Hundred Thousand Degree Gas in the Virgo Cluster of Galaxies

The physical relationship between low-excitation gas filaments at ~10^4 K, seen in optical line emission, and diffuse X-ray emitting coronal gas at ~10^7 K in the centers of many galaxy clusters is not understood. It is unclear whether the ~10^4 K filaments have cooled and condensed from the ambient hot (~10^7 K) medium or have some other origin such as the infall of cold gas in a merger, or the disturbance of an internal cool reservoir of gas by nuclear activity. Observations of gas at intermediate temperatures (~10^5-10^6 K) can potentially reveal whether the central massive galaxies are gaining cool gas through condensation or losing it through conductive evaporation and hence identify plausible scenarios for transport processes in galaxy cluster gas. Here we present spectroscopic detection of ~10^5 K gas spatially associated with the H-alpha filaments in a central cluster galaxy, M87 in the Virgo Cluster. The measured emission-line fluxes from triply ionized carbon (CIV 1549 A) and singly ionized helium (HeII 1640 A) are consistent with a model in which thermal conduction determines the interaction between hot and cold phases.Comment: 10 pages, 2 figures; to appear in ApJ

Continuous Association Schemes and Hypergroups

Classical finite association schemes lead to a finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, the notion of association schemes can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to larger class of examples which are again associated to discrete hypergroups. In this paper we propose a topological generalization of the notion of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by a further locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this new notion and present several classes of examples. It turns out that for a given commutative hypergroup the existence of an associated continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We in particular show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes

Dynamical correlation functions of one-dimensional superconductors and Peierls and Mott insulators

I construct the spectral function of the Luther-Emery model which describes one-dimensional fermions with one gapless and one gapped degree of freedom, i.e. superconductors and Peierls and Mott insulators, by using symmetries, relations to other models, and known limits. Depending on the relative magnitudes of the charge and spin velocities, and on whether a charge or a spin gap is present, I find spectral functions differing in the number of singularities and presence or absence of anomalous dimensions of fermion operators. I find, for a Peierls system, one singularity with anomalous dimension and one finite maximum; for a superconductor two singularities with anomalous dimensions; and for a Mott insulator one or two singularities without anomalous dimension. In addition, there are strong shadow bands. I generalize the construction to arbitrary dynamical multi-particle correlation functions. The main aspects of this work are in agreement with numerical and Bethe Ansatz calculations by others. I also discuss the application to photoemission experiments on 1D Mott insulators and on the normal state of 1D Peierls systems, and propose the Luther-Emery model as the generic description of 1D charge density wave systems with important electronic correlations.Comment: Revtex, 27 pages, 5 figures, to be published in European Physical Journal

Integral representation and sharp asymptotic results for some Heckman-Opdam hypergeometric functions of type BC

The Heckman-Opdam hypergeometric functions of type BC extend classical Jacobi functions in one variable and include the spherical functions of non-compact Grassmann manifolds over the real, complex or quaternionic numbers. There are various limit transitions known for such hypergeometric functions. In the present paper, we use an explicit form of the Harish-Chandra integral representation as well as an interpolated variant, in order to obtain limit results for three continuous classes of hypergeometric functions of type BC which are distinguished by explicit, sharp and uniform error bounds. The first limit realizes the approximation of the spherical functions of infinite dimensional Grassmannians of fixed rank; here hypergeometric functions of type A appear as limits. The second limit is a contraction limit towards Bessel functions of Dunkl type

A Limit Relation for Dunkl-Bessel Functions of Type A and B

We prove a limit relation for the Dunkl-Bessel function of type $B_N$ with multiplicity parameters $k_1$ on the roots $\pm e_i$ and $k_2$ on $\pm e_i\pm e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type $A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Olshanski spherical functions for infinite dimensional motion groups of fixed rank

Consider the Gelfand pairs $(G_p,K_p):=(M_{p,q} \rtimes U_p,U_p)$ associated with motion groups over the fields $\mathbb F=\mathbb R,\mathbb C,\mathbb H$ with $p\geq q$ and fixed $q$ as well as the inductive limit $p\to\infty$,the Olshanski spherical pair $(G_\infty,K_\infty)$. We classify all Olshanski spherical functions of $(G_\infty,K_\infty)$ as functions on the cone $\Pi_q$ of positive semidefinite $q\times q$-matrices and show that they appear as (locally) uniform limits of spherical functions of $(G_p,K_p)$ as $p\to\infty$. The latter are given by Bessel functions on $\Pi_q$. Moreover, we determine all positive definite Olshanski spherical functions and discuss related positive integral representations for matrix Bessel functions. We also extend the results to the pairs $(M_{p,q} \rtimes (U_p\times U_q),(U_p\times U_q))$ which are related to the Cartan motion groups of non-compact Grassmannians. Here Dunkl-Bessel functions of type B (for finite $p$) and of type A (for $p\to\infty$) appear as spherical functions

Limit theorems for radial random walks on pxq-matrices as p tends to infinity

The radial probability measures on $R^p$ are in a one-to-one correspondence with probability measures on $[0,\infty[$ by taking images of measures w.r.t. the Euclidean norm mapping. For fixed $\nu\in M^1([0,\infty[)$ and each dimension p, we consider i.i.d. $R^p$-valued random variables $X_1^p,X_2^p,...$ with radial laws corresponding to $\nu$ as above. We derive weak and strong laws of large numbers as well as a large deviation principle for the Euclidean length processes $S_k^p:=\|X_1^p+...+X_k^p\|$ as k,p\to\infty in suitable ways. In fact, we derive these results in a higher rank setting, where $R^p$ is replaced by the space of $p\times q$ matrices and $[0,\infty[$ by the cone $\Pi_q$ of positive semidefinite matrices. Proofs are based on the fact that the $(S_k^p)_{k\ge 0}$ form Markov chains on the cone whose transition probabilities are given in terms Bessel functions $J_\mu$ of matrix argument with an index $\mu$ depending on p. The limit theorems follow from new asymptotic results for the $J_\mu$ as $\mu\to \infty$. Similar results are also proven for certain Dunkl-type Bessel functions.Comment: 24 page