A Randomized Algorithm for 3-<Emphasis Type="Bold">SAT</Emphasis>

In this work we propose and analyze a simple randomized algorithm for 3-SAT (i.e. an algorithm to find a satisfiable assignment for a Boolean formula in conjunctive normal form (CNF) having at most 3 literals in every clause). Given a k-CNF formula \({\phi}\) on n variables, and \({\alpha \in \{0,1\}^n}\) that satisfies \({\phi}\), a clause of \({\phi}\) is critical if exactly one literal of that clause is satisfied under assignment α. Paturi et al. (Chicago J. Theor. Comput. Sci. 115, 1999) proposed a simple randomized algorithm (PPZ) for k-SAT for which success probability increases with the number of critical clauses (with respect to a fixed satisfiable solution of the input formula). Here, we first describe another simple randomized algorithm DEL which performs better if the number of critical clauses in input formula are less (with respect to a fixed satisfiable solution of the input formula). Subsequently, we combine these two simple algorithms such that the success probability of the combined algorithm is maximum of the success probabilities of PPZ and DEL on every input instance. We show that when the average number of clauses for a variable that appear as unique true literals in one or more critical clauses in \({\phi}\) is between 1 and 2/(3 · log (3/2)), combined algorithm performs better than the PPZ algorithm.     

Palladium-mediated reductive Mizoroki-Heck cyclization strategy for the regioselective formation of dibenzoazocinone framework

The synthesis of dibenzoazocine framework through palladium-mediated reductive Mizoroki-Heck cyclization has been described. The procedure is simple, straightforward, and regioselective.     

A remarkably bent diiron(III)-μ-hydroxo bisporphyrin: unusual stabilization of two spin states of iron in a single molecular framework

A novel diiron(III) bisporphyrin bridged by a hydroxo group between two cofacial Fe centers is reported. X-ray structural characterization revealed the remarkably bent μ-hydroxobis[Fe(III) porphyrin] with the smallest known Fe-O(H)-Fe angle [142.5(2)°] reported to date in an iron porphyrin. The close approach of the two rings in the molecule results in an unequal core deformation, and as a result, the geometrical parameters (such as the Fe-N(p), Fe-O and Fe···Ct(p) distances) are all different for the two Fe(III) centers, leading for the first time to a natural way of stabilizing two different spins of iron in a single molecular framework with complete retention of their own spectroscopic identities in both the solid state and solution. The strong antiferromagnetic coupling between the two Fe(III) centers in the μ-oxo dimer (-J = 126.6 cm(-1)) is attenuated to only 4.5 cm(-1) simply by protonation to give the μ-hydroxo complex.     

Low temperature synthesis of zinc ferrite nanoparticles

Zinc ferrite (ZnFe₂O₄) nanocrystalline powder materials with various particle sizes were prepared by a unique solid-state combustion method. Phase purity of ZnFe₂O₄ was confirmed by X-ray diffraction studies. High resolution transmission electron microscopic analysis and selected area diffraction pattern also confirmed the correct crystalline phase formation. Particle size was determined from both the transmission electron microscopic images and also from the XRD peak broadening analysis. Oxidation states of different elements present in ZnFe₂O₄ were determined by X-ray photoelectron spectroscopy. Frequency dependent dielectric constant and a.c. conductivity were measured as a function of particle size and both of them were found to decrease with decreasing particle size. These studies indicated that good quality zinc ferrite nanocrystalline powdered materials can be synthesized at low temperature.     

Rydberg states of the PO molecule

The absorption spectrum of the PO molecule in the region 2100 to 1550 Å has been studied in detail at high resolution. From a rotational analysis of the spectrum, a number of new electronic states have been discovered. Some of these electronic states have been assigned to Rydberg series of the various nl complexes of PO. An upper limit of the ground state dissociation energy has been lowered to 49090 cm^?1 (X²Π, T₀ = 0). Quantum defects are calculated and the first ionization potential of PO is improved to 67 532 cm^?1.     

One step synthesis of C-dots by microwave mediated caramelization of poly(ethylene glycol)

A rapid, simple and one step microwave mediated method for synthesizing C-dots using poly(ethylene glycol) (PEG) as a precursor and passivating agent is reported. The C-dots possessed low cytotoxicity, were amenable to separation by electrophoresis, photostable and entered cancer cells, making them suitable candidates for bioimaging and biolabelling.     

Matrix factorization method for the Hamiltonian structure of integrable systems

We demonstrate that the process of matrix factorization provides a systematic mathematical method to investigate the Hamiltonian structure of non-linear evolution equations characterized by hereditary operators with Nijenhuis property.