Three-term polynomial progressions in subsets of finite fields

We prove that the group of diffeomorphisms of the interval [0, 1] contains surface groups whose action on (0, 1) has no global fix point and such that only countably many points of the interval (0, 1) have non-trivial stabiliser.     

Simulation of collisions of two droplets containing two different liquids using incompressible smoothed particle hydrodynamics method

Theoretical and Computational Fluid Dynamics - The present study aims to simulate a collision of two droplets containing immiscible liquids by employing a three- dimensional incompressible smoothed...     

Unsymmetrical Pyrene‐Fused Phthalocyanine Derivatives: Synthesis,Structure, and Properties

Novel pyrene‐fused unsymmetrical phthalocyanine derivatives 2,3,9,10,16,17‐hexakis(2,6‐dimethylphenoxy)‐22,25‐diaza(2,7‐di‐tert‐butylpyrene)[4,5]phthalocyaninato zinc complex Zn[Pc(Pz‐pyrene)(OC₈H₉)₆] ( 1 ) and 2,3,9,10‐tra(2,6‐dimethylphenoxy)‐15,18,22,25‐traza(2,7‐di‐tert‐butylpyrene)[4,5]phthalocyaninato zinc compound Zn[Pc(Pz‐pyrene)₂(OC₈H₉)₄] ( 2 ) were isolated for the first time. These unsymmetrical pyrene‐fused phthalocyanine derivatives have been characterized by a wide range of spectroscopic and electrochemical methods. In particular, the pyrene‐fused phthalocyanine structure was unambiguously revealed on the basis of single crystal X‐ray diffraction analysis of 1 , representing the first structurally characterized phthalocyanine derivative fused with an aromatic moiety larger than benzene.     

Formation of Azulene-Embedded Nanographene: Naphthalene to Azulene Rearrangement During the Scholl Reaction

Incorporation of a non-hexagonal ring into a nanographene framework can lead to new electronic properties. During the attempted synthesis of naphthalene-bridged double [6]helicene and heptagon-containing nanographene by the Scholl reaction, an unexpected azulene-embedded nanographene and its triflyloxylated product were obtained, as confirmed by X-ray crystallographic analysis and 2D NMR spectroscopy. A 5/7/7/5 ring-fused substructure containing two formal azulene units is formed, but only one of them shows an azulene-like electronic structure. The formation of this unique structure is explained by arenium ion mediated 1,2-phenyl migration and a naphthalene to azulene rearrangement reaction according to an in-silico study. This report represents the first experimental example of the thermodynamically unfavorable naphthalene to azulene rearrangement and may lead to new azulene-based molecular materials.