A note on tensor categories of Lie type E9E_9

We consider the problem of decomposing tensor powers of the fundamental level 1 highest weight representation $V$ of the affine Kac-Moody algebra \g(E_9). We describe an elementary algorithm for determining the decomposition of the submodule of \Vn whose irreducible direct summands have highest weights which are maximal with respect to the null-root. This decomposition is based on Littelmann's path algorithm and conforms with the uniform combinatorial behavior recently discovered by H. Wenzl for the series $E_N$, $N\not=9$.Comment: Final published versio

Cellular structure of qq-Brauer algebras

In this paper we consider the $q$-Brauer algebra over $R$ a commutative noetherian domain. We first construct a new basis for $q$-Brauer algebras, and we then prove that it is a cell basis, and thus these algebras are cellular in the sense of Graham and Lehrer. In particular, they are shown to be an iterated inflation of Hecke algebras of type $A_{n-1}.$ Moreover, when $R$ is a field of arbitrary characteristic, we determine for which parameters the $q$-Brauer algebras are quasi-heredity. So the general theory of cellular algebras and quasi-hereditary algebras applies to $q$-Brauer algebras. As a consequence, we can determine all irreducible representations of $q$-Brauer algebras by linear algebra methods

Family incidence of endometriosis in first-, second-, and third-degree relatives: case-control study

<p>Abstract</p> <p>Background</p> <p>Initial publications examining the hereditary aspects of endometriosis appeared in the early seventies and demonstrated an up to seven-fold risk for endometriosis in first-degree relatives of endometriosis patients. The aim was to evaluate the influence of hereditary aspects on the endometriosis risk in our patient collective.</p> <p>Methods</p> <p>In a retrospective cohort study we evaluated the incidence of endometriosis among first-, second-, and third-degree relatives of endometriosis patients and compare it with its incidence among first-, second-, and third-degree relatives of patients without endometriosis.</p> <p>Result(s)</p> <p>Eighty patients in whom endometriosis had been confirmed laparoscopically and histologically by biopsy and 60 patients in whom no endometriosis had been found during laparoscopy were given a questionnaire about the presence of symptoms associated with endometriosis and its family incidence. Patients of both the endometriosis and the control group were 37.7 ± 6.2 and 45.9 ± 12.0 years of age at the time of the interview, respectively (p < 0.05). Information about the presence of endometriosis was more readily available for relatives of those in the endometriosis group than for those in the control group (325/749 [43.4%] vs. 239/425 [56.2%], p < 0.05). In 5/136 (3.7%) and 8/134 (6.0%) first-degree relatives of endometriosis patients and the control group, respectively, information about the presence of endometriosis was not available (p = 0.554). Endometriosis was found in 8/136 (5.9%) first-degree relatives of patients and in 4/134 (3.0%) first-degree relatives of controls in the real-case analysis (p = 0.248). When comparing endometriosis characteristics between endometriosis patients with and without a history of familial endometriosis, no significant differences were found.</p> <p>Conclusion(s)</p> <p>There is a trend toward an increased familial incidence of endometriosis. In contrast to the literature, we found a less dramatic increase in familial risk for the development of endometriosis.</p

The Extremely Luminous Quasar Survey in the Pan-STARRS 1 Footprint (PS-ELQS)

We present the results of the Extremely Luminous Quasar Survey in the $3\pi$ survey of the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; PS1). This effort applies the successful quasar selection strategy of the Extremely Luminous Survey in the Sloan Digital Sky Survey footprint ($\sim12,000\,\rm{deg}^2$) to a much larger area ($\sim\rm{21486}\,\rm{deg}^2$). This spectroscopic survey targets the most luminous quasars ($M_{1450}\le-26.5$; $m_{i}\le18.5$) at intermediate redshifts ($z\ge2.8$). Candidates are selected based on a near-infrared JKW2 color cut using WISE AllWISE and 2MASS photometry to mainly reject stellar contaminants. Photometric redshifts ($z_{\rm{reg}}$) and star-quasar classifications for each candidate are calculated from near-infrared and optical photometry using the supervised machine learning technique random forests. We select 806 quasar candidates at $z_{\rm{reg}}\ge2.8$ from a parent sample of 74318 sources. After exclusion of known sources and rejection of candidates with unreliable photometry, we have taken optical identification spectra for 290 of our 334 good PS-ELQS candidates. We report the discovery of 190 new $z\ge2.8$ quasars and an additional 28 quasars at lower redshifts. A total of 44 good PS-ELQS candidates remain unobserved. Including all known quasars at $z\ge2.8$, our quasar selection method has a selection efficiency of at least $77\%$. At lower declinations $-30\le\rm{Decl.}\le0$ we approximately triple the known population of extremely luminous quasars. We provide the PS-ELQS quasar catalog with a total of 592 luminous quasars ($m_{i}\le18.5$, $z\ge2.8$). This unique sample will not only be able to provide constraints on the volume density and quasar clustering of extremely luminous quasars, but also offers valuable targets for studies of the intergalactic medium.Comment: 34 pages, 10 figures, accepted to ApJ

Влияние ширины реза на радиальную проекцию силы резания

В статье обосновывается влияние ширины реза на проекцию составляющей силы резания на плоскость, перпендикулярную оси вращения геохода. Рассмотрена актуальность исследования. Для постановки цели и задач исследования определена проекция составляющей силы резания на плоскость, перпендикулярную оси вращения геохода. На основании проведенного исследования построена зависимость радиальной проекции силы резанию ножевого исполнительного органа геохода (RИО.СВ) от от x до Rг

A New Young Diagrammatic Method For Kronecker Products of O(n) and Sp(2m)

A new simple Young diagrammatic method for Kronecker products of O(n) and Sp(2m) is proposed based on representation theory of Brauer algebras. A general procedure for the decomposition of tensor products of representations for O(n) and Sp(2m) is outlined, which is similar to that for U(n) known as the Littlewood rules together with trace contractions from a Brauer algebra and some modification rules given by King.Comment: Latex, 11 pages, no figure

Quantum Gravity and the Algebra of Tangles

In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links in R^3. The physically correct inner product on this space of states is not yet known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to approach this problem, we consider a larger space H of solutions of the Hamiltonian constraint, which has as a basis isotopy classes of tangles. A certain algebra T, the ``tangle algebra,'' acts as operators on H. The ``empty state'', corresponding to the class of the empty tangle, is conjectured to be a cyclic vector for T. We construct simpler representations of T as quotients of H by the skein relations for the HOMFLY polynomial, and calculate a *-algebra structure for T using these representations. We use this to determine the inner product of certain states of quantum gravity associated to the Jones polynomial (or more precisely, Kauffman bracket).Comment: 16 pages (with major corrections

The induced representations of Brauer algebra and the Clebsch-Gordan coefficients of SO(n)

Induced representations of Brauer algebra $D_{f}(n)$ from $S_{f_{1}}\times S_{f_{2}}$ with $f_{1}+f_{2}=f$ are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients (ORCs) of $S_{f_{1}}\times S_{f_{2}}\uparrow D_{f}(n)$ with $f\leq 4$ up to a normalization factor are derived by using the linear equation method. Weyl tableaus for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining CG coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebra are proposed. Some isoscalar factors of $SO(n)\supset SO(n-1)$ for the resulting irrep $[\lambda_{1},~\lambda_{2},~ \lambda_{3},~\lambda_{4},\dot{0}]$ with $\sum\limits_{i=1}^{4}\lambda_{i}\leq .Comment: 48 pages latex, submitted to Journal of Phys.