M=1,2,\ldots$ and $t<J^2$ is studied, for which a real $2K$ tridiagonal-submatrix component $C(t)$ the Hamiltonian assumed coupled to its other two complex Hermitian $M$ components $A(t)$ $B(t)$. By construction, (i) all submatrices get decoupled at $t=t_M=M\,(2...

A family of sets is said to be symmetric if its automorphism group is transitive, and intersecting if any two sets in the family have nonempty intersection. Our purpose here is to study the following question: for n, k ∈ N with k ≤ n/2, how large can a symmetric intersecting family of k-element subsets of {1, 2, . . . , n} be? As a first step towards a complete answer, we prove that such a fami...

We present deterministic approximation algorithms for the multi-criteria maximum traveling salesman problem (Max-TSP). Our algorithms are faster and simpler than the existing randomized algorithms. We devise algorithms for the symmetric and asymmetric multi-criteria Max-TSP that achieve ratios of 1/2k−ε and 1/(4k−2)−ε, respectively, where k is the number of objective functions. For two objectiv...

C(1)-symmetric isoquinoline-based chiral diaminocarbene ligands (MIQ) have been developed to block three quadrants of the metal coordination sphere, complementing C(2)-symmetric biisoquinoline-based ligands (BIQ). MIQ-Cu complexes catalyzed conjugate borylation of various α,β-unsaturated amides in good yields (82-99%) and enantioselectivities (75-87% ee).