The dimension of a variety

We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic structures are determined.Comment: The results of the paper were presented at the workshop AAA71 and CYA21 at B\c{e}dlewo, Poland on February 11, 2006. the paper is submitted to Discussiones Mathematicae Algebra and Stochastc Methods, special issu

On bb-Whittaker functions

The $b$-Whittaker functions are eigenfunctions of the modular $q$-deformed $\mathfrak{gl}_n$ open Toda system introduced by Kharchev, Lebedev, and Semenov-Tian-Shansky. Using the quantum inverse scattering method, the named authors obtained a Mellin-Barnes integral representation for these eigenfunctions. In the present paper, we develop the analytic theory of the $b$-Whittaker functions from the perspective of quantum cluster algebras. We obtain a formula for the modular open Toda system's Baxter operator as a sequence of quantum cluster transformations, and thereby derive a new modular $b$-analog of Givental's integral formula for the undeformed Whittaker function. We also show that the $b$-Whittaker functions are eigenvectors of the Dehn twist operator from quantum higher Teichm\"uller theory, and obtain $b$-analogs of various integral identities satisfied by the undeformed Whittaker functions, including the continuous Cauchy-Littlewood identity of Stade and Corwin-O'Connell-Sepp\"al\"ainen-Zygouras. Using these results, we prove the unitarity of the $b$-Whittaker transform, thereby completing the analytic part of the proof of the conjecture of Frenkel and Ip on tensor products of positive representations of $U_q(\mathfrak{sl}_n)$, as well as the main step in the modular functor conjecture of Fock and Goncharov. We conclude by explaining how the theory of $b$-Whittaker functions can be used to derive certain hyperbolic hypergeometric integral evaluations found by Rains.Comment: 36 pages, minor changes, references adde

LHCb Results on Semileptonic B/Bs_s/Λb\Lambda_b Decays

Studies of semileptonic decays of b hadrons with the LHCb experiment are reported. In particular, measurements of the b hadron production fractions at the LHC and Bs semileptonic decays in P-wave charmed mesons are presented. An outlook of the LHCb potential in measuring the CKM matrix elements Vub and Vcb, as well as decays involving taus, is given.Comment: Proceedings of CKM 2012, the 7th International Workshop on the CKM Unitarity Triangle, University of Cincinnati, USA, 28 September - 2 October 201

CDF Results on BB Decays

We present recent CDF results on $B$ lifetimes, $B$ meson mass measurements, ratios of branching ratios, and rare decays. In addition, we present the first measurement of time-dependent $B_d$ mixing at CDF. Several results have been updated and a few new ones included since the workshop.Comment: 13 pages, LaTex using world_sci.sty, 18 figures in uuencoded ps file. Contribution to the proceedings of the LISHEP95 Workshop on Heavy Flavor Physics, Rio de Janeiro, Brazil, February 20-22, 1995. Also available at: http://www-cdf.fnal.gov/physics/conf95/cdf3152_rio95_b_decays.p

B decays on the lattice

I review recent developments in lattice calculations of B decay matrix elements and other related quantities.Comment: LATTICE99(plenary), 12 pages, 7 postscript figures, uses espcrc2.sty, references are update