How regular can the boundary of a quadratic Siegel disk be?

In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are but not Jordan curves.

     

Some considerations of fluid interfaces in two dimensions

Basic assumptions of the capillary wave theory of fluid interfaces are examined critically as a function of space dimensionalityd. When the predictions of capillary wave theory are compared with those of the nonclassical Maxwell-van der Waals theory, agreement is found ind=3 and 4, but strong disagreement occurs ind=2. It is shown that the total effective mass density obtained from the Hamiltonian describing the collective capillary wave excitations has a logarithmic divergence ind=2. This result suggests the possibility of anomalous behavior for fluid interfaces ind=2.     

2′‐Ethynyl‐DNA: Synthesis and Pairing Properties

2‐Ethynyl‐DNA was developed as a potential DNA‐selective oligonucleotide analog. The synthesis of 2′‐arabino‐ethynyl‐modified nucleosides was achieved starting from properly protected 2′‐ketonucleosides by addition of lithium (trimethylsilyl)acetylide followed by reduction of the tertiary alcohol. After a series of protecting‐group manipulations, phosphoramidite building blocks suitable for solid‐phase synthesis were obtained. The synthesis of oligonucleotides from these building blocks was successful when a fast deprotection scheme was used. The pairing properties of 2′‐arabino‐ethynyl‐modified oligonucleotides can be summarized as follows: 1) The 2′‐arabino‐ethynyl modification of pyrimidine nucleosides leads to a strong destabilization in duplexes with DNA as well as with RNA. The likely reason is that the ethynyl group sterically influences the torsional preferences around the glycosidic bond leading to a conformation not suitable for duplex formation. 2) If the modification is introduced in purine nucleosides, no such influence is observed. The pairing properties are not or only slightly changed, and, in some cases (deoxyadenosine homo‐polymers), the desired stabilization of the pairing with a DNA complementary strand and destabilization with an RNA complement is observed. 3) In oligonucleotides of alternating deoxycytidine‐deoxyguanosine sequence, the incorporation of 2′‐arabino‐ethynyl deoxyguanosine surprisingly leads to the formation of a left‐handed double helix, irrespective of salt concentration. The rationalization for this behavior is that the ethynyl group locks such duplexes in a left‐handed conformation through steric blockade.     

Intercepting Wacker intermediates with arenes: C-H functionalization and dearomatization

An intramolecular cyclization cascade reaction has been developed utilizing a high valent palladium intermediate that generates a carbon-carbon and carbon-oxygen bond in a single transformation. This method provides rapid access to highly functionalized tricyclic scaffolds, including spirocyclic cyclohexadienones. Good yields and mild conditions are reported with high tolerance toward oxygen and water.     

On the Bieberbach conjecture and holomorphic dynamics

In this note we prove that when is a polynomial of degree with connected Julia set and when belongs to the filled-in Julia set , then . We also show that equality is achieved if and only if is a segment of which one extremity is . In that case, is conjugate to a Tchebycheff polynomial or its opposite. The main tool in our proof is the Bieberbach conjecture proved by de Branges in 1984.

     

Limits of Degenerate Parabolic Quadratic Rational Maps

We investigate the closure in moduli space of the set of quadratic rational maps which possess a degenerate parabolic fixed point.     

New synthesis and structure of a polar manganese coordination polymer

A new synthesis of [N,N-ethylenebis(salicylaldiminiato)((4-pyridylthio)acetate)] manganese(III) (Mn(Salen) polymer) has been developed, and the structure of a methanol adduct of the compound has been determined at 293 K. The compound crystallizes as an infinite chain polymer, where the Mn metal center is bridged by the (4-pyridylthio)acetate ligand. The Mn center is octahedrally coordinated with the equatorial positions being occupied by the Salen ligand. The axial positions are occupied by a pyridyl nitrogen and a carboxylate oxygen. The iteration of this bonding pattern gives the polymer a head-to-tail backbone. The major structural difference between this compound and the one previously reported occurs on the nonmetalated oxygen of the carboxylate functionality (W. Chaing, D.M. Ho, D. Van Engen, M.E. Thompson, Inorg. Chem. 1993, 32, 2886; W. Chaing, Doctoral Thesis, Princeton University, Princeton, 1994). In the original structure, H₂O is hydrogen bonded to the nonmetalated oxygen; however, in the structure reported here methanol is intercalated between chains without any hydrogen bonding. This indicates that this polymer can be viewed as host–guest system and that solvent plays no role in the formation of the structure.