All spaces below are Tychonov. We define the projective πcharacter p πχ(X) of a space X as the supremum of the values πχ(Y ) where Y ranges over all (Tychonov) continuous images of X. Our main result says that every space X has a π-base whose order is ≤ p πχ(X); that is, every point in X is contained in at most p πχ(X)-many members of the π-base. Since p πχ(X) ≤ t(X) for compact X, this is a si...

In 1972 Erich Clar formulated his aromatic π-sextet rule that allows discussing qualitatively the aromatic character of benzenoid species. Now, 40 years later, Clar's aromatic π-sextet rule is still a source of inspiration for many chemists. This simple rule has been validated both experimentally and theoretically. In this review, we select some particular examples to highlight the achievement ...

Let G be a reductive Lie group and O the coadjoint orbit of a hyperbolic element of g. By π is denoted the unitary irreducible representation of G associated with O by the orbit method. We give geometric interpretations in terms of concepts related to O of the constant π(g), for g ∈ Z(G). In the spirit of the orbit method we interpret geometrically the infinitesimal character of the differentia...

Group 8 complexes of π-extended N-heterocyclic ligands present some unexpected results. Benzannulation shifts the lowest energy excitation to blue by impacting character transition rather than affecting frontier orbital energies.

We prove that the Continuum Hypothesis is consistent with the statement that countably compact regular spaces that are hereditarily of countable π–character are either compact or contain an uncountable free sequence. As a corollary we solve a well–known open question by showing that the existence of a compact S–space of size greater than א1 does not follow from the Continuum Hypothesis.

Luminescent silver cluster-based coordination polymers composed of Ag10 or Ag12 cluster cores and multidentate pyridine linkers were synthesized as well luminescent polymorphic silver(I)-based polymers. It was found that these exhibit highly sensitive luminescence quenching effect to nitrobenzene, attributed the π–π stacking interactions between nitrobenzene electron-withdrawing character nitro...

Dispersion and electrostatics are known to stabilize π-π interactions, but the preference for parallel-displaced (PD) and/or twisted (TW) over sandwiched (S) conformations is not well understood. Orbital interactions are generally believed to play little to no role in π-stacking. However, orbital analysis of the dimers of benzene, pyridine, cytosine and several polyaromatic hydrocarbons demonst...

ing this computation, for given k-vectorspace V with non-degenerate alternating form 〈, 〉, put a Lie algebra [2] structure h on V ⊕ k by Lie bracket [v ⊕ z, v′ ⊕ z′] = 0⊕ 〈v, v′〉 In exponential coordinates on H, the exponential map h→ H with H ≈ V ⊕ k is notated exp(v ⊕ z) = v ⊕ z with Lie group structure on H by (v ⊕ z) · (v′ ⊕ z′) = (v + v′)⊕ (z + z′ + 〈v, v ′〉 2 ) (exponential coordinates in...

Let F be a number field with adele ring AF , and let π, π ′ be cuspidal automorphic representations of GL2(AF ), say with the same central character. If the symmetric squares of π and π are isomorphic, we know that π will need to be an abelian, in fact quadratic, twist of π, which amounts to a multiplicity one statement for SL(2) ([Ram1]). It is of interest to ask if the situation is the same f...