On the Erdos-Sos Conjecture for Graphs on n=k+4 Vertices

The Erd\H{o}s-S\'{o}s Conjecture states that if $G$ is a simple graph of order $n$ with average degree more than $k-2,$ then $G$ contains every tree of order $k$. In this paper, we prove that Erd\H{o}s-S\'{o}s Conjecture is true for $n=k+4$.Comment: 18 page

Interpretation of 750 GeV Diphoton Excess at LHC in Singlet Extension of Color-octet Neutrino Mass Model

We propose that the possible 750 GeV diphoton excess can be explained in the color-octet neutrino mass model extended with a scalar singlet $\Phi$. The model generally contains $N_s$ species of color-octet, electroweak doublet scalars $S$ and $N_f$ species of color-octet, electroweak triplet $\chi$ or singlet $\rho$ fermions. While both scalars and fermions contribute to the production of $\Phi$ through gluon fusion, only the charged members induce the diphoton decay of $\Phi$. The diphoton rate can be significantly enhanced due to interference between the scalar and fermion loops. We show that the diphoton cross section can be from 3 to 10 fb for O(TeV) color-octet particles while evading all current LHC limits.Comment: 12 pages, 4 figures; v2: 13 pages, 4 figures, version to appear in EPJC, clarified a few things, updated numerical analysis using the most recent bound on color-octet fermions but without changing conclusions, corrected a mistake when quoting the branching ratio to Z gamma, added some references missed in v

Atomic resolution imaging at 2.5 GHz using near-field microwave microscopy

Atomic resolution imaging is demonstrated using a hybrid scanning tunneling/near-field microwave microscope (microwave-STM). The microwave channels of the microscope correspond to the resonant frequency and quality factor of a coaxial microwave resonator, which is built in to the STM scan head and coupled to the probe tip. We find that when the tip-sample distance is within the tunneling regime, we obtain atomic resolution images using the microwave channels of the microwave-STM. We attribute the atomic contrast in the microwave channels to GHz frequency current through the tip-sample tunnel junction. Images of the surfaces of HOPG and Au(111) are presented.Comment: 9 pages, 5 figures, submitted to Applied Physics Letter

Laspeyres Decomposition of Energy Intensity including Household-energy Factors

AbstractIn the previous study papers the change of the total energy intensity is decomposed to the structure factor and the efficiency factor, not considering the household energy factor, because GDP are not made only by household energy. Household energy is about ten percents in the total energy consumption and the household energy of unit GDP are always being decreased. In this paper the change of the total energy intensity is decomposed to the structure adjustment factor, the efficiency factor and the household energy factor. The result of decomposition shows the efficiency factor contribute most to the reduction of total energy intensity from 1985 to 2007, the household energy factor the next, and structure adjustment factor the last. The result of decomposition also explains the causation of the abnormal ascending of total energy intensity from 2002 to 2005