The invariants of the Weil representation of SL2(Z)\mathrm{SL}_2(\mathbb{Z})

The transformation behaviour of the vector valued theta function of a positive definite even lattice under the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ is described by the Weil representation. This representation plays an important role in the theory of automorphic forms. We show that its invariants are induced from $5$ fundamental invariants

A Comparative Study

Background: The Narcissism Inventory (NI) is a frequently used German inventory for measuring narcissism in clinical settings; an additional short version (NI-90) also exists. Psychometric properties of the NI-90 scales were examined in clinical and non-clinical adolescent samples. Methods: Two adolescent samples were assessed with the NI-90: a non-clinical sample (n = 439, mean age ± SD = 15.05 ± 1.77 years) and a clinical sample (n = 235, 18.26 ± 0.77 years). Confirmatory factor analysis and principle component analysis were used to scrutinize the structure of the scales. Multiple regression analysis was used to predict the scores on two scales (helpless self; negative body self). Results: This study revealed heterogeneity in the NI-90 scales, which in turn explains the wide range seen in Cronbach’s α (from 0.53 to 0.93). The postulated 4-factor structure could not be replicated in both samples. Multiple regression analysis revealed that personality disorder did not significantly predict negative body self or helpless self scores, whereas eating, mood, as well as somatoform and conversion disorders did. One NI-90 scale (greedy for praise and reassurance) showed sufficient psychometric quality for the measurement of narcissism in both samples. Conclusion: Based on the results, the authors recommend revising the NI-90. Items that may be useful for measuring aspects related to affective and body image complaints are presented. The greedy for praise and reassurance scale may be valuable for measuring features of ‘overt’ narcissism

BKM Lie superalgebra for the Z_5 orbifolded CHL string

We study the Z_5-orbifolding of the CHL string theory by explicitly constructing the modular form tilde{Phi}_2 generating the degeneracies of the 1/4-BPS states in the theory. Since the additive seed for the sum form is a weak Jacobi form in this case, a mismatch is found between the modular forms generated from the additive lift and the product form derived from threshold corrections. We also construct the BKM Lie superalgebra, tilde{G}_5, corresponding to the modular form tilde{Delta}_1 (Z) = tilde{Phi}_2 (Z)^{1/2} which happens to be a hyperbolic algebra. This is the first occurrence of a hyperbolic BKM Lie superalgebra. We also study the walls of marginal stability of this theory in detail, and extend the arithmetic structure found by Cheng and Dabholkar for the N=1,2,3 orbifoldings to the N=4,5 and 6 models, all of which have an infinite number of walls in the fundamental domain. We find that analogous to the Stern-Brocot tree, which generated the intercepts of the walls on the real line, the intercepts for the N >3 cases are generated by linear recurrence relations. Using the correspondence between the walls of marginal stability and the walls of the Weyl chamber of the corresponding BKM Lie superalgebra, we propose the Cartan matrices for the BKM Lie superalgebras corresponding to the N=5 and 6 models.Comment: 30 pages, 2 figure

Analytical Solution for the Deformation of a Cylinder under Tidal Gravitational Forces

Quite a few future high precision space missions for testing Special and General Relativity will use optical resonators which are used for laser frequency stabilization. These devices are used for carrying out tests of the isotropy of light (Michelson-Morley experiment) and of the universality of the gravitational redshift. As the resonator frequency not only depends on the speed of light but also on the resonator length, the quality of these measurements is very sensitive to elastic deformations of the optical resonator itself. As a consequence, a detailed knowledge about the deformations of the cavity is necessary. Therefore in this article we investigate the modeling of optical resonators in a space environment. Usually for simulation issues the Finite Element Method (FEM) is applied in order to investigate the influence of disturbances on the resonator measurements. However, for a careful control of the numerical quality of FEM simulations a comparison with an analytical solution of a simplified resonator model is beneficial. In this article we present an analytical solution for the problem of an elastic, isotropic, homogeneous free-flying cylinder in space under the influence of a tidal gravitational force. The solution is gained by solving the linear equations of elasticity for special boundary conditions. The applicability of using FEM codes for these simulations shall be verified through the comparison of the analytical solution with the results gained within the FEM code.Comment: 23 pages, 3 figure

Antibacterial 45S5 Bioglass®-based scaffolds reinforced with genipin cross-linked gelatin for bone tissue engineering

45S5 Bioglass® (BG) scaffolds with high porosity (>90%) were coated with genipin cross-linked gelatin (GCG) and further incorporated with poly(p-xylyleneguanidine) hydrochloride (PPXG). The obtained GCG coated scaffolds maintained the high porosity and well interconnected pore structure. A 26-fold higher compressive strength was provided to 45S5 BG scaffolds by GCG coating, which slightly retarded but did not inhibit the in vitro bioactivity of 45S5 BG scaffolds in SBF. Moreover, the scaffolds were made antibacterial against both Gram-positive and Gram-negative bacteria by using polyguanidine, i.e. PPXG, in this study. Osteoblast-like cells (MG-63) were seeded onto PPXG and GCG coated scaffolds. PPXG was biocompatible with MG-63 cells at a low concentration (10 μg mL−1). MG-63 cells were shown to attach and spread on both uncoated and GCG coated scaffolds, and the mitochondrial activity measurement indicated that GCG coating had no negative influence on the cell proliferation behavior of MG-63 cells. The developed novel antibacterial bioactive 45S5 BG-based composite scaffolds with improved mechanical properties are promising candidates for bone tissue engineering

Internet Use and Perceived Parental Involvement among Adolescents from Lower Socioeconomic Groups in Europe: An Exploration

Internet usage is a salient developmental factor in adolescents' lives. Although relevant correlates of Internet use have been documented earlier, there is a lack of information on lower socioeconomic status groups. This is important, as these adolescents have increased risk of negative online experiences. The current survey aimed to explore Internet use and parental involvement amongst adolescents from areas of socio-economic disadvantage in 30 urban schools across five European countries. A total of 2594 students participated, of whom 90% were 14-16 years. Virtually all adolescents of socioeconomic disadvantage had Internet access, with 88.5% reporting spending more than two hours per day online, often on apps such as Instagram, Snapchat, and YouTube. Almost one-third of adolescents did not talk with their parents about their Internet use and almost two-thirds indicated that their parents were only a little or not interested in their Internet use. A consistent finding across countries was that girls more often talked with their parents about their Internet use and more often reported that their parents were interested in their Internet use than boys. The results suggest that parents have an important task in explicitly showing interest in their adolescents' Internet use, with special attention needed for boys