Non-thermal Emissions from Cool Cores Heated by Cosmic-Rays in Galaxy Clusters

We study non-thermal emissions from cool cores in galaxy clusters. We adopted a recent model, in which cosmic-rays (CRs) prevail in the cores and stably heat them through CR streaming. The non-thermal emissions come from the interaction between CR protons and intracluster medium (ICM). Comparison between the theoretical predictions and radio observations shows that the overall CR spectra must be steep, and most of the CRs in the cores are low-energy CRs. Assuming that the CRs are injected through AGN activities, we study the nature of the shocks that are responsible for the CR acceleration. The steep CR spectra are likely to reflect the fact that the shocks travel in hot ICM with fairly small Much numbers. We also study the dependence on the CR streaming velocity. The results indicate that synchrotron emissions from secondary electrons should be observed as radio mini-halos in the cores. In particular, low-frequency observations (e.g. LOFAR) are promising. On the other hand, the steepness of the spectra makes it difficult to detect non-thermal X-ray and gamma-ray emissions from the cores. The low-energy CRs may be heating optical filaments observed in the cores.Comment: Accepted for publication in Ap

Strong Coupling Limit of Bethe Ansatz Solutions in Massive Thirring Model

We study the strong coupling limit of the Bethe ansatz solutions in the massive Thirring model. We find analytical expressions for the energy eigenvalues for the vacuum state as well as n-particle n- hole states. This formula is compared with the numerical results and is found to achieve a very good agreement. Also, it is found that the 2-particle 2- hole and higher particle-hole states describe n- free bosons states in this limit. The behaviors of the strong coupling limit of the boson mass for various model calculations are examined. We discuss an ambiguity of the coupling constant normalization due to the current regularization.Comment: 15 pages, to appear in Ann. Phys. 281 (2000), Ma

Non-equivalence between Heisenberg XXZ spin chain and Thirring model

The Bethe ansatz equations for the spin 1/2 Heisenberg XXZ spin chain are numerically solved, and the energy eigenvalues are determined for the anti-ferromagnetic case. We examine the relation between the XXZ spin chain and the Thirring model, and show that the spectrum of the XXZ spin chain is different from that of the regularized Thirring model.Comment: 10 pages. 2figure

Simple normal crossing Fano varieties and log Fano manifolds

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r\geq n/2 with \rho(X)\geq 2 or r\geq n-2. This result is a partial generalization of the classification of logarithmic Fano threefolds by Maeda.Comment: 38 pages, minor revision; correct Theorem 2.7 and add reference

A Danilov-type formula for toric origami manifolds via localization of index

We give a direct geometric proof of a Danilov-type formula for toric origami manifolds by using the localization of Riemann-Roch number.Comment: 30 pages, 8 figures : Revision of the introduction. Added references : Typos corrected. Construction in Section 4.1 simplified. Proofs of Lemma 6.2 and Theorem 6.12 corrected. Acknowledgement for the referee added : Final version, to appear in Osaka Journal of Mathematic