Alexander duals of multipermutohedron ideals

An Alexander dual of a multipermutohedron ideal has many combinatorial properties. The standard monomials of an Artinian quotient of such a dual correspond bijectively to some λ-parking functions, and many interesting properties of these Artinian quotients are obtained by Postnikov and Shapiro (Trans. Am. Math. Soc. 356 (2004) 3109–3142). Using the multigraded Hilbert series of an Artinian quotient of an Alexander dual of multipermutohedron ideals, we obtained a simple proof of Steck determinant formula for enumeration of λ-parking functions. A combinatorial formula for all the multigraded Betti numbers of an Alexander dual of multipermutohedron ideals are also obtained.     

A new approach to A/B ring analogue of eleutherobin and sarcodictyns through a sequence of highly diastereofaceselective Diels-Alder reaction and ring opening-ring closing metathesis (RO-RCM)

An approach to the construction of A/B ring analogue of antitumour compounds eleutherobin and sarcodictyns is described. The key steps involve a highly diastereofaceselective Diels-Alder reaction of a dienophile containing a furanosugar moiety with cyclopentadiene and ring opening-ring closing metathesis of the resulting adduct.     

Solutions of coupled KdV-type equations

We consider a family of coupled (Ito type) KdV equations in 1+1 dimensions and use Hlavatý's technique to obtain a class of explicit wave-type solutions.     

Transition metal and base free coupling of N-tosylhydrazones with 1,3-dicarbonyl compound

N-tosylhydrazones derived from a wide variety of aryl, alkyl and heteroaryl aldehydes undergo smooth coupling with 5,5-dimethylcyclohexane-1,3-dione under transition metal and base free conditions to generate tetraketo compounds in high yields. In presence of a base, the coupling reaction generates β-keto enol ether as the major product in a polar aprotic solvent. N-Tosylhydrazone can also be converted to xanthenedione in high yields in one-pot operation under mild acidic conditions.     

A simple derivation of the three magnon bound state equation

A simple derivation of the equation for determining the bound states of three magnons in the Heisenberg linear chain with longitudinal anisotropy is given. The present method utilizes nothing more than the Schrödinger equation and Faddeev’s three body equations, and avoids the introduction of the ideal spin wave Hilbert space.     

Observation of photochemical C-N bond cleavage in CH3N3: a new photochemical route to cyclic N3

We report VUV-photoionization based photofragmentation-translational spectroscopy data, providing a comprehensive study of the collision free photochemistry of methyl azide (CH3N3) at 193 nm. We report the first observation of the production of methyl and the N3 radical and derive the translational energy release distribution of this reaction. The most probable translation energy is only 8%, and the maximum translational energy is only 60% of the available energy, taking CH3 + linear N3 as the zero of energy. However, the maximum translational energy release is quantitatively consistent with production of the higher energy isomer cyclic N3. Threshold photoionization of the N3 fragment using tunable synchrotron radiation shows results consistent with theoretical predictions of the cyclic N3 ionization potential. The secondary dissociation of N3 --> N(2D) + N2 is also observed and its translational energy release is derived. This distribution peaks at approximately 6 and extends to 11 kcal/mol as would be expected from the size of the exit channel barrier for spin-allowed dissociation of cyclic N3 (7 kcal/mol) and, furthermore, inconsistent with the barrier height of the spin-allowed dissociation of linear N3 (3 kcal/mol). A large fraction (approximately 45%) of the N3 does not dissociate on the microsecond time scale of the experiment suggesting methyl azide may be the most attractive photochemical precursor of cyclic N3 yet found.     

A new iterative method for solving split feasibility problem

In this paper, we construct a new iterative algorithm and show that the newly introduced iterative algorithm converges faster than a number of existing iterative algorithms for contractive-like mappings. We present a numerical example followed by graphs to validate our claim. We prove strong and weak convergence results for approximating fixed points of generalized $\alpha$-nonexpansive mappings. Again we reconfirm our results by an example and table. Further, we utilize our proposed algorithm to solve split feasibility problem.