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This paper develops a numerical procedure to accelerate the convergence of the Favre-averaged Non-Linear Harmonic (FNLH) method. The scheme provides a unified mathematical framework for solving the sparse linear systems formed by the mean flow and the time-linearized harmonic flows of FNLH in an explicit or implicit fashion. The approach explores the similarity of the sparse linear systems of FNLH and leads to a memory efficient procedure, so that its memory consumption does not depend on the number of harmonics to compute. The proposed method has been implemented in the industrial CFD solver HYDRA. Two test cases are used to conduct a comparative study of explicit and implicit schemes in terms of convergence, computational efficiency, and memory consumption. Comparisons show that the implicit scheme yields better convergence than the explicit scheme and is also roughly 7 to 10 times more computationally efficient than the explicit scheme with 4 levels of multigrid. Furthermore, the implicit scheme consumes only approximately $50\%$ of the explicit scheme with four levels of multigrid. Compared with the full annulus unsteady Reynolds averaged Navier-Stokes (URANS) simulations, the implicit scheme produces comparable results to URANS with computational time and memory consumption that are two orders of magnitude smaller.

Sterile neutrinos in the keV mass range present a viable candidate for dark matter. They can be detected through single $\beta$ decay, where they cause small spectral distortions. The Karlsruhe Tritium Neutrino (KATRIN) experiment aims to search for keV-scale sterile neutrinos with high sensitivity. To achieve this, the KATRIN beamline will be equipped with a novel multi-pixel silicon drift detector focal plane array named TRISTAN. In this study, we present the performance of a TRISTAN detector module, a component of the eventual 9-module system. Our investigation encompasses spectroscopic aspects such as noise performance, energy resolution, linearity, and stability.

The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call such an invariant of 3-manifolds a Hennings invariant. The work reported in this paper has its background in previous work of the authors. The present paper gives an abstract description of these structures and shows how the Hopf algebraic image of a knot lies in the center of the corresponding Hopf algebra. The paper also shows how all the axiomatic properties of a quasi-triangular Hopf algebra are involved in the topology via a functor from the Tangle Category to the Diagrammatic Category of a Hopf Algebra. The invariants described in this paper generalize to invariants of rotational virtual knots. The contents of this paper are an update of the original 1998 version published in JKTR., Comment: LaTeX document, 36 pages, 23 figures

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. There are several numerical sequences associated with the study of the smallest such integer which are studied in \cite{RadFact} and \cite{RadGamma}. Here we continue the examination of one of them.

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. We show that the smallest such $n$ is one of a pair of possibilities, or is one possibility, which we show how to calculate. There are three interesting numerical sequences which play a central role in our arguments. This paper is based on the improvement on Sterling's approximation of factorials due to Robbins \cite{Robbins} and results of \cite{Radford}

We study a sequence of differences related to the problem of finding the smallest factorial $n!$ greater than or equal to $a^n$, where $a > 1$, using the gamma function.

The KATRIN (Karlsruhe Tritium Neutrino) experiment investigates the energetic endpoint of the tritium beta-decay spectrum to determine the effective mass of the electron anti-neutrino. The collaboration has reported a first mass measurement result at this TAUP-2019 conference. The TRISTAN project aims at detecting a keV-sterile neutrino signature by measuring the entire tritium beta-decay spectrum with an upgraded KATRIN system. One of the greatest challenges is to handle the high signal rates generated by the strong activity of the KATRIN tritium source while maintaining a good energy resolution. Therefore, a novel multi-pixel silicon drift detector and read-out system are being designed to handle rates of about 100 Mcps with an energy resolution better than 300 eV (FWHM). This report presents succinctly the KATRIN experiment, the TRISTAN project, then the results of the first 7-pixels prototype measurement campaign and finally describes the construction of the first TRISTAN module composed of 166 SDD-pixels as well as its implementation in KATRIN experiment.

Sterile neutrinos are a minimal extension of the Standard Model of Particle Physics. If their mass is in the kilo-electron-volt regime, they are viable dark matter candidates. One way to search for sterile neutrinos in a laboratory-based experiment is via tritium-beta decay, where the new neutrino mass eigenstate would manifest itself as a kink-like distortion of the $\beta$-decay spectrum. The objective of the TRISTAN project is to extend the KATRIN setup with a new multi-pixel silicon drift detector system to search for a keV-scale sterile neutrino signal. In this paper we describe the requirements of such a new detector, and present first characterization measurement results obtained with a 7-pixel prototype system.

Neutrinoless double beta decay searches play a major role in determining neutrino properties, in particular the Majorana or Dirac nature of the neutrino and the absolute scale of the neutrino mass. The consequences of these searches go beyond neutrino physics, with implications for Grand Unification and leptogenesis. The \textsc{Majorana} Collaboration is assembling a low-background array of high purity Germanium (HPGe) detectors to search for neutrinoless double-beta decay in $^{76}$Ge. The \textsc{Majorana Demonstrator}, which is currently being constructed and commissioned at the Sanford Underground Research Facility in Lead, South Dakota, will contain 44 kg (30 kg enriched in $^{76}$Ge) of HPGe detectors. Its primary goal is to demonstrate the scalability and background required for a tonne-scale Ge experiment. This is accomplished via a modular design and projected background of less than 3 cnts/tonne-yr in the region of interest. The experiment is currently taking data with the first of its enriched detectors., Comment: Presentation at the DPF 2015 Meeting of the American Physical Society Division of Particles and Fields, Ann Arbor, Michigan, August 4-8, 2015. 6 pages. 5 figures

