Representation Growth of Linear Groups

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function \calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then \calz_\Gamma(s) has an ``Euler factorization". The "factor at infinity" is sometimes called the "Witten zeta function" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa

Clustering via kernel decomposition

Spectral clustering methods were proposed recently which rely on the eigenvalue decomposition of an affinity matrix. In this letter, the affinity matrix is created from the elements of a nonparametric density estimator and then decomposed to obtain posterior probabilities of class membership. Hyperparameters are selected using standard cross-validation methods

Mapping the Asymmetric Thick Disk: The Hercules Thick Disk Cloud

The stellar asymmetry of faint thick disk/inner halo stars in the first quadrant first reported by Larsen & Humphreys (1996) and investigated further by Parker et al. (2003, 2004) has been recently confirmed by SDSS (Juric et al. 2008). Their interpretation of the excess in the star counts as a ringlike structure, however, is not supported by critical complimentary data in the fourth quadrant not covered by SDSS. We present stellar density maps from the Minnesota Automated Plate Scanner (MAPS) Catalog of the POSS I showing that the overdensity does not extend into the fourth quadrant. The overdensity is most probably not a ring. It could be due to interaction with the disk bar, evidence for a triaxial thick disk, or a merger remnant/stream. We call this feature the Hercules Thick Disk Cloud.Comment: 11 pages, 5 figures, to be published in Astrophysical Journal Letter

Space processing of chalcogenide glasses

Chalcogenide glasses are discussed as good infrared transmitters, possessing the strength, corrosion resistance, and scale-up potential necessary for large 10.6-micron windows. The disadvantage of earth-produced chalcogenide glasses is shown to be an infrared absorption coefficient which is unacceptably high relative to alkali halides. This coefficient is traced to optical nonhomogeneities resulting from environmental and container contamination. Space processing is considered as a means of improving the infrared transmission quality of chalcogenides and of eliminating the following problems: optical inhomogeneities caused by thermal currents and density fluctuation in the l-g earth environment; contamination from the earth-melting crucible by oxygen and other elements deleterious to infrared transmission; and, heterogeneous nucleation at the earth-melting crucible-glass interface

Novel Techniques for Constraining Neutron-Capture Rates Relevant for r-Process Heavy-Element Nucleosynthesis

The rapid-neutron capture process ($r$ process) is identified as the producer of about 50\% of elements heavier than iron. This process requires an astrophysical environment with an extremely high neutron flux over a short amount of time ($\sim$ seconds), creating very neutron-rich nuclei that are subsequently transformed to stable nuclei via $\beta^-$ decay. One key ingredient to large-scale $r$-process reaction networks is radiative neutron-capture ($n,\gamma$) rates, for which there exist virtually no data for extremely neutron-rich nuclei involved in the $r$ process. Due to the current status of nuclear-reaction theory and our poor understanding of basic nuclear properties such as level densities and average $\gamma$-decay strengths, theoretically estimated ($n,\gamma$) rates may vary by orders of magnitude and represent a major source of uncertainty in any nuclear-reaction network calculation of $r$-process abundances. In this review, we discuss new approaches to provide information on neutron-capture cross sections and reaction rates relevant to the $r$ process. In particular, we focus on indirect, experimental techniques to measure radiative neutron-capture rates. While direct measurements are not available at present, but could possibly be realized in the future, the indirect approaches present a first step towards constraining neutron-capture rates of importance to the $r$ process.Comment: 62 pages, 24 figures, accepted for publication in Progress in Particle and Nuclear Physic

Logarithmic correction to scaling for multi-spin strings in the AdS_5 black hole background

We find new explicit solutions describing closed strings spinning with equal angular momentum in two independent planes in the $AdS_5$ black hole spacetime. These are $2n$ folded strings in the radial direction and also winding $m$ times around an angular direction. We especially consider these solutions in the long string and high temperature limit, where it is shown that there is a logarithmic correction to the scaling between energy and spin. This is similar to the one-spin case. The strings are spinning, or actually orbiting around the black hole of the $AdS_5$ black hole spacetime, similarly to solutions previously found in black hole spacetimes.Comment: 11 pages, Final version, To appear in IJMP

Exact String Solutions in Nontrivial Backgrounds

We show how the classical string dynamics in $D$-dimensional gravity background can be reduced to the dynamics of a massless particle constrained on a certain surface whenever there exists at least one Killing vector for the background metric. We obtain a number of sufficient conditions, which ensure the existence of exact solutions to the equations of motion and constraints. These results are extended to include the Kalb-Ramond background. The $D1$-brane dynamics is also analyzed and exact solutions are found. Finally, we illustrate our considerations with several examples in different dimensions. All this also applies to the tensionless strings.Comment: 22 pages, LaTeX, no figures; V2:Comments and references added; V3:Discussion on the properties of the obtained solutions extended, a reference and acknowledgment added; V4:The references renumbered, to appear in Phys Rev.

Third Bose Fugacity Coefficient in One Dimension, as a Function of Asymptotic Quantities

In one of the very few exact quantum mechanical calculations of fugacity coefficients, Dodd and Gibbs (\textit{J. Math.Phys}.,\textbf{15}, 41 (1974)) obtained $b_{2}$ and $b_{3}$ for a one dimensional Bose gas, subject to repulsive delta-function interactions, by direct integration of the wave functions. For $b_{2}$, we have shown (\textit{Mol. Phys}.,\textbf{103}, 1301 (2005)) that Dodd and Gibbs' result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms, usually contributing only in 1 dimension. Now, we develop an exact expression for $b_{3}-b_{3}^{0}$ (where $b_{3}^{0}$ is the free particle fugacity coefficient) in terms of sums and differences of 3-body eigenphase shifts. Further, we show that if we obtain these eigenphase shifts in a distorted-Born approximation, then, to first order, we reproduce the leading low temperature behaviour, obtained from an expansion of the two-fold integral of Dodd and Gibbs. The contributions of the oscillating terms cancel. The formalism that we propose is not limited to one dimension, but seeks to provide a general method to obtain virial coefficients, fugacity coefficients, in terms of asymptotic quantities. The exact one dimensional results allow us to confirm the validity of our approach in this domain.Comment: 29 page

Conditional Lower Bounds for Space/Time Tradeoffs

In recent years much effort has been concentrated towards achieving polynomial time lower bounds on algorithms for solving various well-known problems. A useful technique for showing such lower bounds is to prove them conditionally based on well-studied hardness assumptions such as 3SUM, APSP, SETH, etc. This line of research helps to obtain a better understanding of the complexity inside P. A related question asks to prove conditional space lower bounds on data structures that are constructed to solve certain algorithmic tasks after an initial preprocessing stage. This question received little attention in previous research even though it has potential strong impact. In this paper we address this question and show that surprisingly many of the well-studied hard problems that are known to have conditional polynomial time lower bounds are also hard when concerning space. This hardness is shown as a tradeoff between the space consumed by the data structure and the time needed to answer queries. The tradeoff may be either smooth or admit one or more singularity points. We reveal interesting connections between different space hardness conjectures and present matching upper bounds. We also apply these hardness conjectures to both static and dynamic problems and prove their conditional space hardness. We believe that this novel framework of polynomial space conjectures can play an important role in expressing polynomial space lower bounds of many important algorithmic problems. Moreover, it seems that it can also help in achieving a better understanding of the hardness of their corresponding problems in terms of time