On the Kernel of Restriction of Characters*

Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved.     

A Review of Results on Axially Symmetric Navier-Stokes Equations,with Addendum by X. Pan and Q. Zhang

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.     

q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

Generalized logic and the representation of rings

Following the progression towards weaker logics, a number of authors have considered the notion of a sheaf over a quantale or, equivalently, a quantale valued set. In this paper, we use ideas from enriched category theory to motivate the definition of a quantic sheaf. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.     

Theoretical study of the ring‐closing reaction of 5′‐AMP to cAMP and H2O in the gas phase

The ring‐closing reaction of 5′‐adenosine monophosphate (5′‐AMP) to generate cyclic 3′, 5′‐adenosine monophosphate (cAMP) and H₂O was theoretically investigated at the B3LYP/6‐31G**level. It was found that the ring‐closing reaction of 5′‐AMP may proceed in a synchronous way or in a stepwise way. For the latter, the reaction is a multichannel elimination reaction including inner H transfer. The potential energy surface of Path 3 is lowest in all the ring‐closing reaction paths. In addition, H shuttling reaction with the participation of a water molecule to act as a shuttle were also studied at the same level. The calculations indicate that the participation of a water molecule facilitates hydrogen transfer reaction. Our present calculations rationalized all the possible reaction channels. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Useful dual Diels-Alder behavior of 2-azetidinone-tethered aryl imines as azadienophiles or azadienes: a beta-lactam-based stereocontrolled access to optically pure highly functionalized indolizidine systems

Imines derived from 4-oxoazetidine-2-carbaldehydes have been found to be versatile Diels-Alder reagents in that they exhibit two reactivity patterns. 2-Azetidinone-tethered imines undergo diastereoselective reaction with Danishefsky's diene in the presence of different Lewis acids. The effect of the amount of catalyst on the conversion rate as well as on the product ratio has been studied. Under standard reaction conditions, indium(III) chloride and zinc(II) iodide provided the best yields, and indium(III) triflate the highest diastereoselectivity in the Lewis acid promoted aza-Diels-Alder cycloaddition. Treatment of the aforementioned imines with cyclopentadiene, 2,3-dimethyl-1,3-butadiene or 3,4-dihydro-2 H-pyran led to cycloadducts arising from inverse electron-demand condensation involving the beta-lactam-tethered aryl imine as the heterodiene component. In addition, the first methodology for preparing indolizidines from beta-lactams has been developed. This process involves amide bond cleavage of the beta-lactam ring in the aza-Diels-Alder cycloadducts with concomitant cyclization. Full chirality transfer occurs when the reaction is performed with an enantiomerically pure substrate.     

Ring expansion approach for the synthesis of the (3S,4S)-hexahydroazepine core of balanol and ophiocordin

A new and efficient formal total synthesis of (3S,4S)-balanol, a potent protein kinase C inhibitor, was accomplished from tri-O-acetyl-d-glucal. Balanol and ophiocordin consists of a chiral hexahydro azepine-containing fragment and a benzophenone fragment. The azepine core was prepared in chiral form through intramolecular aza Wittig reaction. A triphenylphosphine mediated ring expansion process was employed to form the seven-membered nitrogen heterocycle. The aldehyde equivalent key intermediate was treated with triphenylphosphine to give the azepine core. To demonstrate the applicability of the new route, a synthesis of the balanol is described.     

Drying dissipative patterns of colloidal crystals of silica spheres in organic solvents

Drying dissipative structural patterns formed in the course of drying colloidal crystals of silica spheres (110 nm in diameter) in water, methyl alcohol, ethyl alcohol, 1-propyl alcohol, diethyl ether, and in the mixtures of ethyl alcohol with the other solvents above have been studied on a cover glass. The macroscopic broad rings were formed in the outside edges of the dried film for all the solvents examined. Furthermore, much distinct broad rings appeared in the inner area when the solvents were ethyl alcohol, methyl alcohol, and their mixtures. Profiles of the thickness of the dried films were sensitive to the organic solvents and explained well with changes in the surface tensions, boiling points, and viscosities of the solvents. The macroscopic and microscopic spoke-like crack patterns formed. The drying area (or the drying time) increased (or decreased) as the surface tension of the solvent decreased. However, the absolute values of these drying parameters are determined also by the boiling points of the solvents. Importance of the fundamental properties of the solvents is supported in addition to the characteristics of colloidal particles in the drying dissipative pattern formation.     

Possible conformations and relative stability of the 7-membered ring in 1α,5β,6,7α(H)-and 1β,5α,6,7α(H)-guai-11(13)-en-6,12-olides

Stereoisomers of 1,5,6,7(H)- and 1,5,6,7(H)-guai-11(13)-en-6,12-olides are calculated using molecular mechanics. The possibility of forming various conformations in the 7-membered ring is examined as a function of the fusion to the 5-membered C-ring. The effect of the methyl orientation on the conformation of the 7-membered ring and the relative stability of the conformers are discussed.Insitute of Phytochemistry, Karaganda, fax (3212)-43-37-73. Translated from Khimiya Prirodnykh Soedinenii, No. 1, pp. 54–56, January–February, 2000.