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The present work considers the normal approximation of the binomial distribution and yields estimations of the supremum distance of the distribution functions of the binomial- and the corresponding standardized normal distribution. The type of the estimations correspond to the classical Berry-Esseen theorem, in the special case that all random variables are identically Bernoulli distributed. In this case we state the optimal constant for the Berry-Esseen theorem. In the proof of these estimations several inequalities regarding the density as well as the distribution function of the binomial distribution are presented. Furthermore in the estimations mentioned above the distribution function is replaced by the probability of arbitrary, not only unlimited intervals and in this new situation we also present an upper bound.
Shape optimization is of interest in many fields of application. In particular, shape optimization problems arise frequently in technological processes which are modelled by partial differential equations (PDEs). In a lot of practical circumstances, the shape under investigation is parametrized by a finite number of parameters, which, on the one hand, allows the application of standard optimization approaches, but, on the other hand, unnecessarily limits the space of reachable shapes. Shape calculus presents a way to circumvent this dilemma. However, so far shape optimization based on shape calculus is mainly performed using gradient descent methods. One reason for this is the lack of symmetry of second order shape derivatives or shape Hessians. A major difference between shape optimization and the standard PDE constrained optimization framework is the lack of a linear space structure on shape spaces. If one cannot use a linear space structure, then the next best structure is a Riemannian manifold structure, in which one works with Riemannian shape Hessians. They possess the often sought property of symmetry, characterize well-posedness of optimization problems and define sufficient optimality conditions. In general, shape Hessians are used to accelerate gradient-based shape optimization methods. This thesis deals with shape optimization problems constrained by PDEs and embeds these problems in the framework of optimization on Riemannian manifolds to provide efficient techniques for PDE constrained shape optimization problems on shape spaces. A Lagrange-Newton and a quasi-Newton technique in shape spaces for PDE constrained shape optimization problems are formulated. These techniques are based on the Hadamard-form of shape derivatives, i.e., on the form of integrals over the surface of the shape under investigation. It is often a very tedious, not to say painful, process to derive such surface expressions. Along the way, volume formulations in the form of integrals over the entire domain appear as an intermediate step. This thesis couples volume integral formulations of shape derivatives with optimization strategies on shape spaces in order to establish efficient shape algorithms reducing analytical effort and programming work. In this context, a novel shape space is proposed.
Educational assessment tends to rely on more or less standardized tests, teacher judgments, and observations. Although teachers spend approximately half of their professional conduct in assessment-related activities, most of them enter their professional life unprepared, as classroom assessment is often not part of their educational training. Since teacher judgments matter for the educational development of students, the judgments should be up to a high standard. The present dissertation comprises three studies focusing on accuracy of teacher judgments (Study 1), consequences of (mis-)judgment regarding teacher nomination for gifted programming (Study 2) and teacher recommendations for secondary school tracks (Study 3), and individual student characteristics that impact and potentially bias teacher judgment (Studies 1 through 3). All studies were designed to contribute to a further understanding of classroom assessment skills of teachers. Overall, the results implied that, teacher judgment of cognitive ability was an important constant for teacher nominations and recommendations but lacked accuracy. Furthermore, teacher judgments of various traits and school achievement were substantially related to social background variables, especially the parents" educational background. However, multivariate analysis showed social background variables to impact nomination and recommendation only marginally if at all. All results indicated differentiated but potentially biased teacher judgments to impact their far-reaching referral decisions directly, while the influence of social background on the referral decisions itself seems mediated. Implications regarding further research practices and educational assessment strategies are discussed. The implications on the needs of teachers to be educated on judgment and educational assessment are of particular interest and importance.