Relations in SO (3 ) Supported by Geodetic Angles

We consider rotations A, B of finite order in SO(3), about axes separated by an angle of restricted type, and attempt to classify the possible group relations between A and B . We show that the relations responsible for the symmetries of Platonic solids have consequences far beyond that simple geometric setting. Received April 9, 1999, and in revised form October 11, 1999.     

The torsion of composite tubes and cylinders

Exact solutions are presented for the Saint-Venant torsion of circular tubes and solid cylinders which are reinforced by cylindrical inclusions of different material equally spaced around a concentric circle. The problems simulate those encountered in matrix rods reinforced by longitudinal fibers, and also in corresponding problems of reinforced concrete. Formulae are obtained for the boundary stress distributions and the torsional rigidities.Stress function formulations are made for the torsion of cylinders having multiply connected composite sections. Two systems of polar coordinates are employed, and use is made both of periodicity and symmetry. Three degenerate cases—the respective torsion of a homogeneous tube, ring of circular rods and tube with eccentric circular holes—are deduced for checking purposes. Several numerical examples are worked out and the results presented in tabular and graphical forms.     

Late‐Stage Functionalization of Biologically Active Heterocycles Through Photoredox Catalysis

The direct C? H functionalization of heterocycles has become an increasingly valuable tool in modern drug discovery. However, the introduction of small alkyl groups, such as methyl, by this method has not been realized in the context of complex molecule synthesis since existing methods rely on the use of strong oxidants and elevated temperatures to generate the requisite radical species. Herein, we report the use of stable organic peroxides activated by visible‐light photoredox catalysis to achieve the direct methyl‐, ethyl‐, and cyclopropylation of a variety of biologically active heterocycles. The simple protocol, mild reaction conditions, and unique tolerability of this method make it an important tool for drug discovery.     

A Series of Potent CREBBP Bromodomain Ligands Reveals an Induced‐Fit Pocket Stabilized by a Cation–π Interaction

The benzoxazinone and dihydroquinoxalinone fragments were employed as novel acetyl lysine mimics in the development of CREBBP bromodomain ligands. While the benzoxazinone series showed low affinity for the CREBBP bromodomain, expansion of the dihydroquinoxalinone series resulted in the first potent inhibitors of a bromodomain outside the BET family. Structural and computational studies reveal that an internal hydrogen bond stabilizes the protein‐bound conformation of the dihydroquinoxalinone series. The side chain of this series binds in an induced‐fit pocket forming a cation–π interaction with R1173 of CREBBP. The most potent compound inhibits binding of CREBBP to chromatin in U2OS cells.     

Cyclopentadienyl mesityl complexes of chromium(II) and chromium(III)

Chromium(III) mesityl complexes were synthesized by protonolysis of chromocene with 1,3-diisopropylimidazolium chloride or DBU hydrochloride, salt metathesis with MesMgBr, and single electron oxidation with iodine.     

Synthesis and Diels-Alder Reactions of a New Kind of Chiral Dienophiles: Cyclic Vinyl-p-tolylsulfilimines

A new kind of chiral dienophiles, cyclic vinyl-p-tolylsulfilimines (2a and 2b), were obtained from the corresponding (Z)-sulfinylacrylonitriles with HBF(4) and methanol. The asymmetric Diels-Alder reaction of optically pure 2a with cyclopentadiene under mild thermal or catalyzed conditions afforded only the endo-4a adduct with complete endo and pi-facial selectivities. The ability of the sulfilimine moiety to enhance the dienophilic reactivity of the double bond is similar to that of the sulfinyl group.     

Reagent Selector: using Synthon Analysis to visualize reagent properties and assist in combinatorial library design

Reagent Selector is an intranet-based tool that aids in the selection of reagents for use in combinatorial library construction. The user selects an appropriate reagent group as a query, for example, primary amines, and further refines it on the basis of various physicochemical properties, resulting in a list of potential reagents. The results of this selection process are, in turn, converted into synthons: the fragments or R-groups that are to be incorporated into the combinatorial library. The Synthon Analysis interface graphically depicts the chemical properties for each synthon as a function of the topological bond distance from the scaffold attachment point. Displayed in this fashion, the user is able to visualize the property space for the universe of synthons as well as that of the synthons selected. Ultimately, the reagent list that embodies the selected synthons is made available to the user for reagent procurement. Application of the approach to a sample reagent list for a G-protein coupled receptor targeted library is described.     

Indefinite integrals of special functions from inhomogeneous differential equations

A method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions satisfying second-order homogeneous equations. Sample results are derived for Bessel functions, parabolic cylinder functions, Gauss hypergeometric functions and the six classical orthogonal polynomials. For the orthogonal polynomials the method gives indefinite integrals which reduce to the usual orthogonality conditions on the usual orthogonality intervals. These indefinite integrals for the orthogonal polynomials appear to be new. All results have been checked with Mathematica.