@phdthesis{Helbig2009, author = {Helbig, Michael}, title = {Lifting of Nichols Algebras}, doi = {10.5282/edoc.10378}, school = {Technische Hochschule Rosenheim}, pages = {107}, year = {2009}, abstract = {Nichols algebras are a fundamental building block of pointed Hopf algebras. Part of the classi cation program of nite-dimensional pointed Hopf algebras with the lifting method of Andruskiewitsch and Schneider [6] is the determination of the liftings, i.e., all possible deformations of a given Nichols algebra. The classi cation was carried out in this way in [11] when the group of group-like elements is abelian and the prime divisors of the order of the group are > 7. In this case the appearing Nichols algebras are of Cartan type. Based on recent work of Heckenberger about diagonal Nichols algebras [29, 28, 27] we compute explicitly the liftings of some Nichols algebras not treated in [11]; namely we lift - all Nichols algebras with Cartan matrix of type A2 (Theorem 6.3.3), - some Nichols algebras with Cartan matrix of type B2 (Theorem 6.4.3), and - some Nichols algebras of two Weyl equivalence classes of non-standard type (Theorem 6.5.3), giving new classes of nite-dimensional pointed Hopf algebras. Crucial is the knowledge of a good presentation of the Nichols algebra and its liftings: We want to have an explicit description in terms of generators and (non-redundant) relations, and a basis; this requires new ideas and methods that generalize those in [11]. In this spirit, we describe Hopf algebras generated by skew-primitive elements and an abelian group with action given via characters (including Nichols algebras and their liftings) in Theorem 5.4.1. The relations form a Grobner basis and are given by a combinatorial property involving the theory of Lyndon words. Furthermore, in Theorem 7.3.1 we give a necessary and su cient criterion to check whether a given set of iterated q-commutators establishes a restricted PBW basis for a given realization of the relations. Also with the help of this criterion we determine the redundant relations in the examined Nichols algebras and their liftings.}, language = {en} } @phdthesis{Schoepfer2019, author = {Sch{\"o}pfer, Fabian}, title = {Vibroacoustics of timber-frame structures excited by structure-borne sound sources}, school = {Technische Hochschule Rosenheim}, year = {2019}, abstract = {This thesis investigates the measurement and prediction of machinery noise in timber-frame buildings. To quantify the structure-borne sound power input from multi-point sources, simplified approaches were assessed that reduce the required data for out-of-plane force excitation. This identified approaches that give estimates within ±5 dB from 20 Hz to 2000 Hz. To investigative the importance of out-of-plane moment excitation, inverse methods were used to determine the power input; these were affected by noise but processing was used to overcome this shortcoming. A series of experimental investigations were carried out on a timber-frame structure undergoing mechanical point excitation. The driving-point mobility showed orthotropic plate characteristics at low frequencies, ribbed-plate characteristics in a narrow frequency band and infinite plate characteristics in mid- and high-frequency ranges. The moment mobility above or in-between studs was similar to infinite beam or plate theory with interpolation between these theories where necessary. The experimental work indicated the potential to use Statistical Energy Analysis (SEA) to predict sound transmission. The first experimental finding was that above the mass-spring-mass resonance frequency, the vibrational response of the wall leaves was uncorrelated. The second was a significant decrease in vibration across the wall from the excitation point, with structural intensity showing a decrease in net power flow across successive timber studs. The third was that tongue and groove connections between chipboard sheets significantly reduce the vibration transmission above 500 Hz. This led to different SEA models being used to model a timber-frame wall undergoing mechanical point excitation. A 41-subsystem model was found to be necessary to reproduce the measured vibration levels on both leaves within 10 dB. As there is a significant decrease in vibration with distance in the mid- and high-frequency range, the region close to the excitation point is particularly important and the SEA model has better accuracy in this region. An alternative engineering approach to the prediction of machinery noise in timber-frame buildings was introduced and validated that used measured transmission functions between the injected power and the spatial-average sound pressure level in a receiving room. A field survey and case studies indicate this is a feasible and practical approach.}, language = {en} }