13,242 research outputs found
Connectivity of Direct Products of Graphs
Let be the connectivity of and the direct product
of and . We prove that for any graphs and with ,
, which was conjectured
by Guji and Vumar.Comment: 5 pages, accepted by Ars Com
The Gierer-Meinhardt system on a compact two-dimensional Riemannian Manifold: Interaction of Gaussian curvature and Green's function
In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold
without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity is small enough. We show that for a threshold ratio of the activator diffusivity and the inhibitor diffusivity, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o(1) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O(1) eigenvalues which all have negative part in this case.RGC of Hong Kon
From Equivalence Principles to Cosmology: Cosmic Polarization Rotation, CMB Observation, Neutrino Number Asymmetry, Lorentz Invariance and CPT
In this paper, we review the approach leading to cosmic polarization rotation
observation and present the current status with an outlook. In the study of the
relations among equivalence principles, we found that long-range
pseudoscalar-photon interaction is allowed. Pseudoscalar-photon interaction
would induce a rotation of linear polarization of electromagnetic wave
propagating with cosmological/astrophysical distance. In 2002, DASI
successfully observed the polarization of the cosmological microwave background
radiation. In 2003, WMAP observed the correlation of polarization with
temperature anisotropy at more than 10 sigma in the cosmological microwave
background. From this high polarization-temperature correlation in WMAP
observation, we put a limit of 0.1 rad on the rotation of linear polarization
of cosmological microwave background (CMB) propagation. Pseudoscalar-photon
interaction is proportional to the gradient of the pseudoscalar field. From
phenomenological point of view, this gradient could be neutrino number
asymmetry current, other density current, or a constant vector. In these
situations, Lorentz invariance or CPT may or may not effectively be violated.
In this paper, we review and compile various results. Better accuracy in CMB
polarization observation is expected from PLANCK mission to be launched next
year. A dedicated CMB polarization observer in the future would probe this
fundamental issue more deeply.Comment: 9 pages, 2 figures, a few references with corresponding text change
added in this version, invited talk given in VIII Asia-Pacific International
Conference on Gravitation and Astophysics (ICGA8), August 29 - September 1,
2007, Nara Women's University, Japan, submitted to Progress of Theoretical
Physics Supplemen
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