Non-degenerate colorings in the Brook's Theorem

Let $c\geq 2$ and $p\geq c$ be two integers. We will call a proper coloring of the graph $G$ a \textit{$(c,p)$-nondegenerate}, if for any vertex of $G$ with degree at least $p$ there are at least $c$ vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem. Let $D\geq 3$ and $G$ be a graph without cliques on $D+1$ vertices and the degree of any vertex in this graph is not greater than $D$. Then for every integer $c\geq 2$ there is a proper $(c,p)$-nondegenerate vertex $D$-coloring of $G$, where $p=(c^3+8c^2+19c+6)(c+1).$ During the primary proof, some interesting corollaries are derived.Comment: 18 pages, 10 figure

Quantum dense coding in multiparticle entangled states via local measurements

In this paper, we study quantum dense coding between two arbitrarily fixed particles in a (N+2)-particle maximally-entangled states through introducing an auxiliary qubit and carrying out local measurements. It is shown that the transmitted classical information amount through such an entangled quantum channel usually is less than two classical bits. However, the information amount may reach two classical bits of information, and the classical information capacity is independent of the number of the entangled particles in the initial entangled state under certain conditions. The results offer deeper insights to quantum dense coding via quantum channels of multi-particle entangled states.Comment: 3 pages, no figur

Dynamical behavior of interacting dark energy in loop quantum cosmology

The dynamical behaviors of interacting dark energy in loop quantum cosmology are discussed in this paper. Based on defining three dimensionless variables, we simplify the equations of the fixed points. The fixed points for interacting dark energy can be determined by the Friedmann equation coupled with the dynamical equations {in Einstein cosmology}. But in loop quantum cosmology, besides the Friedmann equation, the conversation equation also give a constrain on the fixed points. The difference of stability properties for the fixed points in loop quantum cosmology and the ones in Einstein cosmology also have been discussed.Comment: 7 pages, 5 figure

Enzymes and genes involved in the betalain biosynthesis in higher plants

Betalains, a class of water-soluble nitrogen-containing pigments, replace anthocyanins and serve the analogous functions in 13 families of the order, caryophyllales. They modulate the attractive appearance of plants and protect them against destructive oxidative damage. Their antioxidant roles, radicalscavenging properties in human health and their potential uses in food and pharmaceutical industries have made significant progress achieved in the detection, purification, quantification, structure elucidation of betalains, and in particular in the understanding of biosynthetic pathways of the pigments,the enzymes and their genes involved in the pathways. In this paper, major progress in betalain biosynthesis and the enzymes and genes involved in the biosynthetic pathways in higher plant are reviewed, and the perspectives discussed

Magnetic properties of a novel Pr Fe Ti phase

In a systematic study of the (Pr1−xTix)Fe5 alloy series, the (Pr0.65Ti0.35)Fe5 alloy has been found to have a dominant phase with either the rhombohedral Th2Zn17 structure or the newly discovered Nd2(Fe,Ti)19 (S. J. Collocott, R. K. Day, J. B. Dunlop, and R. L. Davis, in Proceedings of the Seventh International Symposium on Magnetic Anisotropy and Coercivity in R‐T Alloys, Canberra, July 1992, p. 437) structure, depending on the annealing procedure. Powder‐x‐ray‐diffraction patterns and scanning electron microscopy show that the sample annealed at a temperature of 850 °C followed by 1000 °C has the 2:17 structure whereas annealing at 1000 °C directly leads to the new 2:19 structure. Energy‐dispersive x‐ray analysis yields Pr:Fe:Ti ratios of 10.7:86.2:3.1 for the Pr2(Fe,Ti)17 phase and 9.2:85.9:4.9 for the Pr2(Fe,Ti)19 phase. 57 Fe Mössbauer spectroscopy (at 295 K) gives values for the average 57 Fe hyperfine field of 15.7 T for the 2:17 phase and 17.5 T for the 2:19 phase, respectively

Observation of orbital ordering and origin of the nematic order in FeSe

To elucidate the origin of nematic order in FeSe, we performed field-dependent 77Se-NMR measurements on single crystals of FeSe. We observed orbital ordering from the splitting of the NMR spectra and Knight shift and a suppression of it with magnetic field B0 up to 16 T applied parallel to the Fe-planes. There is a significant change in the distribution and magnitude of the internal magnetic field across the orbital ordering temperature Torb while stripe-type antiferromagnetism is absent. Giant antiferromagnetic (AFM) spin fluctuations measured by the NMR spin-lattice relaxation are gradually developed starting at ~ 40 K, which is far below the nematic ordering temperature Tnem. These results demonstrate that orbital ordering is the origin of the nematic order, and the AFM spin fluctuation is the driving mechanism of superconductivity in FeSe under the presence of the nematic order.Comment: 6 pages, 4 figure

Spin Squeezing under Non-Markovian Channels by Hierarchy Equation Method

We study spin squeezing under non-Markovian channels, and consider an ensemble of $N$ independent spin-1/2 particles with exchange symmetry. Each spin interacts with its own bath, and the baths are independent and identical. For this kind of open system, the spin squeezing under decoherence can be investigated from the dynamics of the local expectations, and the multi-qubit dynamics can be reduced into the two-qubit one. The reduced dynamics is obtained by the hierarchy equation method, which is a exact without rotating-wave and Born-Markov approximation. The numerical results show that the spin squeezing displays multiple sudden vanishing and revival with lower bath temperature, and it can also vanish asymptotically.Comment: 7 pages, 4 figure

Density-functional fidelity approach to quantum phase transitions

We propose a new approach to quantum phase transitions in terms of the density-functional fidelity, which measures the similarity between density distributions of two ground states in parameter space. The key feature of the approach, as we will show, is that the density-functional fidelity can be measured easily in experiments. Both the validity and versatility of the approach are checked by the Lipkin-Meshkov-Glick model and the one-dimensional Hubbard model.Comment: 4 pages, 2 figures, submitted to Chin. Phys. Let