We study certain subgroups of the full group of Hopf algebra automorphisms of a biproduct. In the process interesting subgroups of certain permutation groups come into play., Comment: 43 pages

Hydra is a full-scale industrial CFD application used for the design of turbomachinery at Rolls Royce plc. It consists of over 300 parallel loops with a code base exceeding 50K lines and is capable of performing complex simulations over highly detailed unstructured mesh geometries. Unlike simpler structured-mesh applications, which feature high speed-ups when accelerated by modern processor architectures, such as multi-core and many-core processor systems, Hydra presents major challenges in data organization and movement that need to be overcome for continued high performance on emerging platforms. We present research in achieving this goal through the OP2 domain-specific high-level framework. OP2 targets the domain of unstructured mesh problems and follows the design of an active library using source-to-source translation and compilation to generate multiple parallel implementations from a single high-level application source for execution on a range of back-end hardware platforms. We chart the conversion of Hydra from its original hand-tuned production version to one that utilizes OP2, and map out the key difficulties encountered in the process. To our knowledge this research presents the first application of such a high-level framework to a full scale production code. Specifically we show (1) how different parallel implementations can be achieved with an active library framework, even for a highly complicated industrial application such as Hydra, and (2) how different optimizations targeting contrasting parallel architectures can be applied to the whole application, seamlessly, reducing developer effort and increasing code longevity. Performance results demonstrate that not only the same runtime performance as that of the hand-tuned original production code could be achieved, but it can be significantly improved on conventional processor systems. Additionally, we achieve further..., Comment: Submitted to ACM Transactions on Parallel Computing

We revisit a class of examples described in the original paper on biproducts, expand the class, and provide a detailed analysis of the coalgebra and algebra structures of many of these examples. Connections with the semisimple Hopf algebras of dimension a power of two determined by Kashina are examined. The finite-dimensional non-trivial semisimple cosemisimple Hopf algebras we construct are shown to be lower cosolvable. Some of these have one proper normal Hopf subalgebra and are not lower solvable.

In this paper we study oriented quantum coalgebras which are structures closely related to oriented quantum algebras. We study the relationship between oriented quantum coalgebras and oriented quantum algebras and the relationship between oriented quantum coalgebras and quantum coalgebras. We show that there are regular isotopy invariants of oriented 1-1 tangles and of oriented knots and links associated to oriented and twist oriented quantum coalgebras respectively. There are many parallels between the theory of oriented quantum coalgebras and the theory of quantum coalgebras

We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category., Comment: 44 pages

We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair of characters $\rho \in \Alg(U, k)$ and $\chi \in \Alg(A, k)$ we define a left $H$-module $L(\rho, \chi)$. Under reasonable hypotheses the correspondence $(\rho, \chi) \mapsto L(\rho, \chi)$ determines a bijection between character pairs and the isomorphism classes of objects in a certain category ${}_H\underline{\mathcal M}$ of left $H$-modules. In many cases the finite-dimensional objects of ${}_H\underline{\mathcal M}$ are the finite-dimensional irreducible left $H$-modules. In math.QA/0603269 we apply the results of this paper and show that the finite-dimensional irreducible representations of a wide class of pointed Hopf algebras are parameterized by pairs of characters., Comment: amstex, 33 pages

We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work of N. Andruskiewitsch and the second author. Special cases are the multiparameter deformations of the enveloping algebras of semisimple Lie algebras where the deforming parameters are not roots of unity and some of their finite-dimensional versions in the root of unity case., Comment: amstex, 35 pages

Let $H$ be a Hopf algebra with bijective antipode over a field $k$ and suppose that $R{#}H$ is a bi-product. Then $R$ is a bialgebra in the Yetter--Drinfel'd category ${}_H^H{\mathcal YD}$. We describe the bialgebras $(R{#}H)^{op}$ and $(R{#}H)^o$ explicitly as bi-products $R^{\UOP}{#}H^{op}$ and $R^{\UO}{#}H^o$ respectively where $R^{\UOP}$ is a bialgebra in ${}^{H^{op}}_{H^{op}}{\mathcal YD}$ and $R^{\UO}$ is a bialgebra in ${}^{H^o}_{H^o}{\mathcal YD}$. We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products., Comment: amstex, 29 pages

This study was designed to investigate the effect on teachers and their students of a model of professional development that immerses teachers in scientific inquiry appropriate to their classrooms while modeling the use of reform-based pedagogical strategies for teaching science. The course was designed and team taught as a collaborative effort between scientists and science educators. Quantitative data were collected on teachers' pre and post course science content knowledge, attitudes toward science, and science process skills. Quantitative data on the process skills and attitudes toward science of students of project participants and matched groups of students and teachers not participating in the project were collected and compared. Qualitative data from open-ended questionnaires, journals, and learning logs were collected from both students and teachers. The program was found to have a statistically significant impact on teachers, their classrooms, and their students. It is recommended that future science professional development projects incorporate collaboration between scientists and science educators. These collaborative efforts should focus on integrating scientific investigations designed and carried out by participants with appropriate pedagogical models for incorporating inquiry-based learning in the classroom. Contains 16 references. (Author)

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Joyce (which, in turn, is a generalization of the fundamental group of the knot or link). We then introduce the concept of a bi-oriented quantum algebra which provides an algebraic context for the associated quantum invariant., Comment: LaTeX document, 34 pages, 13 figures, in text LaTeX graphics

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a more algebraic perspective, and develop a more detailed theory for them and their associated invariants., Comment: LAteX document, 45 pages, 17 figures

This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show that all known quantum link invariants can be put into this framework., Comment: LateX document, 47 pages, 21 figures