On the Hadwiger's conjecture for graph products

The Hadwiger number η(G) of a graph G is the largest integer h such that the complete graph on h nodes K_h is a minor of G. Equivalently, η(G) is the largest integer such that any graph on at most η(G) nodes is a minor of G. The Hadwiger's conjecture states that for any graph G, η(G)?χ(G), where χ(G) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists a unique prime factorization with respect to Cartesian graph products. If the unique prime factorization of G is given as G₁□G₂□?□G_k, where each G_i is prime, then we say that the product dimension of G is k. Such a factorization can be computed efficiently.In this paper, we study the Hadwiger's conjecture for graphs in terms of their prime factorization. We show that the Hadwiger's conjecture is true for a graph G if the product dimension of G is at least . In fact, it is enough for G to have a connected graph M as a minor whose product dimension is at least , for G to satisfy the Hadwiger's conjecture. We show also that if a graph G is isomorphic to F^d for some F, then η(G)?χ(G)^{⌊(d-1)/2⌋}, and thus G satisfies the Hadwiger's conjecture when d?3. For sufficiently large d, our lower bound is exponentially higher than what is implied by the Hadwiger's conjecture.Our approach also yields (almost) sharp lower bounds for the Hadwiger number of well-known graph products like d-dimensional hypercubes, Hamming graphs and the d-dimensional grids. In particular, we show that for the d-dimensional hypercube H_d, . We also derive similar bounds for G^d for almost all G with n nodes and at least edges.     

Significance of Stefan blowing effect on flow and heat transfer of Casson nanofluid over a moving thin needle

The current mathematical model explains the influence of non-linear thermal radiation on the Casson liquid flow over a moving thin needle by considering Buongiorno's nanofluid model.The influences of Stefan blowing, Dufour and Soret effects are also considered in the model. The equations which represent the described flow pattern are reduced to ordinary differential equations(ODEs) by using apt similarity transformations and then they are numerically solved with Runge–Kutta-Fehlberg's fourth fifth-order method(RKF-45) with shooting process. The impacts of pertinent parameters on thermal, mass and velocity curves are deliberated graphically.Skin friction, rate of heat and mass transfer are also discussed graphically. Results reveal that, the increase in values of Brownian motion, thermophoresis, Dufour number, heating and radiative parameters improves the heat transfer. The increasing values of the Schmidt number deteriorates the mass transfer but a converse trend is seen for increasing values of the Soret number. Finally,the escalating values of the radiative parameter decays the rate of heat transfer.     

Relativistic Effects on Stimulated Brillouin Scattering of Self-Focused q-Gaussian Laser Beams in Plasmas with Axial Density Ramp

Journal of Russian Laser Research - We investigate the phenomenon of stimulated Brillouin scattering (SBS) of q-Gaussian laser beams nonlinearly-interacting with underdense plasmas. When an intense...     

Intramolecular Aminocyanation of Alkenes by NCN Bond Cleavage

A metal‐free, Lewis acid promoted intramolecular aminocyanation of alkenes was developed. B(C₆F₅)₃ activates N‐sulfonyl cyanamides, thus leading to a formal cleavage of the N? CN bonds in conjunction with vicinal addition of sulfonamide and nitrile groups across an alkene. This method enables atom‐economical access to indolines and tetrahydroquinolines in excellent yields, and provides a complementary strategy for regioselective alkene difunctionalizations with sulfonamide and nitrile groups. Labeling experiments with ¹³C suggest a fully intramolecular cyclization pattern due to the lack of label scrambling in double crossover experiments. Catalysis with Lewis acid is realized and the reaction can be conducted under air.     

Controlling Aqueous Self‐Assembly Mechanisms by Hydrophobic Interactions

We report an innovative template‐assisted synthetic protocol for the selective functionalization of terminal triple bonds in oligophenyleneethynylenes (OPE) by pre‐organization in aqueous solution. By this approach, three new OPE‐based bolaamphiphiles substituted with hydrophilic poly(2‐ethyl‐2‐oxazoline) (PEtOx) chains of different length have been synthesized. The chain length was observed to strongly influence the aqueous supramolecular polymerization: bolaamphiphiles with longer hydrophilic chains aggregate into spherical nanoparticles in a stepwise fashion, whereas 2D anisotropic platelets are formed cooperatively if shorter PEtOx chains are used. Our results demonstrate that hydrophobic interactions can be strong enough to trigger cooperative effects in aqueous self‐assembly processes.     

Intramolecular Aminocyanation of Alkenes by N?CN Bond Cleavage

A metal‐free, Lewis acid promoted intramolecular aminocyanation of alkenes was developed. B(C₆F₅)₃ activates N‐sulfonyl cyanamides, thus leading to a formal cleavage of the N CN bonds in conjunction with vicinal addition of sulfonamide and nitrile groups across an alkene. This method enables atom‐economical access to indolines and tetrahydroquinolines in excellent yields, and provides a complementary strategy for regioselective alkene difunctionalizations with sulfonamide and nitrile groups. Labeling experiments with ¹³C suggest a fully intramolecular cyclization pattern due to the lack of label scrambling in double crossover experiments. Catalysis with Lewis acid is realized and the reaction can be conducted under air.     

Synthesis of trans-1,8,12,13-tetraoxadispiro[4.1.4.2]tridecanes—a new class of peroxides

The synthesis of trans-1,8,12,13-tetraoxadispiro[4.1.4.2]tridecanes, a new class of peroxide skeletons using Birch reduction of aromatic compounds followed by ozonolysis and acid catalysed cyclisation is described.     

Palladium‐Catalyzed Annulation of Diarylamines with Olefins through C?H Activation: Direct Access to N‐Arylindoles

A palladium‐catalyzed dehydrogenative coupling between diarylamines and olefins has been discovered for the synthesis of substituted indoles. This intermolecular annulation approach incorporates readily available olefins for the first time and obviates the need of any additional directing group. An ortho palladation, olefin coordination, and β‐migratory insertion sequence has been proposed for the generation of olefinated intermediate, which is found to produce the expected indole moiety.     

Synthetic Methodology,Spectral Elucidation,and Antioxidative Properties of Benzothianes and Their Sulfones

This article reflects the synthetic strategies and spectral investigation of 4H-1,4-benzothiazines. 4H-1,4-benzothiazines have been prepared by the condensation and oxidative cyclization of substituted 2-aminobenzenethiol with β-diketones/β-ketoesters in dimethyl sulfoxide and with the oxidation of 4H-1,4-benzothiazines by 30% hydrogen peroxide in glacial acetic acid, which results in the formation of 4H-1,4-benzothiazine sulfones. The compounds were evaluated for their antioxidative properties through in vitro and in vivo studies in Swiss albino mice. The structural assignments of compounds were made on the basis of spectroscopic data and elemental analysis.     

Synthesis of Biological Active Chromene Benzothiadiazole Derivatives

In this article, we describe the synthesis of various substituted Chromene-[1, 3]thiazolo [2, 1, 3] benzothiadiazoles 4a-l by conventional (Method A) and microwave-assisted (Method B) which achieved reductions in reaction times, higher yields, and a cleaner reaction than the conventional method. A comparative study of these two methods has been discussed.