Probing Graph Proper Total Colorings With Additional Constrained Conditions

Graph colorings are becoming an increasingly useful family of mathematical models for a broad range of applications, such as time tabling and scheduling, frequency assignment, register allocation, computer security and so on. Graph proper total colorings with additional constrained conditions have been investigated intensively in the last decade year. In this article some new graph proper total colorings with additional constrained conditions are defined, and approximations to the chromatic numbers of these colorings are researched, as well as some graphs having these colorings have been verified

Phase Transition of Trapped Nuclear Exciton of Long-lived Rhodium Mossbauer States

We report experimental observations of the long-lived rhodium Mossbauer emissions by the time- and energy-resolved spectroscopy. The extraordinary observations manifest the open-up of photonic band gap in analogy to the superconducting gap of remarkable symmetry breakings at transition point. These observations are of potential importance for detecting gravitational waves and development of the two-photon gamma laser. Firstly, phase transitions shown by spectral evolution of characteristic emissions reveal the different aggregate exciton orderings at room temperature. Six different phases are identified by spectra profiles emitted from the color centers. Secondly, the cascade branching of the multipolar nuclear transition is discovered being the spontaneous cascade down-conversion to generate entangled gammas. The macroscopic angular distribution of entangled gammas from the polycrystalline sample manifests a global photon-nucleus-photon bound state across the grain boundaries. Thirdly, the gamma-energy distributions depending on exciton phases reveal the photonic band gap typically on the order of several hundred eV.Comment: Revised by additional discover

Gamma Standing Wave in the Photonic Crystal of Resonant Rh Nuclei

In a previous report, we have shown that the rhodium lattice consisting of resonant nuclei is an ideal photonic crystal in nature. Plenty of extraordinary observations are attributed to the collective down conversion of the multipolar nuclear transition; in particular the spontaneous open up of photonic band gap. Emissions of directionality depending on the macroscopic geometry manifest that the standing wave is global in the polycrystalline sample. In this work, further observations of the directional emissions are summarized. By applying an external magnetic field at room temperature, not only the predicted macroscopic nuclear polarization but also its strong directionality are demonstrated. The standing wave lasts for more than hundred hours at room temperature in the single crystal despite its natural half-life of one hour. The so-called nuclear Raman Effect between two M4 transitions of 193mIr and 195mPt and the E3 transition of 103mRh is discovered, which gives the brand new aspects to detect gravitational waves.Comment: Subsequent work to 0707.0960v

Anomalous emissions of 103mRh biphoton transitions

In this report, the anomalous emissions, centered on the one half transition energy 39.76/2 keV, are observed from the long-lived Mossbauer state of 103mRh excited by bremsstrahlung exposure. Strong coupling with identical nuclei in Rh crystals opens cascade channels for biphoton transitions

Graph Homomorphisms Based On Particular Total Colorings of Graphs and Graphic Lattices

Lattice-based cryptography is not only for thwarting future quantum computers, and is also the basis of Fully Homomorphic Encryption. Motivated from the advantage of graph homomorphisms we combine graph homomorphisms with graph total colorings together for designing new types of graph homomorphisms: totally-colored graph homomorphisms, graphic-lattice homomorphisms from sets to sets, every-zero graphic group homomorphisms from sets to sets. Our graph-homomorphism lattices are made up by graph homomorphisms. These new homomorphisms induce some problems of graph theory, for example, Number String Decomposition and Graph Homomorphism Problem

A Note On Vertex Distinguishing Edge colorings of Trees

A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by \chi\,'_s(G) called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds \chi\,'_s(T)=n_1(T)+1 if its diameter $D(T)=3$ or one of two particular trees with $D(T) =4$, and \chi\,'_s(T)=n_1(T) otherwise; furthermore \chi\,'_{es}(T)=\chi\,'_s(T) when $|E(T)|\leq 2(n_1(T)+1)$, where \chi\,'_{es}(T) is the equitable vdec chromatic number of $T$

Distributions of full and non-full words in beta-expansions

The structures of full words and non-full for $\beta$-expansions are completely characterized in this paper. We obtain the precise lengths of all the maximal runs of full and non-full words among admissible words with same order

Rhodium Mossbauer Supperadiance Induced by Liquid-Nitrogen Cooling

In the previous report, we have demonstrated cascade branching channels of the multipolar E3 transition of rhodium Mossbauer gamma via the time- and energy-resolved spectroscopy. Moreover, superradiance in the Borrmann channel from inverted nuclei gives gamma entanglement. In this letter, we report further four observations of superradiance and its associated gamma entanglement at the liquid-nitrogen temperature, i.e. (i) speed-up decay, (ii) immediate recovery of the speed-up decay after quenching, (iii) simultaneous suppression of gamma and K lines, and (iv) enhanced multiple ionizations. Anisotropic superradiant channels open by quenching and recover back immediately after quenching. Enhanced K satellites and K hypersatellites induced by cooling are attributed to the inelastic scattering of more than three entangled gammas.Comment: 10 page

Improper Graceful and Odd-graceful Labellings of Graph Theory

In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings

Slipping magnetic reconnections with multiple flare ribbons during an X-class solar flare

With the observations of the Solar Dynamics Observatory, we present the slipping magnetic reconnections with multiple flare ribbons (FRs) during an X1.2 eruptive flare on 2014 January 7. A center negative polarity was surrounded by several positive ones, and there appeared three FRs. The three FRs showed apparent slipping motions, and hook structures formed at their ends. Due to the moving footpoints of the erupting structures, one tight semi-circular hook disappeared after the slippage along its inner and outer edge, and coronal dimmings formed within the hook. The east hook also faded as a result of the magnetic reconnection between the arcades of a remote filament and a hot loop that was impulsively heated by the under flare loops. Our results are accordant with the slipping magnetic reconnection regime in 3D standard model for eruptive flares. We suggest that complex structures of the flare is likely a consequence of the more complex flux distribution in the photosphere, and the eruption involves at least two magnetic reconnections