Weyl metals host relativistic chiral quasiparticles, which display quantum anomalies in the presence of external electromagnetic fields. Here, we study the manifestations of chiral anomalies in the longitudinal and planar magnetotransport coefficients of Weyl metals, in the presence of a quantizing magnetic field. We present a general framework for calculating all the transport coefficients in the regime where multiple Landau levels are occupied. We explicitly show that all the longitudinal and planar transport coefficients show quantum oscillations which are periodic in 1/B. Our calculations recover the quadratic-B dependence in the semiclassical regime and predict a linear-B dependence in the ultraquantum limit for all the transport coefficients.

Coloring the faces of a two-dimensional square lattice with k distinct colors such that no two adjacent faces have the same color is considered by establishing connection between the kcoloring problem and a generalized vertex model. Associating the colors with k distinct species of particles with an infinite repulsive force between nearest neighbors of the same type and zero chemical potential µ associated with each species, the number of ways [W (k)] N for large N is related to the entropy of the hard square lattice gas at close packing of the lattice, where N is the number of lattice sites. We discuss the evaluation of W (k) using transfer matrix method with non-periodic boundary conditions imposed along at least one direction, and, show the characteristic Toeplitz block structure of the transfer matrix. Using this result, we present some analytical calculations for non-periodic models that remain finite in one dimension. The case k = 3 is found to approach the exact result obtained by Lieb for the residual entropy of ice with periodic boundary conditions. Finally, we show, by explicit calculation of the contribution of subgraphs and the series expansion of W (k), that the generalized Pauling type estimate (which is based on mean-field approximation) dominates at large values of k. We thus also provide an alternative series expansion for the chromatic polynomial of a regular square graph.

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Biodiesel is a unique diverse fuel source that aims to enhance the worth of fossil fuels and the endurance and hygiene of diesel engines. This cuts requirement on overseas fuels and decreases greenhouse gas productions because of closed carbon chain. The increased nitrogen oxide emissions and their irreconcilability with cold weather environments and the many advantages of biodiesel such as regular fuel filters, fuel reservoirs and fuel fuels for engine components are outweighed. The growth and the inclusion of nanoparticles for fuel additives have another potential to overcome the shortcomings. Cashews are high in tropical biomass surplus used to generate energy such as cerium oxide. Cobalt oxide and aluminums oxide can increase the property of non-additive CNSL and reduce deficiencies.